diff --git a/docs/01-paper.md b/docs/01-paper.md index dc2c01a5..98c329af 100755 --- a/docs/01-paper.md +++ b/docs/01-paper.md @@ -1,5 +1,5 @@ --- -title: Connectome-Based Attractor Dynamics Guide Brain Activity in Rest, Task, and Disease +title: Connectome-Based Attractor Dynamics Underly Brain Activity in Rest, Task, and Disease subject: manuscript draft #subtitle: Optional Subtitle short_title: Manuscript @@ -328,7 +328,7 @@ Strongest differences were found on the "action-perception" axis ({numref}`tab-c **A** The distribution of time-frames on the fcHNN-projection separately for ASD patients and typically developing control (TDC) participants.

**B** We quantified attractor state activations in the Autism Brain Imaging Data Exchange datasets ([study 7](tab-samples)) as the individual-level mean activation of all time-frames belonging to the same attractor state. This analysis captured alterations similar to those previously associated to ASD-related perceptual atypicalities (visual, auditory and somatosensory cortices) as well as atypical integration of information about the “self” and the “other” (default mode network regions). All results are corrected for multiple comparisons across brain regions and attractor states (122*4 comparisons) with Bonferroni-correction. See {numref}`tab-clinical-results` and {numref}`Supplementary Figure %s ` for detailed results.

-**C** The comparison of data generated by fcHNNs initialized with ASD and TDC connectomes, respectively, revealed a characteristic pattern of differences in the system's dynamics, with increased pull towards (and potentially a higher separation between) the action and perception attractors attractors and a lower tendency of trajectories going towards the internal and external attractors.

+**C** The comparison of data generated by fcHNNs initialized with ASD and TDC connectomes, respectively, revealed a characteristic pattern of differences in the system's dynamics, with increased pull towards (and potentially a higher separation between) the action and perception attractors and a lower tendency of trajectories going towards the internal and external attractors.

***Abbreviations**: MCC: middle cingulate cortex, ACC: anterior cingulate cortex, pg: perigenual, PFC: prefrontal cortex, dm: dorsomedial, dl: dorsolateral, STG: superior temporal gyrus, ITG: inferior temporal gyrus, Caud/Acc: caudate-accumbens, SM: sensorimotor, V1: primary visual, A1: primary auditory, SMA: supplementary motor cortex, ASD: autism spectrum disorder, TDC: typically developing control.* ::: @@ -457,5 +457,5 @@ https://github.com/pni-lab/connattractor https://pni-lab.github.io/connattractor/ ## Data availability -Study 1,2 and 4 is available at openneuro.org (ds002608, ds002608, ds000140). Data for study 3 is available upon request. Data for study 5-6 is available at the github page of the project: https://github.com/pni-lab/connattractor. Study 7 is available at https://fcon_1000.projects.nitrc.org/indi/abide/, preprocessed data is available at http://preprocessed-connectomes-project.org/. +Study 1, 2 and 4 is available at openneuro.org (ds002608, ds002608, ds000140). Data for study 3 is available upon request. Data for study 5-6 is available at the github page of the project: https://github.com/pni-lab/connattractor. Study 7 is available at https://fcon_1000.projects.nitrc.org/indi/abide/, preprocessed data is available at http://preprocessed-connectomes-project.org/. +++ diff --git a/docs/02-methods.md b/docs/02-methods.md index 361b270f..4cbc453f 100644 --- a/docs/02-methods.md +++ b/docs/02-methods.md @@ -46,7 +46,7 @@ bibliography: ### Data We obtained functional MRI data from 7 different sources ([](tab-samples)). -We included three resting state studies with healthy volunteers (study 1, study 2, study 3, $n_{total}=118$), one task-based study (study 4, $n_{total}=33$ participants, 9 runs each), an individual participant meta-analytic pain activation map of pain (study 5, $n_{total}=603$ from 20 different studies), 8 task-based activation patterns obtained from coordinate-based meta-analyses via Neurosynth (study 6, 14371 studies in total, see {numref}`Supplementary Table %s `) and a resting state dataset focusing on ASD from the ABIDE (Autism Brain Imaging Data Exchange, study 6, $n_{total}=1112$ {cite}`di2014autism`). +We included three resting state studies with healthy volunteers (study 1, study 2, study 3, $n_{total}=118$), one task-based study (study 4, $n_{total}=33$ participants, 9 runs each), an individual participant meta-analytic pain activation map of pain (study 5, $n_{total}=603$ from 20 different studies), 8 task-based activation patterns obtained from coordinate-based meta-analyses via Neurosynth (study 6, 14371 studies in total, see {numref}`Supplementary Table %s `) and a resting state dataset focusing on ASD from the ABIDE (Autism Brain Imaging Data Exchange, study 6, $n_{total}=1112$, {cite}`di2014autism`). ```{list-table} **Datasets and studies.** The table includes details about the study modality, analysis aims, sample size used for analyses, mean age, gender ratio, and references. :header-rows: 1 @@ -110,9 +110,9 @@ We included three resting state studies with healthy volunteers (study 1, study - {cite}`di2014autism` ``` -Study 1 was used to evaluate the ability of the proposed approach to reconstruct resting state brain activity. Study 2 and 3 served es replications studies for these analyses. Further details on study 1-3 are described in [](10.1038/s41467-019-13785-z). The ability of the proposed approach to model task-based perturbation of brain dynamics was evaluated in Study 4, which consisted of nine task-based fMRI runs for each of the 33 healthy volunteers. In all runs, participants received heat pain stimulation. Each stimulus lasted 12.5 seconds, with 3-second ramp-up and 2-second ramp-down periods and 7.5 seconds at target temperature. Six levels of temperature were administered to the participants (level 1: 44.3°C; level 2: 45.3°C; level 3: 46.3°C; level 4: 47.3°C; level 5: 48.3°C; level 6: 49.3°C). In this analysis we used run 1 (passive experience) run 3 (down-regulation) and run 7 (up-regulation). Participants were asked to cognitively “increase” (regulate-up) or “decrease” (regulate-down) pain intensity. See {cite:t}`woo2015distinct` for details. +Study 1 was used to evaluate the ability of the proposed approach to reconstruct resting state brain activity. Study 2 and 3 served as replications studies for these analyses. Further details on study 1-3 are described in [](10.1038/s41467-019-13785-z). The ability of the proposed approach to model task-based perturbation of brain dynamics was evaluated in Study 4, which consisted of nine task-based fMRI runs for each of the 33 healthy volunteers. In all runs, participants received heat pain stimulation. Each stimulus lasted 12.5 seconds, with 3-second ramp-up and 2-second ramp-down periods and 7.5 seconds at target temperature. Six levels of temperature were administered to the participants (level 1: 44.3°C; level 2: 45.3°C; level 3: 46.3°C; level 4: 47.3°C; level 5: 48.3°C; level 6: 49.3°C). In this analysis we used run 1 (passive experience), run 3 (down-regulation) and run 7 (up-regulation). Participants were asked to cognitively “increase” (regulate-up) or “decrease” (regulate-down) pain intensity. See {cite:t}`woo2015distinct` for details. Pain control signal for our task-based trajectory analyses on data from study 4 was derived from our participant-level meta-analysis of 20 pain fMRI studies (study 5, n=603). For details, see {cite:t}`zunhammer2021meta`. -To obtain fMRI activation maps for other tasks, we used Neurosynth([](https://doi.org/10.3389/conf.fninf.2011.08.00058)), a web-based platform for large-scale, automated synthesis of functional magnetic resonance imaging (fMRI) data. We performed 8 different coordinate-based meta-analyses with the terms "motor", "auditory", "visual", "face", "autobiographical", "theory mind", "language" and "pain" ({numref}`Supplementary Table %s `) and obtained the z-scores maps from a two-way ANOVA, comparing the coordinates reported for studies with and without the term of interest, and testing for the presence of a non-zero association between term use and voxel activation. +To obtain fMRI activation maps for other tasks, we used Neurosynth([](https://doi.org/10.3389/conf.fninf.2011.08.00058)), a web-based platform for large-scale, automated synthesis of functional magnetic resonance imaging (fMRI) data. We performed 8 different coordinate-based meta-analyses with the terms "motor", "auditory", "visual", "face", "autobiographical", "theory mind", "language" and "pain" ({numref}`Supplementary Table %s `) and obtained the Z-score maps from a two-way ANOVA, comparing the coordinates reported for studies with and without the term of interest, and testing for the presence of a non-zero association between term use and voxel activation. In study 7 (ABIDE), we obtained preprocessed regional timeseries data from the Preprocessed Connectome Project {cite:p}`craddock2013towards`, as shared (https://osf.io/hc4md) by {cite:t}`dadi2019benchmarking`. All preprocessed timeseries data were obtained with the 122-region version of the BASC (Bootstrap Analysis of Stable Clusters) brain atlas {cite:p}`bellec2010multi`. ### Preprocessing and timeseries extraction @@ -160,7 +160,8 @@ During the stochastic relaxation procedure, we add weak Gaussian noise to each n where $ \epsilon \sim \mathcal{N}(\mathbf{\mu}, \sigma)$, with $\sigma$ regulating the amount of noise added and $\mathbf{\mu}$ set to 0, by default. -In this work we propose functional connectome-based Hopfield neural networks (fcHNNs) as a model for large-scale brain dynamics. fcHNNs are continuous-state Hopfield neural networks with each node representing a brain region and weights initialized with a group-level functional connectivity matrix. The weights are scaled to zero mean and unit standard deviation. +In this work we propose functional connectome-based Hopfield neural networks (fcHNNs) as a model for large-scale brain dynamics. +FcHNNs are continuous-state Hopfield neural networks with each node representing a brain region and weights initialized with a group-level functional connectivity matrix. The weights are scaled to zero mean and unit standard deviation. In studies 1-3, we obtained the finite number of attractor states for all fcHNNs by repeatedly ($10^5$ times) initializing the fcHNN with random activations and relaxing them until convergence. @@ -177,38 +178,37 @@ We obtained the four attractor states in study 1, as described above. We then co ### Evaluation: resting state dynamics Analogously to the methodology of the fcHNN projection, we performed PCA on the preprocessed fMRI time-frames from study 1 (based on the empirical regional timeseries data). -To compare the explanatory power of the first two PCs derived from fcHNN-generated data and real fMRI data, we fit linear regression models which used the first two fcHNN or real data-based PCs as regressors to reconstruct the real fMRI time-frames. In-sample explained variances and the corresponding confidence intervals were calculated for both models with bootstrapping (100 samples). To evaluate the out-of-sample generalization of the PCAs (fcHNN- and real data-based) from study 1, we calculated how much variance they can explain in study 2. +To compare the explanatory power of the first two PCs derived from fcHNN-generated data and real fMRI data, we fitted linear regression models which used the first two fcHNN or real data-based PCs as regressors to reconstruct the real fMRI time-frames. In-sample explained variances and the corresponding confidence intervals were calculated for both models with bootstrapping (100 samples). To evaluate the out-of-sample generalization of the PCs (fcHNN- and real data-based) from study 1, we calculated how much variance they can explain in study 2. -To calculate fractional time occupancies of the attractor states in real timeseries vs simulated data, we used each real and simulated timeframe as an input to the fcHNN of study 1 and obtained the corresponding attractor state. Statistical inference on the similarity of the real fractional occupancies and the fcHNN prediction was performed with two different null models. Null model #1 was constructed by a spatial autocorrelation preserving randomization of all time-frames in the real data. Null model #2 was constructed by random sampling from a multivariate normal distributions, with the covariance matrix set based on the functional connectome (partial correlations). More detail on the null-models can be found in {numref}`Supplementary figure %s `. +To calculate fractional time occupancies of the attractor states in real timeseries vs simulated data, we used each real and simulated timeframe as an input to the fcHNN of study 1 and obtained the corresponding attractor state. Statistical inference on the similarity of the real fractional occupancies and the fcHNN prediction was performed with two different null models. Null model #1 was constructed by a spatial autocorrelation preserving randomization of all time-frames in the real data. Null model #2 was constructed by random sampling from a multivariate normal distribution, with the covariance matrix set based on the functional connectome (partial correlations). More detail on the null-models can be found in {numref}`Supplementary figure %s `. To confirm that the real and fcCHNN temporal sequences (from the stochastic relaxation) on display similar temporal autocorrelation properties, we compared both to their randomly shuffled variant with a "flow analysis". First we calculated the direction on the projection plane between each successive TR (a vector on the fcHNN projection plane for each TR transition), both for the empirical and the shuffled data. -Next, we obtained a two-dimensional binned means for both the x and y coordinates of these transition vectors (pooled across all participants), calculated over a total of 100 uniformly distributed bins in the [-6,6] range (arbitrary units) and applied a Gaussian smoothing with a $\sigma$ of 5 bins. -Finally, we visualized the difference between the binned-mean trajectories of the empirical and the shuffled data. +Next, we obtained a two-dimensional binned means for both the x and y coordinates of these transition vectors (pooled across all participants), calculated over a 2-dimensional grid of 100×100 uniformly distributed bins in the [-6,6] range (arbitrary units) and applied a Gaussian smoothing with a $\sigma$ of 5 bins. +Finally, we visualized the difference between the binned-mean trajectories of the empirical and the shuffled data as a "streamplot", with the Python package matplotlib. The same approach was repeated with the fcHNN-generated data. ### Evaluation: task-based dynamics We used study 4 to evaluate the ability of the fcHNN approach to capture and predict task-induced alterations in large-scale brain dynamics. +sFirst, runs 1, 3 and 7 from all participants, which investigated the passive experience and the down- and up-regulation of pain, respectively, was preprocessed with the same workflow used to preprocess studies 1-3. Regional timeseries data was grouped to "pain" and "rest" blocks, with a 6-second delay to adjust for the hemodynamic response function. All activation timeframes were transformed to the fcHNN projection plane obtained from study 1. Within-participant differences of the average location on the fcHNN projection was calculated and visualized with radial plots, showing the participant-level mean trajectory on the projection plane from rest to pain, denoted with circles, as well as the group level trajectory (arrow). The significance of the position difference and energy difference of the participant-level mean activations in the projection plane was tested with a permutation test. We used the L2 norm of the two-dimensional position difference and the absolute energy difference, respectively, as test statistics. The permutation tests were run with 1000 permutations, randomly swapping the conditions. -First, runs 1, 3 and 7 from all participants, which investigated the passive experience and the down- and up-regulation of pain, respectively, was preprocessed with the same workflow used to preprocess studies 1-3. Regional timeseries data was grouped to "pain" and "rest" blocks, taking into account a 6-second delay to adjust for the hemodynamic response function. All activation timeframes were transformed to the fcHNN projection plane obtained from study 1. Within-participant differences of the average location on the fcHNN projection was calculated and visualized with radial plots, showing the participant-level mean trajectory on the projection plane from rest to pain, denoted with circles, as well as the group level trajectory (arrow). The significance of the position difference and energy difference of the participant-level mean activations in the projection plane was tested with a permutation test. We used the L2 norm of the two-dimensional position difference and the absolute energy difference, respectively, as test statistics. The permutation tests were run with 1000 permutations, randomly swapping the conditions. - -To further highlight the difference between the task and rest conditions, a "flow analysis" was performed to investigate the dynamic trajectory differences between the conditions rest and pain. First we calculated the direction in the projection plane between each successive TR during the rest conditions (a vector on the fcHNN projection plane for each TR transition). Next, we obtained a two-dimensional binned means for both the x and y coordinates of these transition vectors (pooled across all participants), calculated over a total of 100 uniformly distributed bins in the [-6,6] range (arbitrary units) and applied Gaussian smoothing with a $\sigma$ 5 bins. -The same procedure was repeated for the pain condition and the difference in direction between the two conditions was visualized as streamplots (using matplotlib). We used the same approach the difference in characteristic state transition trajectories between the up- and downregulation conditions. The empirically estimated trajectory differences were contrasted to the trajectory differences predicted by the fcHNN model from study 1. +To further highlight the difference between the task and rest conditions, a "flow analysis" was performed to investigate the dynamic trajectory differences between the conditions rest and pain. The analysis method was identical to the flow analysis of resting sate data. First we calculated the direction in the projection plane between each successive TR during the rest conditions (a vector on the fcHNN projection plane for each TR transition). Next, we obtained a two-dimensional binned means for both the x and y coordinates of these transition vectors (pooled across all participants), calculated over a 2-dimensional grid of 100×100 uniformly distributed bins in the [-6,6] range (arbitrary units) and applied Gaussian smoothing with a $\sigma$ 5 bins. +The same procedure was repeated for the pain condition and the difference in the mean directions between the two conditions was visualized as “streamplots” (using Python’s matplotlib). We used the same approach to quantify the difference in characteristic state transition trajectories between the up- and downregulation conditions. The empirically estimated trajectory differences (from real fMRI data) were contrasted to the trajectory differences predicted by the fcHNN model from study 1. To obtain fcHNN-simulated state transitions in resting conditions, we used the stochastic relaxation procedure ({numref}`hopfield-update-matrix-stochastic`), with $\mathbf{\mu}$ set zero. -To simulate the effect of pain-related activation on large-scale brain dynamics, we set $\mu_i$ during the stochastic relaxation procedure simulate pain-elicited activity in region i. The region-wise activations were obtained calculating the parcel-level mean activations from the meta-analytic pain activation map from {cite:p}`zunhammer2021meta`, which contained Hedges' g effect sizes from an individual participant-level meta-analysis of 20 pain studies, encompassing a total of n=603 participants. The whole activation map was scaled with five different values ranging from $10^{-3}$ to $10^{-1}$, spaced logarithmically, to investigate various signal-to-noise scenarios. +To simulate the effect of pain-related activation on large-scale brain dynamics, we set $\mu_i$ during the stochastic relaxation procedure to a value representing pain-elicited activity in region i. The region-wise activations were obtained calculating the parcel-level mean activations from the meta-analytic pain activation map from {cite:p}`zunhammer2021meta`, which contained Hedges' g effect sizes from an individual participant-level meta-analysis of 20 pain studies, encompassing a total of n=603 participants. The whole activation map was scaled with five different values ranging from $10^{-3}$ to $10^{-1}$, spaced logarithmically, to investigate various signal-to-noise scenarios. We obtained the activity patterns of $10^5$ iterations from this stochastic relaxation procedure and calculated the state transition trajectories with the same approach used with the empirical data. -Next we calculated the simulated difference between the rest and pain conditions and compared it to the actual difference through a permutation test with 1000 permutations, using the Pearson's correlation coefficient as the test statistic. +Next we calculated the simulated difference between the rest and pain conditions and compared it to the actual difference through a permutation test with 1000 permutations, using Pearson's correlation coefficient as test statistic. From the five investigated signal-to-noise values, we chose the one that provided the highest similarity to the real pain vs. rest trajectory difference. -When comparing the simulated and real trajectory differences between pain up- and downregulation, we used the same procedure, with two differences. First, when calculating the simulated state transition vectors for the self-regulation conditions, we used the same procedure as for the pain condition, but introduced and additional signal in the nucleus accumbens, with a negative and positive sign, for up- and downregulation, respectively.We did not optimize the signal-to-noise ratio for the nucleus accumbens signal but, instead, simply used the value optimized for the pain vs. rest contrast (For a robustness analysis, see {numref}`Supplementary figure %s `). +When comparing the simulated and real trajectory differences between pain up- and downregulation, we used the same procedure, with two differences. First, when calculating the simulated state transition vectors for the self-regulation conditions, we used the same procedure as for the pain condition, but introduced and additional signal in the nucleus accumbens, with a negative and positive sign, for up- and downregulation, respectively. We did not optimize the signal-to-noise ratio for the nucleus accumbens signal but, instead, simply used the value optimized for the pain vs. rest contrast (For a robustness analysis, see {numref}`Supplementary figure %s `). ### Clinical data -To demonstrate the sensitivity of the fcHNN approach to clinically relevant alterations of large-scale brain dynamics in Autism Spectrum Disorder (ASD), we obtained data from n=172 individuals acquired at the New York University Langone Medical Center, New York, NY, USA (NYU) as shared in the Autism Brain Imaging Data Exchange dataset ([study 7](tab-samples): ABIDE, {cite:p}`di2014autism`. We focused on the largest ABIDE imaging center to ensure that our results are not biased by center effects. We excluded high motion cases similarly to our approach in studies 1-4, i.e. by ignoring ("scrubbing") volumes with FD>0.15 and excluding participants with more than 50% of data scrubbed. Timeseries data was pooled and visualized on the fcHNN projection of study 1, separately for ASD and control participants. +To demonstrate the sensitivity of the fcHNN approach to clinically relevant alterations of large-scale brain dynamics in Autism Spectrum Disorder (ASD), we obtained data from n=172 individuals, acquired at the New York University Langone Medical Center, New York, NY, USA (NYU) as shared in the Autism Brain Imaging Data Exchange dataset ([study 7](tab-samples): ABIDE, {cite:p}`di2014autism`. We focused on the largest ABIDE imaging center to ensure that our results are not biased by center effects. We excluded high motion cases similarly to our approach in studies 1-4, i.e. by ignoring ("scrubbing") volumes with FD>0.15 and excluding participants with more than 50% of data scrubbed. Timeseries data was pooled and visualized on the fcHNN projection of study 1, separately for ASD and control participants. Next, for each participant, we grouped the timeframes from the regional timeseries data according to the corresponding attractor states (obtained with the fcHNN model from study 1) and averaged timeframes corresponding to the same attractor state to calculated participant-level mean attractor activations. We assessed mean attractor activity differences between the patient groups with a permutation test, randomly re-assigning the group labels 50000 times. We adjusted the significance threshold with a Bonferroni-correction, accounting for tests across 4 states and 122 regions, resulting in $\alpha = 0.0001$. diff --git a/docs/03-supplement.md b/docs/03-supplement.md index 4164acac..8bd45b5e 100644 --- a/docs/03-supplement.md +++ b/docs/03-supplement.md @@ -34,7 +34,7 @@ activation.** See [supplemental_material.ipynb](https://github.com/pni-lab/conn :name: si_state_occupancy_null_models **Statistical inference of the fcHNN state occupancy prediction with different null models.** **A** Results with a spatial autocorrelation-preserving null model for the empirical activity patterns. See [null_models.ipynb](https://github.com/pni-lab/connattractor/blob/master/notebooks/null_models.ipynb) for more details. -**B** Results where simulated samples are randomly sampled from a multivariate normal distribution, with te functional connectome as the covariance matrix, and compared to the fcHNN performance. See [supplemental_material.ipynb](https://github.com/pni-lab/connattractor/blob/master/notebooks/supplemental_material.ipynb) for details. +**B** Results where simulated samples are randomly sampled from a multivariate normal distribution, with the functional connectome as the covariance matrix, and compared to the fcHNN performance. See [supplemental_material.ipynb](https://github.com/pni-lab/connattractor/blob/master/notebooks/supplemental_material.ipynb) for details. ::: :::{figure} figures/supplement/si_pain_ghost_attractor_sim.png diff --git a/docs/_build/site/config.json b/docs/_build/site/config.json index 8401fcee..51476747 100644 --- a/docs/_build/site/config.json +++ b/docs/_build/site/config.json @@ -1 +1 @@ -{"title":"ConnAttractor","logo":"/logo-fchnn-262bc19d2ed396aea18fe22dd66afe96.png","logo_dark":"/logo-fchnn-262bc19d2ed396aea18fe22dd66afe96.png","logo_text":"fcHNN","analytics_google":"G-16X8EW2S89","myst":"v1","nav":[],"actions":[{"title":"PNI-lab","url":"https://pni-lab.github.io/"}],"projects":[{"license":{"content":{"id":"CC-BY-4.0","name":"Creative Commons Attribution 4.0 International","free":true,"CC":true,"url":"https://creativecommons.org/licenses/by/4.0/"}},"title":"ConnAttractor Website","short_title":"ConnAttractor","description":"Webpage for functional connectome-based Hopfield Neural 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The challenge for such models lies in modelling the relation between the structural wiring of the brain and functional connectivity.\nThe \"neuroconnectionist\" approach, on the other hand, ","key":"OAhQ1udNOo"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"doerig2023neuroconnectionist","identifier":"doerig2023neuroconnectionist","children":[{"type":"text","value":"Doerig ","key":"rknc8VxOwM"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"ltHeL33voY"}],"key":"WeqSlT8eix"},{"type":"text","value":", 2023","key":"RVP0HCl08X"}],"key":"iRyaCZbP3D"}],"key":"G9Y77Cpdj1"},{"type":"text","value":" aims at \"cognitive/behavioral fidelity\" ","key":"NVaucz4Q1X"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"kriegeskorte2018cognitive","identifier":"kriegeskorte2018cognitive","children":[{"type":"text","value":"Kriegeskorte & Douglas, 2018","key":"Dy4naiiIia"}],"key":"xIlBJWffM1"}],"key":"I0PoL0Ihrp"},{"type":"text","value":", by using artificial neural networks (","key":"m4XiURM5Cl"},{"type":"abbreviation","title":"Artificial Neural Network","children":[{"type":"text","value":"ANN","key":"ttunVXvBnG"}],"key":"MEjF4JRHLy"},{"type":"text","value":"s) that are trained to perform various tasks, as brain models. However, the need to train ","key":"gLlEHf9PvV"},{"type":"abbreviation","title":"Artificial Neural Network","children":[{"type":"text","value":"ANN","key":"sk9izxGdXi"}],"key":"t5k3FgqxA0"},{"type":"text","value":"s for specific tasks inherently limits their ability to explain task-independent, spontaneous neural dynamics ","key":"aBL1jVzolO"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"richards2019deep","identifier":"richards2019deep","children":[{"type":"text","value":"Richards ","key":"o8FuGbIXlR"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"aHf94815rJ"}],"key":"ZWHjiqZRBx"},{"type":"text","value":", 2019","key":"wezQ0x6Fff"}],"key":"e5qwH8uk80"}],"key":"WrBrbQE9p8"},{"type":"text","value":".","key":"pKAUcH1I4s"}],"key":"rkFXg2GMqr"},{"type":"paragraph","position":{"start":{"line":143,"column":0},"end":{"line":146,"column":0}},"children":[{"type":"text","value":"Here we propose a novel approach that combines the advantages of large-scale network models and neuroconnectionism, to investigate brain dynamics.\nSimilar to neuroconnectionism, we utilize an ","key":"tqMAITCDzO"},{"type":"abbreviation","title":"Artificial Neural Network","children":[{"type":"text","value":"ANN","key":"drRBYTr5TT"}],"key":"gMf5espZSf"},{"type":"text","value":" as a high-level computational model of the brain.\nHowever, our model is not explicitly trained for a specific task. Instead, we set its weights empirically, with data based on the \"activity flow\" ","key":"JOmA7zJ0It"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"cole2016activity","identifier":"cole2016activity","children":[{"type":"text","value":"Cole ","key":"gNeAFz6TUC"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"hhDeP0SV1u"}],"key":"s6HfNDCFW7"},{"type":"text","value":", 2016","key":"XWkM41hyOr"}],"key":"iI6QlYqvbS"},{"type":"cite","label":"ito2017cognitive","identifier":"ito2017cognitive","children":[{"type":"text","value":"Ito ","key":"LB86nzIL7O"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"IVdRHKMwsm"}],"key":"iCHgI8fhpk"},{"type":"text","value":", 2017","key":"YHNhT3Zvmo"}],"key":"ljpxDxbEEE"}],"key":"whmLRXQ7G4"},{"type":"text","value":" across regions within the functional brain connectome, as measured with functional magnetic resonance imaging (","key":"dN1YnMbduP"},{"type":"abbreviation","title":"functional Magnetic Resonance Imaging","children":[{"type":"text","value":"fMRI","key":"kRpS6cWUiN"}],"key":"FfUurf3sVC"},{"type":"text","value":", ","key":"iVwV25TNxQ"},{"type":"crossReference","kind":"figure","identifier":"concept","label":"concept","children":[{"type":"text","value":"Figure ","key":"oe3WDI7R4Z"},{"type":"text","value":"1","key":"xMoDxKasGS"}],"template":"Figure %s","enumerator":"1","resolved":true,"html_id":"concept","key":"PSj6BnYPdx"},{"type":"text","value":"B).","key":"Gofxw83BNU"}],"key":"pKaKlzD7M7"},{"type":"paragraph","position":{"start":{"line":147,"column":0},"end":{"line":151,"column":0}},"children":[{"type":"text","value":"Specifically, we employ a continuous-space Hopfield Neural Network (","key":"RL3o53pBv1"},{"type":"abbreviation","title":"Hopfield Neural Network","children":[{"type":"text","value":"HNN","key":"zJrB2BEuxg"}],"key":"uSc0h1dtE6"},{"type":"text","value":") ","key":"f3yEoNFZO7"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"hopfield1982neural","identifier":"hopfield1982neural","children":[{"type":"text","value":"Hopfield, 1982","key":"DNlCZReyUE"}],"key":"uBNZUKZzp3"},{"type":"cite","label":"krotov2023new","identifier":"krotov2023new","children":[{"type":"text","value":"Krotov, 2023","key":"e7Uqarmgd8"}],"key":"nP7qbDoz3E"}],"key":"CcvHxg5gUy"},{"type":"text","value":", with its nodes representing large-scale brain areas, and its weights initialized with the functional connectivity values between these areas.\nBased on the topology of the functional connectome, this architecture establishes an energy level for any arbitrary activation patterns and determines a \"trajectory of least action\" towards one of the finite number of stable patterns, known as ","key":"nt1JKMKzaB"},{"type":"emphasis","children":[{"type":"text","value":"attractor states","key":"d6hHDbOx0R"}],"key":"PpOtCFNzNR"},{"type":"text","value":", that minimize this energy.\nIn this simplistic yet powerful framework, brain dynamics can be conceptualized as an intricate, high-dimensional path on the energy landscape (","key":"Sm1wVydukz"},{"type":"crossReference","kind":"figure","identifier":"concept","label":"concept","children":[{"type":"text","value":"Figure ","key":"Ue728HGfdG"},{"type":"text","value":"1","key":"vbxjeackle"}],"template":"Figure %s","enumerator":"1","resolved":true,"html_id":"concept","key":"jbUTE5ch9o"},{"type":"text","value":"C), arising from the activity flow ","key":"ghjBvEiFzF"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"cole2016activity","identifier":"cole2016activity","children":[{"type":"text","value":"Cole ","key":"i0Xs4waIi7"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"bZso6Gr2oZ"}],"key":"pWfxCrLHtx"},{"type":"text","value":", 2016","key":"PZOv8Ic4ah"}],"key":"RaZaUQQdze"}],"key":"lOn43X1Yr2"},{"type":"text","value":" within the functional connectome and constrained by the \"gravitational pull\" of the attractor states of the system.\nGiven its generative nature, the proposed model offers testable predictions for the effect of various perturbations and alterations of these dynamics, from task-induced activity, to changes related to brain disorders.","key":"iHBT2WydNE"}],"key":"D83QLUTvzg"},{"type":"container","kind":"figure","identifier":"concept","label":"concept","children":[{"type":"image","url":"/concept-fc61e809920bee5c5ec3407459032fc2.png","alt":"Connectome-based Hopfield networks as models of macro-scale brain dynamics.

\nA Hopfield artificial neural networks (HNNs) are a form of recurrent artificial neural networks that serve as content-addressable (\"associative\") memory systems. Hopfield networks can be trained to store a finite number of patterns (e.g. via Hebbian learning a.k.a. \"fire together - wire together\"). During the training procedure, the weights of the HNN are trained so that the stored\npatterns become stable attractor states of the network. Thus, when the trained network is presented partial, noisy or corrupted variations of the stored patterns, it can effectively reconstruct the original pattern via an iterative relaxation procedure that converges to the attractor states.\nB We consider regions of the brain as nodes of a Hopfield network. Instead of training the Hopfield network to\nspecific tasks, we set its weights empirically, with the interregional activity flow estimated via functional\nbrain connectivity. Capitalizing on strong analogies between the relaxation rule of Hopfield networks and the\nactivity flow principle that links activity to connectivity in brain networks, we propose the resulting\nfunctional connectome-based Hopfield neural network (fcHNN) as a computational model for macro-scale brain dynamics.C The proposed computational framework assigns an energy level, an attractor state and a position in a\nlow-dimensional embedding to brain activation patterns. Additionally, it models how the entire state-space of viable activation patterns is restricted by the dynamics of the system and how alterations in activity and/or connectivity modify these dynamics.","urlSource":"figures/concept.png","urlOptimized":"/concept-fc61e809920bee5c5ec3407459032fc2.webp","key":"jcBnntnIIg"},{"type":"caption","children":[{"type":"paragraph","position":{"start":{"line":154,"column":0},"end":{"line":164,"column":0}},"children":[{"type":"captionNumber","kind":"figure","label":"concept","identifier":"concept","html_id":"concept","enumerator":"1","children":[{"type":"text","value":"Figure ","key":"TZGaEuKgbX"},{"type":"text","value":"1","key":"q1mlOBCsCf"},{"type":"text","value":":","key":"kqL0AgrVp1"}],"template":"Figure %s:","key":"AfmDVHUjUU"},{"type":"strong","children":[{"type":"text","value":"Connectome-based Hopfield networks as models of macro-scale brain dynamics.","key":"UYMCzm2k1I"}],"key":"FVemfmVhQF"},{"type":"text","value":" ","key":"FUI6mmUK9X"},{"type":"break","key":"WLmGu7TRDg"},{"type":"break","key":"OKgCYlf4w7"},{"type":"text","value":"\n","key":"NZIS2lwY6b"},{"type":"strong","children":[{"type":"text","value":"A","key":"eARdyFpdkD"}],"key":"WRQftdSWgA"},{"type":"text","value":" Hopfield artificial neural networks (","key":"SoVg9W0cci"},{"type":"abbreviation","title":"Hopfield Neural Network","children":[{"type":"text","value":"HNN","key":"V0qvLwdPxx"}],"key":"v9h8so77he"},{"type":"text","value":"s) are a form of recurrent artificial neural networks that serve as content-addressable (\"associative\") memory systems. Hopfield networks can be trained to store a finite number of patterns (e.g. via Hebbian learning a.k.a. \"fire together - wire together\"). During the training procedure, the weights of the ","key":"HNjs26TdqR"},{"type":"abbreviation","title":"Hopfield Neural Network","children":[{"type":"text","value":"HNN","key":"waf52VzzMM"}],"key":"M41M1nUZ7c"},{"type":"text","value":" are trained so that the stored\npatterns become stable attractor states of the network. Thus, when the trained network is presented partial, noisy or corrupted variations of the stored patterns, it can effectively reconstruct the original pattern via an iterative relaxation procedure that converges to the attractor states.\n","key":"Ds1alOlSNN"},{"type":"strong","children":[{"type":"text","value":"B","key":"qKhKiUT3WF"}],"key":"HAVAwiwa3y"},{"type":"text","value":" We consider regions of the brain as nodes of a Hopfield network. Instead of training the Hopfield network to\nspecific tasks, we set its weights empirically, with the interregional activity flow estimated via functional\nbrain connectivity. Capitalizing on strong analogies between the relaxation rule of Hopfield networks and the\nactivity flow principle that links activity to connectivity in brain networks, we propose the resulting\nfunctional connectome-based Hopfield neural network (","key":"bXKZzHq7ID"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"MrtHoXz9wZ"}],"key":"gDQTbJB0lN"},{"type":"text","value":") as a computational model for macro-scale brain dynamics.","key":"Vvj6aJ6l7X"},{"type":"break","key":"TqkERqw8hB"},{"type":"strong","children":[{"type":"text","value":"C","key":"tG1dafa26H"}],"key":"ynuoepflQc"},{"type":"text","value":" The proposed computational framework assigns an energy level, an attractor state and a position in a\nlow-dimensional embedding to brain activation patterns. Additionally, it models how the entire state-space of viable activation patterns is restricted by the dynamics of the system and how alterations in activity and/or connectivity modify these dynamics.","key":"S9SLRaddnu"}],"key":"evOssPyFFW"}],"key":"bQBTK11Bhs"}],"enumerator":"1","html_id":"concept","key":"DVdzePomJZ"},{"type":"paragraph","position":{"start":{"line":166,"column":0},"end":{"line":168,"column":0}},"children":[{"type":"text","value":"In the present work, we use ","key":"vhTwI9uFEc"},{"type":"abbreviation","title":"Hopfield Neural Network","children":[{"type":"text","value":"HNN","key":"Dxb80I21zf"}],"key":"fggPywdhh0"},{"type":"text","value":"s to explore the functional connectome's attractor-dynamics with the aid of a streamlined, low-dimensional representation of the energy landscape.\nSubsequently, we use a diverse set of experimental, clinical and meta-analytic studies to evaluate our model's ability to reconstruct various characteristics of resting state brain dynamics, as well as its capacity to detect and explain changes induced by experimental tasks or alterations in brain disorders.","key":"iDZ9oNkzK3"}],"key":"Awego0IoBE"},{"type":"heading","depth":2,"position":{"start":{"line":169,"column":0},"end":{"line":170,"column":0}},"children":[{"type":"text","value":"Results","key":"PwSkBXzCQ7"}],"identifier":"results","label":"Results","html_id":"results","implicit":true,"key":"lesMRDGap4"},{"type":"heading","depth":3,"position":{"start":{"line":171,"column":0},"end":{"line":172,"column":0}},"children":[{"type":"text","value":"Connectome-based Hopfield network as a model of brain dynamics","key":"EwjpnXetWt"}],"identifier":"connectome-based-hopfield-network-as-a-model-of-brain-dynamics","label":"Connectome-based Hopfield network as a model of brain dynamics","html_id":"connectome-based-hopfield-network-as-a-model-of-brain-dynamics","implicit":true,"key":"GLnImwkRyc"},{"type":"paragraph","position":{"start":{"line":173,"column":0},"end":{"line":180,"column":0}},"children":[{"type":"text","value":"First, we explored the attractor states of the functional connectome in a sample of n=41 healthy young\nparticipants (","key":"cHP0a3h94O"},{"type":"crossReference","children":[{"type":"text","value":"study 1","key":"Z5B3KZBywO"}],"urlSource":"#tab-samples","identifier":"tab-samples","label":"tab-samples","kind":"table","template":"Table %s","enumerator":"1","resolved":true,"html_id":"tab-samples","remote":true,"url":"/methods","dataUrl":"/methods.json","key":"zQ2Fbmo8uQ"},{"type":"text","value":", see Methods ","key":"KsenjH3vIq"},{"type":"crossReference","children":[{"type":"text","value":"Table ","key":"LLAfuiDllt"},{"type":"text","value":"1","key":"mjMRmSRMNp"}],"urlSource":"#tab-samples","identifier":"tab-samples","label":"tab-samples","kind":"table","template":"Table %s","enumerator":"1","resolved":true,"html_id":"tab-samples","remote":true,"url":"/methods","dataUrl":"/methods.json","key":"y9VhUn4uj9"},{"type":"text","value":" for details). We estimated interregional activity flow ","key":"QkKYiBZfzc"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"cole2016activity","identifier":"cole2016activity","children":[{"type":"text","value":"Cole ","key":"kggHvhUg7x"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"zfIQr6vOnK"}],"key":"i8e2wnDvMR"},{"type":"text","value":", 2016","key":"VKjVayT1So"}],"key":"g6jN8WIHCA"},{"type":"cite","label":"ito2017cognitive","identifier":"ito2017cognitive","children":[{"type":"text","value":"Ito ","key":"dOGJxFk5ox"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"e4zYDw0D1j"}],"key":"oYKm4XVXpp"},{"type":"text","value":", 2017","key":"XfvlbV5PYy"}],"key":"JFK0py7GaU"}],"key":"iX0KNJUymK"},{"type":"text","value":"\nas the study-level average of regularized partial correlations among the resting state ","key":"uI1Wd90COo"},{"type":"abbreviation","title":"functional Magnetic Resonance Imaging","children":[{"type":"text","value":"fMRI","key":"p2HgmnC85x"}],"key":"uw0Rqm6q2Z"},{"type":"text","value":" timeseries of m = 122\nfunctionally defined brain regions (see ","key":"Tw1KmTV9fR"},{"type":"crossReference","children":[{"type":"text","value":"Methods","key":"kqTT7wGXgZ"}],"urlSource":"#Functional-connectome","identifier":"functional-connectome","label":"Functional-connectome","kind":"heading","template":"{name}","resolved":true,"html_id":"functional-connectome","remote":true,"url":"/methods","dataUrl":"/methods.json","key":"hIhnqFUkwA"},{"type":"text","value":" for details). We then used the standardized\nfunctional connectome as the ","key":"YusYYyMR5u"},{"type":"inlineMath","value":"w_{ij}","html":"wijw_{ij}wij​","key":"QkhbCsDxGd"},{"type":"text","value":" weights of a fully connected recurrent ","key":"tiHnXLZMXd"},{"type":"abbreviation","title":"Artificial Neural Network","children":[{"type":"text","value":"ANN","key":"r9oXF84Dhh"}],"key":"aICSDQQdzD"},{"type":"text","value":", specifically a continuous-state Hopfield network ","key":"PV1LPbzucO"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"hopfield1982neural","identifier":"hopfield1982neural","children":[{"type":"text","value":"Hopfield, 1982","key":"Mooaa6eeMG"}],"key":"f2OchNzl8y"},{"type":"cite","label":"koiran1994dynamics","identifier":"koiran1994dynamics","children":[{"type":"text","value":"Koiran, 1994","key":"RGwumdptvE"}],"key":"yUsHmJU2cX"}],"key":"sllUnKxaxK"},{"type":"text","value":", consisting of ","key":"B9S4NKO6AL"},{"type":"inlineMath","value":"m","html":"mmm","key":"zxFongWyJp"},{"type":"text","value":" neural units, each having an activity\n","key":"CVTAvB1bj1"},{"type":"inlineMath","value":"a_i \\in [-1,1] \\subset \\mathbb{R})","html":"ai∈[−1,1]⊂R)a_i \\in [-1,1] \\subset \\mathbb{R})ai​∈[−1,1]⊂R)","key":"TS6M0dMOjF"},{"type":"text","value":". Hopfield networks can be initialized by an arbitrary activation pattern (consisting of ","key":"lgo1i963Li"},{"type":"inlineMath","value":"m","html":"mmm","key":"EuvdBEzxbv"},{"type":"text","value":" activation values) and iteratively updated (i.e. \"relaxed\") until their energy converges a local minimum, that is, to one of the finite number of attractor states (see ","key":"tcteCh6YxH"},{"type":"crossReference","children":[{"type":"text","value":"Methods","key":"fWCO98rjKJ"}],"urlSource":"#connectome-based-hopfield-networks","identifier":"connectome-based-hopfield-networks","label":"connectome-based-hopfield-networks","kind":"heading","template":"{name}","resolved":true,"html_id":"connectome-based-hopfield-networks","remote":true,"url":"/methods","dataUrl":"/methods.json","key":"VCZyRiOg0I"},{"type":"text","value":"). The relaxation procedure is based on a simple rule; in each iteration, the activity of a region is constructed as the weighted average of the activities of all other regions, with weights defined by the connectivity between them. The average is then transformed by a sigmoidal activation function, to keep it in the desired [-1,1] interval.\nThis can be expressed by the following equation:","key":"Lgp86UkRxy"}],"key":"cfgHolwRCo"},{"type":"math","identifier":"hopfield-update","label":"hopfield-update","value":"\\dot{a}_i = S(\\beta \\sum_{j=1}^m w_{ij}a_j - b_i)","html":"a˙i=S(β∑j=1mwijaj−bi)\\dot{a}_i = S(\\beta \\sum_{j=1}^m w_{ij}a_j - b_i)a˙i​=S(βj=1∑m​wij​aj​−bi​)","enumerator":"1","html_id":"hopfield-update","key":"CIRdEhIN0w"},{"type":"paragraph","position":{"start":{"line":186,"column":0},"end":{"line":189,"column":0}},"children":[{"type":"text","value":"where ","key":"hY7FG6gBif"},{"type":"inlineMath","value":"\\dot{a}_i","html":"a˙i\\dot{a}_ia˙i​","key":"o83MVxKsPJ"},{"type":"text","value":" is the activity of neural unit ","key":"RVRf7OZ32N"},{"type":"inlineMath","value":"i","html":"iii","key":"vmb0NxkSGg"},{"type":"text","value":" in the next iteration and ","key":"QqS2Go5s2u"},{"type":"inlineMath","value":"S(a_j)","html":"S(aj)S(a_j)S(aj​)","key":"Bh7vp481bZ"},{"type":"text","value":" is the sigmoidal activation\nfunction (","key":"Of1dnnsNpD"},{"type":"inlineMath","value":"S(a) = tanh(a)","html":"S(a)=tanh(a)S(a) = tanh(a)S(a)=tanh(a)","key":"BQcKIZ33oM"},{"type":"text","value":" in our implementation) and ","key":"zRgMv3XrbA"},{"type":"inlineMath","value":"b_i","html":"bib_ibi​","key":"ZcDJgyCyLl"},{"type":"text","value":" is the bias of unit ","key":"vldhJW0WWP"},{"type":"inlineMath","value":"i","html":"iii","key":"U7IrMBe7La"},{"type":"text","value":" and ","key":"ttEcFFyMuM"},{"type":"inlineMath","value":"\\beta","html":"β\\betaβ","key":"A78PyChAxu"},{"type":"text","value":" is the so-called temperature parameter. For the sake of simplicity, we set ","key":"JpGqYypoWO"},{"type":"inlineMath","value":"b_i=0","html":"bi=0b_i=0bi​=0","key":"Y1dpwxozVc"},{"type":"text","value":" in all our experiments. We refer to this architecture as a functional connectivity-based Hopfield Neural Network (","key":"iPg0oj2EQE"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"qztstun2vV"}],"key":"u8gmF9cqVk"},{"type":"text","value":").\nThe relaxation of a ","key":"CY8j37hems"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"ORzDbfOSjq"}],"key":"R3ShRt5VKo"},{"type":"text","value":" model can be conceptualized as the repeated application of the activity flow principle ","key":"K5iEVu7CkF"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"cole2016activity","identifier":"cole2016activity","children":[{"type":"text","value":"Cole ","key":"NEfRQoV32d"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"b6vPxJGFIC"}],"key":"ykjZaIHETf"},{"type":"text","value":", 2016","key":"mpcEQvpGkD"}],"key":"hf7AnSu18w"},{"type":"cite","label":"ito2017cognitive","identifier":"ito2017cognitive","children":[{"type":"text","value":"Ito ","key":"Uzr5dkveM4"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"XZT7VqvIfF"}],"key":"lTXa9jHP5O"},{"type":"text","value":", 2017","key":"pQR4RFaF3l"}],"key":"P2Wk09uf8U"}],"key":"TYzDvZpUm8"},{"type":"text","value":" , simultaneously for all regions: ","key":"ggLOy3zLki"},{"type":"inlineMath","value":"\\dot{a}_i = \\sum_{j=1}^m w_{ij}a_j","html":"a˙i=∑j=1mwijaj\\dot{a}_i = \\sum_{j=1}^m w_{ij}a_ja˙i​=∑j=1m​wij​aj​","key":"TWP2ZegGzW"},{"type":"text","value":". The update rule also exhibits analogies with network control theory ","key":"otl4IOfbUM"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"gu2015controllability","identifier":"gu2015controllability","children":[{"type":"text","value":"Gu ","key":"BQ3QoTuMkT"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"xXpkpgxaJq"}],"key":"bxHwNjjTkz"},{"type":"text","value":", 2015","key":"YjLaxPXTFE"}],"key":"sHbKot6gQ6"}],"key":"i1y80FF7nV"},{"type":"text","value":" and the inner workings of neural mass models, as applied e.g. in dynamic causal modeling ","key":"yx4Gtf3ggn"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"daunizeau2012stochastic","identifier":"daunizeau2012stochastic","children":[{"type":"text","value":"Daunizeau ","key":"SMhhK6X4s9"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"wXu2XiNRtK"}],"key":"NKq8wGAT4h"},{"type":"text","value":", 2012","key":"i3of5xGnf3"}],"key":"CNmPtKppbf"}],"key":"SfJDjnXwPF"},{"type":"text","value":".","key":"ljS7VFxUTK"}],"key":"RdRraBSmjb"},{"type":"paragraph","position":{"start":{"line":190,"column":0},"end":{"line":193,"column":0}},"children":[{"type":"text","value":"Hopfield networks assign an energy value to each possible activity configuration ","key":"cudsx4SY9h"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"hopfield1982neural","identifier":"hopfield1982neural","children":[{"type":"text","value":"Hopfield, 1982","key":"dtkdQBc5jX"}],"key":"Pvy098ZJNG"},{"type":"cite","label":"koiran1994dynamics","identifier":"koiran1994dynamics","children":[{"type":"text","value":"Koiran, 1994","key":"eu4PvuP1di"}],"key":"VE2g320Y1v"}],"key":"dwSed6Qe1T"},{"type":"text","value":", which decreases during the relaxation procedure until reaching an equilibrium state with minimal energy (","key":"MOai11Rpld"},{"type":"crossReference","kind":"figure","identifier":"attractors","label":"attractors","children":[{"type":"text","value":"Figure ","key":"moep3wzEbc"},{"type":"text","value":"2","key":"Mpv4cJUMyV"}],"template":"Figure %s","enumerator":"2","resolved":true,"html_id":"attractors","key":"c5aDPNSd3s"},{"type":"text","value":"A, top panel).\nWe used a large number of random initializations to obtain all possible attractor states of the connectome-based\nHopfield network in study 1 (","key":"nzvRUL8rix"},{"type":"crossReference","kind":"figure","identifier":"attractors","label":"attractors","children":[{"type":"text","value":"Figure ","key":"rWGRFdkQnW"},{"type":"text","value":"2","key":"GA2VgX9hdg"}],"template":"Figure %s","enumerator":"2","resolved":true,"html_id":"attractors","key":"jXpg03dts2"},{"type":"text","value":"A, bottom panel).","key":"mdEA6yrsm4"}],"key":"cdoH2ZldDB"},{"type":"container","kind":"figure","identifier":"attractors","label":"attractors","children":[{"type":"image","url":"/embedding_method-78f8071a1c322ba464e9c83b5777e057.png","alt":"Attractor states and state-space dynamics of connectome-based Hopfield networks

\nA Top: During so-called relaxation procedure, activities in the nodes of an fcHNN model are iteratively updated based on the activity of all other regions and the connectivity between them. The energy of a\nconnectome-based Hopfield network decreases during the relaxation procedure until reaching an equilibrium state with\nminimal energy, i.e. an attractor state. Bottom: Four attractor states of the fcHNN derived from the\ngroup-level functional connectivity matrix from study 1 (n=44).\nB Top: In presence of weak noise (stochastic update), the system\ndoes not converge to equilibrium anymore. Instead, activity transverses on the state landscape in a way\nrestricted by the topology of the connectome and the \"gravitational pull\" of the attractor states. Bottom: We sample\nthe \"state landscape\" by running the stochastic relaxation procedure for an extended amount of time (e.g. 100.000 consecutive\nstochastic updates), each point representing an activation configuration or state. To construct a\nlow-dimensional representation of the state space, we take the first two principal components of the simulated activity\npatterns. The first two principal components explain approximately 58-85% of the variance of state energy (depending\non the noise parameter \\sigma, see Supplementary Figure %s).\nC We map all states of the state space sample to their corresponding attractor state, with the conventional\nHopfield relaxation procedure (A). The four attractor states are also visualized in their corresponding position on the\nPCA-based projection. The first two principal components yield a clear separation of the attractive state basins\n(cross-validated classification accuracy: 95.5%, Supplementary Figure %s). We refer to the resulting visualization\nas the fcHNN projection and use it to visualize fcHNN-derived and empirical brain dynamics throughout the rest of\nthe manuscript.\nE At its simplest form, the fcHNN framework entails only two free hyperparameters: the temperature parameter\n\\beta (left) that controls the number of attractor states and the noise parameter of the stochastic relaxation\n\\sigma. To avoid overfitting these parameters to the empirical data, we set \\beta=0.04 and \\sigma=0.37 for the\nrest of the paper (dotted boxes).","urlSource":"figures/embedding_method.png","urlOptimized":"/embedding_method-78f8071a1c322ba464e9c83b5777e057.webp","key":"GyP6Djno8i"},{"type":"caption","children":[{"type":"paragraph","position":{"start":{"line":196,"column":0},"end":{"line":219,"column":0}},"children":[{"type":"captionNumber","kind":"figure","label":"attractors","identifier":"attractors","html_id":"attractors","enumerator":"2","children":[{"type":"text","value":"Figure ","key":"bsq11sUVEF"},{"type":"text","value":"2","key":"wEBrhb2U4g"},{"type":"text","value":":","key":"fH2Ib6lr5C"}],"template":"Figure %s:","key":"Z9qzS7mB4e"},{"type":"strong","children":[{"type":"text","value":"Attractor states and state-space dynamics of connectome-based Hopfield networks","key":"YZ6keDUfXo"}],"key":"HkI0u7eHBB"},{"type":"text","value":" ","key":"VTMdxP4AWw"},{"type":"break","key":"MGBZcm86t3"},{"type":"break","key":"QFfTrJrbCv"},{"type":"text","value":"\n","key":"SSgS9oxryf"},{"type":"strong","children":[{"type":"text","value":"A","key":"E7ozxoWh0e"}],"key":"OmHnjGwW1g"},{"type":"text","value":" Top: During so-called relaxation procedure, activities in the nodes of an ","key":"pKGWmCKLGf"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"BPyDL4RtBm"}],"key":"Yqzwe2wYj7"},{"type":"text","value":" model are iteratively updated based on the activity of all other regions and the connectivity between them. The energy of a\nconnectome-based Hopfield network decreases during the relaxation procedure until reaching an equilibrium state with\nminimal energy, i.e. an attractor state. Bottom: Four attractor states of the ","key":"LZX5EF9IhP"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"qKFOIZXfaa"}],"key":"JgNcTQ9IRk"},{"type":"text","value":" derived from the\ngroup-level functional connectivity matrix from ","key":"qXYk2OFnEy"},{"type":"crossReference","children":[{"type":"text","value":"study 1","key":"YKdJeufr7i"}],"urlSource":"#tab-samples","identifier":"tab-samples","label":"tab-samples","kind":"table","template":"Table %s","enumerator":"1","resolved":true,"html_id":"tab-samples","remote":true,"url":"/methods","dataUrl":"/methods.json","key":"EulprMo9Pj"},{"type":"text","value":" (n=44).\n","key":"PxG3DjZxlJ"},{"type":"strong","children":[{"type":"text","value":"B","key":"eE6y0NWkBt"}],"key":"CP9ulICg6X"},{"type":"text","value":" Top: In presence of weak noise (stochastic update), the system\ndoes not converge to equilibrium anymore. Instead, activity transverses on the state landscape in a way\nrestricted by the topology of the connectome and the \"gravitational pull\" of the attractor states. Bottom: We sample\nthe \"state landscape\" by running the stochastic relaxation procedure for an extended amount of time (e.g. 100.000 consecutive\nstochastic updates), each point representing an activation configuration or state. To construct a\nlow-dimensional representation of the state space, we take the first two principal components of the simulated activity\npatterns. The first two principal components explain approximately 58-85% of the variance of state energy (depending\non the noise parameter ","key":"af5Xs8MJcl"},{"type":"inlineMath","value":"\\sigma","html":"σ\\sigmaσ","key":"cpIIbj8yoB"},{"type":"text","value":", see ","key":"L8xNBzU44Q"},{"type":"crossReference","kind":"figure","identifier":"si_expl_variance_energy","label":"si_expl_variance_energy","children":[{"type":"text","value":"Supplementary Figure ","key":"IB0gdDN1lA"},{"type":"text","value":"1","key":"f3A6cjOM1M"}],"template":"Figure %s","enumerator":"1","resolved":true,"html_id":"si-expl-variance-energy","remote":true,"url":"/supplement","dataUrl":"/supplement.json","key":"nMzFxMROnb"},{"type":"text","value":").\n","key":"ZWfMJmtmBU"},{"type":"strong","children":[{"type":"text","value":"C","key":"q4pQSoXQHH"}],"key":"T8Jmmnbwgb"},{"type":"text","value":" We map all states of the state space sample to their corresponding attractor state, with the conventional\nHopfield relaxation procedure (A). The four attractor states are also visualized in their corresponding position on the\n","key":"pB6jFJXVmh"},{"type":"abbreviation","title":"Principal Component","children":[{"type":"text","value":"PC","key":"NPGL1U7i6U"}],"key":"CAdFc23fmM"},{"type":"text","value":"A-based projection. The first two principal components yield a clear separation of the attractive state basins\n(cross-validated classification accuracy: 95.5%, ","key":"T8nqoBnl5j"},{"type":"crossReference","kind":"figure","identifier":"si_classification_acc_state_basins","label":"si_classification_acc_state_basins","children":[{"type":"text","value":"Supplementary Figure ","key":"vmBnYJ3Gwp"},{"type":"text","value":"2","key":"CzQbjHU3V5"}],"template":"Figure %s","enumerator":"2","resolved":true,"html_id":"si-classification-acc-state-basins","remote":true,"url":"/supplement","dataUrl":"/supplement.json","key":"t7r5yIPnRr"},{"type":"text","value":"). We refer to the resulting visualization\nas the ","key":"evxZSvZyfJ"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"HvCBNvBvPM"}],"key":"hFHFX6wJ8d"},{"type":"text","value":" projection and use it to visualize ","key":"Qope6eLJxi"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"T816IJ6jvQ"}],"key":"u0cHuEmHPo"},{"type":"text","value":"-derived and empirical brain dynamics throughout the rest of\nthe manuscript.\n","key":"CsS2N0rNlk"},{"type":"strong","children":[{"type":"text","value":"E","key":"Zvm4IX4Kgq"}],"key":"mnlq6BXVTu"},{"type":"text","value":" At its simplest form, the ","key":"BtlP1nzOcP"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"mx3Cgp9GPf"}],"key":"MxB5GKGYCu"},{"type":"text","value":" framework entails only two free hyperparameters: the temperature parameter\n","key":"voolOGXBW9"},{"type":"inlineMath","value":"\\beta","html":"β\\betaβ","key":"EwYHKF6Bxg"},{"type":"text","value":" (left) that controls the number of attractor states and the noise parameter of the stochastic relaxation\n","key":"JOisyKf9dU"},{"type":"inlineMath","value":"\\sigma","html":"σ\\sigmaσ","key":"RMe4oazQXV"},{"type":"text","value":". To avoid overfitting these parameters to the empirical data, we set ","key":"WiUvD9Q2Ig"},{"type":"inlineMath","value":"\\beta=0.04","html":"β=0.04\\beta=0.04β=0.04","key":"HrPwuczoay"},{"type":"text","value":" and ","key":"otxSPoRdjZ"},{"type":"inlineMath","value":"\\sigma=0.37","html":"σ=0.37\\sigma=0.37σ=0.37","key":"r5ZWkFujZo"},{"type":"text","value":" for the\nrest of the paper (dotted boxes).","key":"qYfVxUvjpD"}],"key":"kqo8i9nSOI"}],"key":"G03pNXk8w9"}],"enumerator":"2","html_id":"attractors","key":"k4TyVlaZvp"},{"type":"paragraph","position":{"start":{"line":221,"column":0},"end":{"line":225,"column":0}},"children":[{"type":"text","value":"Consistent with theoretical expectations, we observed that increasing the temperature parameter ","key":"jqvBB7e0Yy"},{"type":"inlineMath","value":"\\beta","html":"β\\betaβ","key":"hLi70Odd0L"},{"type":"text","value":" led to an\nincreasing number of attractor states (","key":"BpMGdrmcbF"},{"type":"crossReference","kind":"figure","identifier":"attractors","label":"attractors","children":[{"type":"text","value":"Figure ","key":"gI1MePqI5f"},{"type":"text","value":"2","key":"vNvptdrXgm"}],"template":"Figure %s","enumerator":"2","resolved":true,"html_id":"attractors","key":"gOuHfOQvra"},{"type":"text","value":"E, left, ","key":"Tn7G0TVqVr"},{"type":"crossReference","kind":"figure","identifier":"si_att_state_emergence_over_beta","label":"si_att_state_emergence_over_beta","children":[{"type":"text","value":"Supplementary Figure ","key":"HuwJU67n5z"},{"type":"text","value":"3","key":"tcstOba8qj"}],"template":"Figure %s","enumerator":"3","resolved":true,"html_id":"si-att-state-emergence-over-beta","remote":true,"url":"/supplement","dataUrl":"/supplement.json","key":"ttY7zM1Tzl"},{"type":"text","value":"), appearing in symmetric pairs\n(i.e. ","key":"TWn6uhj4K6"},{"type":"inlineMath","value":"a_i^{(1)} = -a_i^{(2)}","html":"ai(1)=−ai(2)a_i^{(1)} = -a_i^{(2)}ai(1)​=−ai(2)​","key":"JTva5tXeVl"},{"type":"text","value":"). For simplicity, we set the temperature parameter for the rest of the paper to a value\nresulting in 4 distinct attractor states (","key":"lxS2M6XoZB"},{"type":"inlineMath","value":"\\beta=0.4","html":"β=0.4\\beta=0.4β=0.4","key":"gQzA78lokT"},{"type":"text","value":").","key":"Dtd9yHUob1"}],"key":"RWSOc0TLap"},{"type":"paragraph","position":{"start":{"line":226,"column":0},"end":{"line":229,"column":0}},"children":[{"type":"text","value":"Fc","key":"GCFQt3FweE"},{"type":"abbreviation","title":"Hopfield Neural Network","children":[{"type":"text","value":"HNN","key":"qNO0K1zrmD"}],"key":"wLroEG6UHq"},{"type":"text","value":"s, without any modifications, always converge to an equilibrium state.\nTo incorporate stochastic fluctuations in neuronal activity ","key":"wyogG7mNn6"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"robinson2005multiscale","identifier":"robinson2005multiscale","children":[{"type":"text","value":"Robinson ","key":"brX6GAcQxX"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"NHOMygbkah"}],"key":"YtJhbuunJG"},{"type":"text","value":", 2005","key":"FxFiQO5xHX"}],"key":"UQYhA5LBv3"}],"key":"RDlgEBobsJ"},{"type":"text","value":", we introduced weak\nGaussian noise to the ","key":"LNpuurgJeU"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"AMZWaCzZho"}],"key":"GsK5j4bXeO"},{"type":"text","value":" relaxation procedure. This procedure, referred to as stochastic relaxation, prevents the system from reaching equilibrium and, somewhat similarly to stochastic DCM ","key":"R8gykTzdw0"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"daunizeau2012stochastic","identifier":"daunizeau2012stochastic","children":[{"type":"text","value":"Daunizeau ","key":"PWaCiCEYtL"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"FZqgZappbk"}],"key":"fmgctwqj4V"},{"type":"text","value":", 2012","key":"xACza4uNLI"}],"key":"Y7PPRTpbf7"}],"key":"QYclBNXzBK"},{"type":"text","value":", induces complex system dynamics (","key":"f1ZeK1Qvs0"},{"type":"crossReference","kind":"figure","identifier":"attractors","label":"attractors","children":[{"type":"text","value":"Figure ","key":"jlBEdmkNzl"},{"type":"text","value":"2","key":"K2jn4AWvjV"}],"template":"Figure %s","enumerator":"2","resolved":true,"html_id":"attractors","key":"mGSpxkTIpM"},{"type":"text","value":"B).","key":"Wl4dhTViie"}],"key":"x2P5SKEcWv"},{"type":"paragraph","position":{"start":{"line":230,"column":0},"end":{"line":236,"column":0}},"children":[{"type":"text","value":"In order to enhance interpretability, we obtained the first two principal components (","key":"sBRUDzsBlM"},{"type":"abbreviation","title":"Principal Component","children":[{"type":"text","value":"PC","key":"xZx0TKO65R"}],"key":"spjJN54HAe"},{"type":"text","value":"s) of the states sampled from the stochastic relaxation procedure.\nThe resulting two-dimensional embedding (","key":"o2ZlVCPHgt"},{"type":"crossReference","kind":"figure","identifier":"attractors","label":"attractors","children":[{"type":"text","value":"Figure ","key":"SqXy8PB78h"},{"type":"text","value":"2","key":"gq1hvLkLQc"}],"template":"Figure %s","enumerator":"2","resolved":true,"html_id":"attractors","key":"bUHvIbsWM2"},{"type":"text","value":"B, bottom plot) exhibited high consistency across different values of ","key":"Tf3yhjWA3p"},{"type":"inlineMath","value":"\\beta","html":"β\\betaβ","key":"nSEA3Vny2K"},{"type":"text","value":" and ","key":"zPmscbpySI"},{"type":"inlineMath","value":"\\sigma","html":"σ\\sigmaσ","key":"ojbBaY4Jjn"},{"type":"text","value":" (","key":"vueQZgELRX"},{"type":"crossReference","kind":"figure","identifier":"attractors","label":"attractors","children":[{"type":"text","value":"Figure ","key":"Nlw1Vtq2tY"},{"type":"text","value":"2","key":"zHOpgaUFnF"}],"template":"Figure %s","enumerator":"2","resolved":true,"html_id":"attractors","key":"gC1js6hsik"},{"type":"text","value":"E).\nFor all subsequent analyses, we set ","key":"tjCu8N72AL"},{"type":"inlineMath","value":"\\sigma=0.37","html":"σ=0.37\\sigma=0.37σ=0.37","key":"dboNgbh0xv"},{"type":"text","value":" (based a coarse optimization procedure aimed at reconstructing the bimodal distribution of empirical data, ","key":"u79IFgzFy3"},{"type":"crossReference","kind":"figure","identifier":"attractors","label":"attractors","children":[{"type":"text","value":"Figure ","key":"xTvvjEWUo7"},{"type":"text","value":"2","key":"dRQXCOXxZ2"}],"template":"Figure %s","enumerator":"2","resolved":true,"html_id":"attractors","key":"gGZ7tXqsle"},{"type":"text","value":"E right). On the low-dimensional embedding, which we refer to as the ","key":"vhBMcJbB9H"},{"type":"emphasis","children":[{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"kJ9tYcbWq1"}],"key":"QMo7h6AJCG"},{"type":"text","value":" projection","key":"Qf2mVtQB3g"}],"key":"EvY7WhlKSe"},{"type":"text","value":", we observed a clear separation of the attractor states (","key":"qVoPcc8U0C"},{"type":"crossReference","kind":"figure","identifier":"attractors","label":"attractors","children":[{"type":"text","value":"Figure ","key":"BvBLtUaop6"},{"type":"text","value":"2","key":"AvQow61W9W"}],"template":"Figure %s","enumerator":"2","resolved":true,"html_id":"attractors","key":"kx9ZLRTfQA"},{"type":"text","value":"C), with the two symmetric pairs of attractor states located at the extremes of the first and second ","key":"N6ZtTXa2L4"},{"type":"abbreviation","title":"Principal Component","children":[{"type":"text","value":"PC","key":"YLu7jqWft1"}],"key":"CoIdJAlQA5"},{"type":"text","value":".\nTo map the attractor basins on the space spanned by the first two ","key":"TRXIqXeOIl"},{"type":"abbreviation","title":"Principal Component","children":[{"type":"text","value":"PC","key":"R9YkCxcAi6"}],"key":"wHE0Z1SOpC"},{"type":"text","value":"s (","key":"a9AZtV7aoe"},{"type":"crossReference","kind":"figure","identifier":"attractors","label":"attractors","children":[{"type":"text","value":"Figure ","key":"TEi19K4FVK"},{"type":"text","value":"2","key":"zHTnAupm20"}],"template":"Figure %s","enumerator":"2","resolved":true,"html_id":"attractors","key":"uLVsxu3lPF"},{"type":"text","value":"C), we obtained the attractor state of each point visited during the stochastic relaxation and fit a multinomial logistic regression model to predict the attractor state from the first two ","key":"I4dq6i0sdO"},{"type":"abbreviation","title":"Principal Component","children":[{"type":"text","value":"PC","key":"G4UgOOsOeZ"}],"key":"EuwywmOkPc"},{"type":"text","value":"s.\nThe resulting model accurately predicted attractor states of arbitrary brain activity patterns, achieving a cross-validated accuracy of 96.5%.\nThe attractor basins were visualized by using the decision boundaries obtained from this model. (","key":"DUdtVJY8nO"},{"type":"crossReference","kind":"figure","identifier":"attractors","label":"attractors","children":[{"type":"text","value":"Figure ","key":"jsAXUIM2rj"},{"type":"text","value":"2","key":"xr1zrJHmtA"}],"template":"Figure %s","enumerator":"2","resolved":true,"html_id":"attractors","key":"NpBwKgwIGu"},{"type":"text","value":"C). We propose the 2-dimensional ","key":"eiyVZBQil0"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"KqxaDTpTOL"}],"key":"XDMVJUvjJ5"},{"type":"text","value":" projection depicted on (","key":"lvYnDTLZCx"},{"type":"crossReference","kind":"figure","identifier":"attractors","label":"attractors","children":[{"type":"text","value":"Figure ","key":"kVPTGe0vJr"},{"type":"text","value":"2","key":"Rz4F8TAPfC"}],"template":"Figure %s","enumerator":"2","resolved":true,"html_id":"attractors","key":"swSlxsne1A"},{"type":"text","value":"C) as a simplified representation of brain dynamics, and use it as a basis for all subsequent analyses in this work.","key":"EtvB2rtqLt"}],"key":"FavrfR8pkL"},{"type":"heading","depth":3,"position":{"start":{"line":237,"column":0},"end":{"line":238,"column":0}},"children":[{"type":"text","value":"Reconstruction of resting state brain dynamics","key":"nxcFwzlVfN"}],"identifier":"reconstruction-of-resting-state-brain-dynamics","label":"Reconstruction of resting state brain dynamics","html_id":"reconstruction-of-resting-state-brain-dynamics","implicit":true,"key":"H3yo0OQj1V"},{"type":"paragraph","position":{"start":{"line":239,"column":0},"end":{"line":241,"column":0}},"children":[{"type":"text","value":"The spatial patterns of the obtained attractor states exhibit high neuroscientific relevance and closely resemble previously described large-scale brain systems. 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The first pair of attractors (mapped on ","key":"uVDpoWjnit"},{"type":"abbreviation","title":"Principal Component","children":[{"type":"text","value":"PC","key":"jN4zjTnkiq"}],"key":"wBxC4vpYgj"},{"type":"text","value":"1, horizontal axis) resemble the two complementary “macro” systems described, among others, by ","key":"i9LpAAqbDw"},{"type":"citeGroup","kind":"narrative","children":[{"type":"cite","label":"golland2008data","identifier":"golland2008data","children":[{"type":"text","value":"Golland ","key":"TyXmHmRBuc"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"OxyKM8i6YH"}],"key":"eLCAmC8rPl"},{"type":"text","value":" (2008)","key":"n10Vn5pZtS"}],"key":"Tywavgdvtm"}],"key":"PVCVMp77mR"},{"type":"text","value":" and ","key":"u9rNnX0y1s"},{"type":"citeGroup","kind":"narrative","children":[{"type":"cite","label":"cioli2014differences","identifier":"cioli2014differences","children":[{"type":"text","value":"Cioli ","key":"cXw765iWL5"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"vdFA4llcvC"}],"key":"oku23ZxadT"},{"type":"text","value":" (2014)","key":"eWoor2ps45"}],"key":"x7jSGpuikk"}],"key":"zcgtCH5ixF"},{"type":"text","value":" as well as the two \"primary\" brain states observed by ","key":"HxvIRs1Uv6"},{"type":"citeGroup","kind":"narrative","children":[{"type":"cite","label":"chen2018human","identifier":"chen2018human","children":[{"type":"text","value":"Chen ","key":"lmzoSj8tWL"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"t7Yqan9Ace"}],"key":"Y6mNE3AAm5"},{"type":"text","value":" (2018)","key":"Ul3HYbepV6"}],"key":"sqVhCSKD7F"}],"key":"zKT8463Kwj"},{"type":"text","value":" and the 'unimodal-to-transmodal' principal gradient of ","key":"n8tXhghDIt"},{"type":"citeGroup","kind":"narrative","children":[{"type":"cite","label":"margulies2016situating","identifier":"margulies2016situating","children":[{"type":"text","value":"Margulies ","key":"FIv2Q8YeJA"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"c1fdZCDq9U"}],"key":"FEi1aHKQLR"},{"type":"text","value":" (2016)","key":"yfPP6TQhxA"}],"key":"EdOCZIUOy7"}],"key":"jnRrKDNAop"},{"type":"text","value":" and ","key":"FD1cLTtell"},{"type":"citeGroup","kind":"narrative","children":[{"type":"cite","label":"huntenburg2018large","identifier":"huntenburg2018large","children":[{"type":"text","value":"Huntenburg ","key":"juhWegdBrS"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"hQEbWnn9g0"}],"key":"ROfSvG5JMS"},{"type":"text","value":" (2018)","key":"gcTS0z3SKW"}],"key":"mPLjpylOyG"}],"key":"u2jioHAMeV"},{"type":"text","value":". A common interpretation of these two patterns is that they represent (i) an “extrinsic” system linked to the immediate sensory environment and (ii) an \"intrinsic\" system for higher-level internal context.\nThe other pair of attractors spans an orthogonal axis, and resemble to patterns commonly associated with perception–action cycles ","key":"JNSItIm3oy"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"fuster2004upper","identifier":"fuster2004upper","children":[{"type":"text","value":"Fuster, 2004","key":"l2tgAhsKME"}],"key":"sOvFLqZj9I"}],"key":"QNpWznaLDq"},{"type":"text","value":", and described as a gradient across sensory-motor modalities ","key":"xOpoER4Yud"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"huntenburg2018large","identifier":"huntenburg2018large","children":[{"type":"text","value":"Huntenburg ","key":"XYWli365uh"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"tlslpD6wqk"}],"key":"BVG4AHpJMW"},{"type":"text","value":", 2018","key":"ZGwYZ1LZH6"}],"key":"kdXYDKUFqz"}],"key":"WWvAZSbgVB"},{"type":"text","value":", recruiting regions associated with active inference (e.g. motor cortices) and perceptual inference (e.g visual areas).","key":"peQNaZJ2kc"}],"key":"piyYMaB7IV"},{"type":"container","kind":"figure","identifier":"rest-validity","label":"rest-validity","children":[{"type":"image","url":"/face_validity-85a51599b8224a0d26a8283f9370e3a0.png","alt":"Connectome-based Hopfield networks reconstruct characteristics of real resting state brain activity.

\nA The four attractor states of the fcHNN model from study 1 reflect brain activation\npatterns with high neuroscientific relevance, representing sub-systems previously associated with \"internal context\"\n(blue), \"external context\" (yellow), \"action\" (red) and \"perception\" (green)\n.\nB The attractor states show excellent replicability in two external datasets (study 2 and 3, mean correlation 0.93).\nC The fcHNN projection (first two PCs of the fcHNN state space) explains significantly more variance (p<0.0001) in the real\nresting state fMRI data than principal components derived from the real resting state data itself and generalizes\nbetter (p<0.0001) to out-of-sample data (study 2). Error bars denote 99% bootstrapped confidence intervals.\nD The fcHNN of study 1 seeded with real activation maps (gray dots) of an example participant. All activation maps converge to one of the four attractor states during the relaxation procedure (without noise) and the system reaches equilibrium. Trajectories are colored by attractor state.\nE Illustration of the stochastic relaxation procedure in the same fcHNN model, seeded from a single starting point (activation pattern). The system does not converge to an attractor state but instead transverses the state space in a way restricted by the topology of the connectome and the \"gravitational pull\" of the attractor states. The shade of the trajectory changes with increasing number of iterations. The trajectory is smoothed with a moving average over 10 iterations for visualization purposes.\nF Real resting state fMRI data of an example participant from study 1, plotted on the fcHNN projection. The shade of the trajectory changes with increasing number of iterations.\nG Flow map of the mean trajectories (i.e. the timeframe-to-timeframe transition direction) in fcHNN-generated data, as compared to a shuffled null model representing zero temporal autocorrelation. The flow map reveals that the \"gravitational pull\" of the attractor states gives rise to a characteristic temporal autocorrelation structure.\nH A similar pattern can be found in real data (flow analysis of all participants from study 1 pooled, as compared to a shuffled null model representing zero temporal autocorrelation).\nI The fcHNN analysis accurately predicts (p<0.0001) the fraction of time spent on the basis of the four attractor\nstates in real restring state fMRI data (study 1) and,\nJ, reconstructs the characteristic bimodal distribution of the real resting state data.\nK Stochastic fcHNNs are capable of self-reconstruction: the timeseries resulting from the stochastic relaxation procedure\nmirror the co-variance structure of the functional connectome the fcHNN model was initialized 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Error bars denote 99% bootstrapped confidence intervals.\n","key":"gCZwfi7loE"},{"type":"strong","children":[{"type":"text","value":"D","key":"GfT3ilBByk"}],"key":"tQLh6VzQUo"},{"type":"text","value":" The ","key":"fuSarX3a0w"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"T4namCpIkf"}],"key":"cvisCQdziQ"},{"type":"text","value":" of study 1 seeded with real activation maps (gray dots) of an example participant. All activation maps converge to one of the four attractor states during the relaxation procedure (without noise) and the system reaches equilibrium. Trajectories are colored by attractor state.\n","key":"ab30fHBujr"},{"type":"strong","children":[{"type":"text","value":"E","key":"AahIw10HaB"}],"key":"tUokTj7Qu0"},{"type":"text","value":" Illustration of the stochastic relaxation procedure in the same ","key":"ZUowudRIby"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"FUN9ryml1E"}],"key":"DN8H6aMbVu"},{"type":"text","value":" model, seeded from a single starting point (activation pattern). The system does not converge to an attractor state but instead transverses the state space in a way restricted by the topology of the connectome and the \"gravitational pull\" of the attractor states. The shade of the trajectory changes with increasing number of iterations. The trajectory is smoothed with a moving average over 10 iterations for visualization purposes.\n","key":"PLHa9pnwzX"},{"type":"strong","children":[{"type":"text","value":"F","key":"bUVLewpBis"}],"key":"l047VlIXrk"},{"type":"text","value":" Real resting state ","key":"AcxsRsHdtD"},{"type":"abbreviation","title":"functional Magnetic Resonance Imaging","children":[{"type":"text","value":"fMRI","key":"IUvF33YGgk"}],"key":"DMgPS9cFA2"},{"type":"text","value":" data of an example participant from study 1, plotted on the ","key":"OWQEiAMSH2"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"wy5UmCDJB6"}],"key":"dm9xgOaClK"},{"type":"text","value":" projection. The shade of the trajectory changes with increasing number of iterations.\n","key":"vDWdySSaOQ"},{"type":"strong","children":[{"type":"text","value":"G","key":"J7hv3jnVOu"}],"key":"EzFdRS4o8V"},{"type":"text","value":" Flow map of the mean trajectories (i.e. the timeframe-to-timeframe transition direction) in ","key":"e9OiaETKRC"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"jZgLtMz8AJ"}],"key":"O5R1MeLSN3"},{"type":"text","value":"-generated data, as compared to a shuffled null model representing zero temporal autocorrelation. The flow map reveals that the \"gravitational pull\" of the attractor states gives rise to a characteristic temporal autocorrelation structure.\n","key":"dEImMgX2yq"},{"type":"strong","children":[{"type":"text","value":"H","key":"QHveHmGAg4"}],"key":"B5daj8pTBc"},{"type":"text","value":" A similar pattern can be found in real data (flow analysis of all participants from study 1 pooled, as compared to a shuffled null model representing zero temporal autocorrelation).\n","key":"GbJmQ7aJPs"},{"type":"strong","children":[{"type":"text","value":"I","key":"DuAQNQoEYT"}],"key":"haEixHAyd7"},{"type":"text","value":" The ","key":"wQqoXZhWM9"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"PwQw1sWR3t"}],"key":"di007kHYST"},{"type":"text","value":" analysis accurately predicts (p<0.0001) the fraction of time spent on the basis of the four attractor\nstates in real restring state ","key":"aSR1WNoUx4"},{"type":"abbreviation","title":"functional Magnetic Resonance Imaging","children":[{"type":"text","value":"fMRI","key":"hUWczfZ0M4"}],"key":"UABxL5lOWm"},{"type":"text","value":" data (study 1) and,\n","key":"UBXUYKIYFw"},{"type":"strong","children":[{"type":"text","value":"J","key":"bztgBaB1t0"}],"key":"tH5ocB2Ss4"},{"type":"text","value":", reconstructs the characteristic bimodal distribution of the real resting state data.\n","key":"ecV9t4zXHw"},{"type":"strong","children":[{"type":"text","value":"K","key":"gUqjeENVvX"}],"key":"rrOq38uKnG"},{"type":"text","value":" Stochastic ","key":"EuSFVON8t3"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"TT7t9FVOc2"}],"key":"DPur6tEEkA"},{"type":"text","value":"s are capable of self-reconstruction: the timeseries resulting from the stochastic relaxation procedure\nmirror the co-variance structure of the functional connectome the ","key":"rB08vUYBoj"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"FlCBnAtK7Q"}],"key":"vWxIgGHDue"},{"type":"text","value":" model was initialized with.","key":"hZDRfIK4Di"}],"key":"o0XCVmuaOc"}],"key":"F2rLI2GMTE"}],"enumerator":"3","html_id":"rest-validity","key":"FSJtszbKTx"},{"type":"paragraph","position":{"start":{"line":265,"column":0},"end":{"line":268,"column":0}},"children":[{"type":"text","value":"The discovered attractor states demonstrate remarkable replicability (mean Pearson's\ncorrelation 0.93) across the discovery dataset (study 1) and two independent replication datasets\n(","key":"mTDQPc2og9"},{"type":"crossReference","children":[{"type":"text","value":"study 2 and 3","key":"zyI7kN2h6K"}],"urlSource":"#tab-samples","identifier":"tab-samples","label":"tab-samples","kind":"table","template":"Table %s","enumerator":"1","resolved":true,"html_id":"tab-samples","remote":true,"url":"/methods","dataUrl":"/methods.json","key":"psf5ozIhYK"},{"type":"text","value":", ","key":"PpvwyBcNG5"},{"type":"crossReference","kind":"figure","identifier":"rest-validity","label":"rest-validity","children":[{"type":"text","value":"Figure ","key":"rHskpwgj1u"},{"type":"text","value":"3","key":"GvLmIIbILq"}],"template":"Figure %s","enumerator":"3","resolved":true,"html_id":"rest-validity","key":"hkrOAg6YJH"},{"type":"text","value":"C). Moreover, they were found to be robust to noise added to the connectome (","key":"kn01PKR123"},{"type":"crossReference","kind":"figure","identifier":"si_noise_robustness_weights","label":"si_noise_robustness_weights","children":[{"type":"text","value":"Supplementary Figure ","key":"X5XvHZrCHm"},{"type":"text","value":"7","key":"NtxiWGe5dX"}],"template":"Figure %s","enumerator":"7","resolved":true,"html_id":"si-noise-robustness-weights","remote":true,"url":"/supplement","dataUrl":"/supplement.json","key":"vKIwACJv5o"},{"type":"text","value":").","key":"b1LlhTo4O7"}],"key":"Ljn8KkF5JZ"},{"type":"paragraph","position":{"start":{"line":269,"column":0},"end":{"line":272,"column":0}},"children":[{"type":"text","value":"Further analysis in study 1 showed that connectome-based Hopfield models accurately reconstructed multiple\ncharacteristics of true resting-state data.\nFirst, the first two components of the ","key":"Qb6ZwO9wi2"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"L2DTuiL07U"}],"key":"sD0DrQ29N0"},{"type":"text","value":" projection accounted for a substantial amount of variance in the real resting-state ","key":"lKap0YmJvh"},{"type":"abbreviation","title":"functional Magnetic Resonance Imaging","children":[{"type":"text","value":"fMRI","key":"dniajtkrug"}],"key":"jHz4VLeGCS"},{"type":"text","value":" data in study 1 (mean ","key":"kf6l1d8YAs"},{"type":"inlineMath","value":"R^2=0.399","html":"R2=0.399R^2=0.399R2=0.399","key":"NI4FPnBjCv"},{"type":"text","value":") and generalized well to out-of-sample data (study 2, mean ","key":"WV4rcbszj9"},{"type":"inlineMath","value":"R^2=0.396","html":"R2=0.396R^2=0.396R2=0.396","key":"A5ifso2rJ3"},{"type":"text","value":") (","key":"o9n7FVQ6lp"},{"type":"crossReference","kind":"figure","identifier":"rest-validity","label":"rest-validity","children":[{"type":"text","value":"Figure ","key":"agwnu4UR5f"},{"type":"text","value":"3","key":"ufSlKKp3BY"}],"template":"Figure %s","enumerator":"3","resolved":true,"html_id":"rest-validity","key":"NvUISuDac5"},{"type":"text","value":"E). Remarkably, the explained variance of the ","key":"n5LLdnIt1A"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"yOvii9Zk5q"}],"key":"FoffUF5qsG"},{"type":"text","value":" projection significantly exceeded that of a ","key":"JAoGKWsJJD"},{"type":"abbreviation","title":"Principal Component","children":[{"type":"text","value":"PC","key":"UURt57UE4U"}],"key":"q10gIAVVRW"},{"type":"text","value":"A performed directly on the real resting-state ","key":"LB94OHQ86e"},{"type":"abbreviation","title":"functional Magnetic Resonance Imaging","children":[{"type":"text","value":"fMRI","key":"sZvnFibi2B"}],"key":"R8ZaKV99m0"},{"type":"text","value":" data itself (","key":"qkRApkLPGy"},{"type":"inlineMath","value":"R^2=0.37","html":"R2=0.37R^2=0.37R2=0.37","key":"Pb4VGZ1Gw9"},{"type":"text","value":" and ","key":"bt5HqfTXuD"},{"type":"inlineMath","value":"0.364","html":"0.3640.3640.364","key":"M5vOaEdwJq"},{"type":"text","value":" for in- and out-of-sample analyses).","key":"f7cYE1OEjP"}],"key":"yyEl6jK0y3"},{"type":"paragraph","position":{"start":{"line":273,"column":0},"end":{"line":275,"column":0}},"children":[{"type":"text","value":"Second, ","key":"ZD4Hz2N1TP"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"EWG2cTCByo"}],"key":"zKRU6f7qzl"},{"type":"text","value":" analyses accurately reconstructed various aspects of true resting state brain dynamics.\nPanel D on ","key":"XH7gEbTuoI"},{"type":"crossReference","kind":"figure","identifier":"rest-validity","label":"rest-validity","children":[{"type":"text","value":"Figure ","key":"rGQ7WULDC6"},{"type":"text","value":"3","key":"cpr3Lyn8QX"}],"template":"Figure %s","enumerator":"3","resolved":true,"html_id":"rest-validity","key":"sgyOByIMwt"},{"type":"text","value":" shows that, with the conventional Hopfield relaxation procedure, individual activation maps converge to one of the four attractor states. When weak noise is introduced to the system (stochastic relaxation, panel E), the system does not converge to an attractor state but the resulting path is still influenced by the attractor states' gravity. The empirical timeseries data exhibits a similar pattern not only visually (panel F), but also when quantifying the average trajectories of flow, as compared to null-models of zero temporal autocorrelation (randomized timeframe order), reflecting the \"gravitational pull\" of attractor states (","key":"KfURXcYxyc"},{"type":"crossReference","kind":"figure","identifier":"rest-validity","label":"rest-validity","children":[{"type":"text","value":"Figure ","key":"v3vxazXS0S"},{"type":"text","value":"3","key":"YAWIQaN30e"}],"template":"Figure %s","enumerator":"3","resolved":true,"html_id":"rest-validity","key":"Ui0Xl9XrLp"},{"type":"text","value":" G and H, see ","key":"ZiznQz4LF0"},{"type":"crossReference","children":[{"type":"text","value":"Methods","key":"GRgMZkEZyG"}],"urlSource":"#evaluation-resting-state-dynamics","identifier":"evaluation-resting-state-dynamics","label":"evaluation-resting-state-dynamics","kind":"heading","template":"{name}","resolved":true,"html_id":"evaluation-resting-state-dynamics","remote":true,"url":"/methods","dataUrl":"/methods.json","key":"Dhe44TqbAw"},{"type":"text","value":" fro analysis details).","key":"WwmnYSM5Rk"}],"key":"pqSPAry0Oy"},{"type":"paragraph","position":{"start":{"line":276,"column":0},"end":{"line":277,"column":0}},"children":[{"type":"text","value":"During stochastic relaxation, the ","key":"I5FFe3vXBI"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"dnYPEdu22q"}],"key":"xN4fwsGrQ2"},{"type":"text","value":" model was found to spend approximately three-quarters of the time on the basis of the first two attractor states and one-quarter on the basis of the second pair of attractor states (approximately equally distributed between pairs). We observed strikingly similar temporal occupancies in the real data ","key":"UTIsgseUKt"},{"type":"crossReference","kind":"figure","identifier":"rest-validity","label":"rest-validity","children":[{"type":"text","value":"Figure ","key":"BNe60ZdwnO"},{"type":"text","value":"3","key":"j985YicbtM"}],"template":"Figure %s","enumerator":"3","resolved":true,"html_id":"rest-validity","key":"orthMmVLvD"},{"type":"text","value":"D), statistically significant with various null models (","key":"Qn4Cn2VtW7"},{"type":"crossReference","kind":"figure","identifier":"si_state_occupancy_null_models","label":"si_state_occupancy_null_models","children":[{"type":"text","value":"Supplementary Figure ","key":"ehxO7gSAnw"},{"type":"text","value":"4","key":"o8wc93MRVV"}],"template":"Figure %s","enumerator":"4","resolved":true,"html_id":"si-state-occupancy-null-models","remote":true,"url":"/supplement","dataUrl":"/supplement.json","key":"CrYNKbuta5"},{"type":"text","value":"). Fine-grained details of the bimodal distribution observed in the real resting-state ","key":"lbVqjrwxGP"},{"type":"abbreviation","title":"functional Magnetic Resonance Imaging","children":[{"type":"text","value":"fMRI","key":"ARm6ACopRg"}],"key":"BUkACIwAq9"},{"type":"text","value":" data were also convincingly reproduced by the ","key":"FFas9bMAKc"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"FDjAzdBUhh"}],"key":"sYkaMP4dpY"},{"type":"text","value":" model (","key":"R2n2v7vzsy"},{"type":"crossReference","kind":"figure","identifier":"rest-validity","label":"rest-validity","children":[{"type":"text","value":"Figure ","key":"opXQjncmdt"},{"type":"text","value":"3","key":"jjpbvQZSSE"}],"template":"Figure %s","enumerator":"3","resolved":true,"html_id":"rest-validity","key":"vm7QDR2xT6"},{"type":"text","value":"F and ","key":"gHm1O5ZN1j"},{"type":"crossReference","kind":"figure","identifier":"attractors","label":"attractors","children":[{"type":"text","value":"Figure ","key":"faNQ1UIT7C"},{"type":"text","value":"2","key":"vDQq2BC4Or"}],"template":"Figure %s","enumerator":"2","resolved":true,"html_id":"attractors","key":"ho8WLkkWyr"},{"type":"text","value":"E).","key":"QlanIVYJnx"}],"key":"AAqRjRu66E"},{"type":"paragraph","position":{"start":{"line":278,"column":0},"end":{"line":279,"column":0}},"children":[{"type":"text","value":"Finally, ","key":"uDd2Ieh2NL"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"QaUSxDnMjx"}],"key":"I0pY0PC45T"},{"type":"text","value":"s were found to generate signal that preserves the covariance structure of the real functional connectome, indicating that dynamic systems of this type (including the brain) inevitably \"leak\" their underlying structure into the activity time series, strengthening the construct validity of our approach (","key":"AQlpd1GDCL"},{"type":"crossReference","kind":"figure","identifier":"rest-validity","label":"rest-validity","children":[{"type":"text","value":"Figure ","key":"QpMzxpjmLM"},{"type":"text","value":"3","key":"ku4pEgh4og"}],"template":"Figure %s","enumerator":"3","resolved":true,"html_id":"rest-validity","key":"LSzuHC4FVx"},{"type":"text","value":"D).","key":"uFhDEL7OFX"}],"key":"P3jfTDocIh"},{"type":"heading","depth":3,"position":{"start":{"line":280,"column":0},"end":{"line":281,"column":0}},"children":[{"type":"text","value":"An explanatory framework for task-based brain activity","key":"ciJm7pbLYv"}],"identifier":"an-explanatory-framework-for-task-based-brain-activity","label":"An explanatory framework for task-based brain activity","html_id":"an-explanatory-framework-for-task-based-brain-activity","implicit":true,"key":"H0VFB0SzYF"},{"type":"paragraph","position":{"start":{"line":282,"column":0},"end":{"line":283,"column":0}},"children":[{"type":"text","value":"Next to reproducing various characteristics of spontaneous brain dynamics, ","key":"KKg6WO9ESk"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"OpOkGfpWwz"}],"key":"SauYhBLFRc"},{"type":"text","value":"s can also be used to model responses to various perturbations. We obtained task-based ","key":"CDVQS2phUy"},{"type":"abbreviation","title":"functional Magnetic Resonance Imaging","children":[{"type":"text","value":"fMRI","key":"ZmhCExAHkA"}],"key":"i5GqaY1D7Q"},{"type":"text","value":" data from a study by ","key":"PYyvIUTxJg"},{"type":"citeGroup","kind":"narrative","children":[{"type":"cite","label":"woo2015distinct","identifier":"woo2015distinct","children":[{"type":"text","value":"Woo ","key":"iwLNVrWx35"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"WB4vxUfaha"}],"key":"UgVLU104s0"},{"type":"text","value":" (2015)","key":"ILcVamhjnQ"}],"key":"Jaq3LqY8an"}],"key":"n8Hb7N8zVZ"},{"type":"text","value":" (","key":"dJan5M1y4R"},{"type":"crossReference","children":[{"type":"text","value":"study 4","key":"yauA2AGr5N"}],"urlSource":"#tab-samples","identifier":"tab-samples","label":"tab-samples","kind":"table","template":"Table %s","enumerator":"1","resolved":true,"html_id":"tab-samples","remote":true,"url":"/methods","dataUrl":"/methods.json","key":"IFWHm1C6X3"},{"type":"text","value":", n=33, see ","key":"XjlT6MPVVx"},{"type":"crossReference","kind":"figure","identifier":"rest-validity","label":"rest-validity","children":[{"type":"text","value":"Figure ","key":"iWGeHJitDc"},{"type":"text","value":"3","key":"aHEOdTgc61"}],"template":"Figure %s","enumerator":"3","resolved":true,"html_id":"rest-validity","key":"pHxcsDUMXu"},{"type":"text","value":"), investigating the neural correlates of pain and its self-regulation.","key":"PoeZsLTCmp"}],"key":"OW1FneBWYQ"},{"type":"paragraph","position":{"start":{"line":284,"column":0},"end":{"line":285,"column":0}},"children":[{"type":"text","value":"We found that activity changes due to pain (taking into account hemodynamics, see ","key":"X7Gey9ya47"},{"type":"crossReference","children":[{"type":"text","value":"Methods","key":"htN8W1s8Z1"}],"urlSource":"#evaluation-task-based-dynamics","identifier":"evaluation-task-based-dynamics","label":"evaluation-task-based-dynamics","kind":"heading","template":"{name}","resolved":true,"html_id":"evaluation-task-based-dynamics","remote":true,"url":"/methods","dataUrl":"/methods.json","key":"ro2iU4KW5b"},{"type":"text","value":") were characterized on the ","key":"x8FG5txNEP"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"aeNiOiypx6"}],"key":"FEkN8qQC3f"},{"type":"text","value":" projection by a shift towards the attractor state of action/execution (permutation test for mean projection difference by randomly swapping conditions, p<0.001, ","key":"NgluQzIz8P"},{"type":"crossReference","kind":"figure","identifier":"task-validity","label":"task-validity","children":[{"type":"text","value":"Figure ","key":"I3ZKbGyjrx"},{"type":"text","value":"4","key":"B88IHUUqH8"}],"template":"Figure %s","enumerator":"4","resolved":true,"html_id":"task-validity","key":"Oo8uU8lXpB"},{"type":"text","value":"A, left). Energies, as defined by the ","key":"TloHykTuml"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"COlgJ47W3J"}],"key":"aUEE6nFvUj"},{"type":"text","value":", were also significantly different between the two conditions (p<0.001), with higher energies during pain stimulation.","key":"t5qVAd9TyM"}],"key":"lh68PO6Fvq"},{"type":"paragraph","position":{"start":{"line":286,"column":0},"end":{"line":287,"column":0}},"children":[{"type":"text","value":"When participants were instructed to up- or downregulate their pain sensation (resulting in increased and decreased pain reports and differential brain activity in the nucleus accumbens, NAc (see ","key":"be1hf3id39"},{"type":"cite","label":"woo2015distinct","identifier":"woo2015distinct","children":[{"type":"text","value":"Woo ","key":"AGPTfLW3TZ"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"nodPPFga01"}],"key":"S8lZHIQmcL"},{"type":"text","value":", 2015","key":"IuOcWP5gZ3"}],"key":"Rg2K0VSpVN"},{"type":"text","value":" for details), we observed further changes of the location of momentary brain activity patterns on the ","key":"AAyO3tHMDY"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"R5er7tEeAM"}],"key":"ZAPLdVzm6o"},{"type":"text","value":" projection (p<0.001, ","key":"FoMqoIG2Jc"},{"type":"crossReference","kind":"figure","identifier":"task-validity","label":"task-validity","children":[{"type":"text","value":"Figure ","key":"upySQPvytZ"},{"type":"text","value":"4","key":"LTK4mWksjQ"}],"template":"Figure %s","enumerator":"4","resolved":true,"html_id":"task-validity","key":"U7UAN6cWku"},{"type":"text","value":"A, right), with down-regulation pulling brain dynamics towards the attractor state of internal context and perception. Interestingly, self-regulation did not trigger significant energy changes (p=0.36).","key":"GhKa2j7z3f"}],"key":"qX5B2YTjH3"},{"type":"container","kind":"figure","identifier":"task-validity","label":"task-validity","children":[{"type":"image","url":"/task_validity-e7bb5a8057fc47eba1be0aee743c3783.png","alt":"Empirical Hopfield-networks reconstruct real task-based brain activity.

\nA Functional MRI time-frames during pain stimulation from study 4 (second fcHNN projection plot)\nand self-regulation (third and fourth) are distributed differently on the fcHNN projection than brain states\nduring rest (first projection, permutation test, p<0.001 for all). Energies, as defined by the Hopfield model, are also\nsignificantly different between rest and the pain conditions (permutation test, p<0.001), with higher energies during\npain stimulation. Triangles denote participant-level mean activations in the various blocks (corrected for\nhemodynamics). Small circle plots show the directions of the change for each individual (points) as well as the mean direction\nacross participants (arrow), as compared to the reference state (downregulation for the last circle plot, rest for all\nother circle plots).\nB Flow-analysis (difference in the average timeframe-to-timeframe transition direction) reveals a non-linear difference in brain dynamics during pain and rest (left). When introducing weak pain-related signal in the fcHNN model during stochastic relaxation, it accurately reproduces these non-linear flow differences (right).\nC Simulating activity in the Nucleus Accumbens (NAc) (the region showing significant activity differences in ) reconstructs the observed non-linear flow difference between up- and downregulation (left).\nD Schematic representation of brain dynamics during pain and its up- and downregulation, visualized on the fcHNN projection. In the proposed framework, pain does not simply elicit a direct response in certain regions, but instead, shifts spontaneous brain dynamics towards the \"action\" attractor, converging to a characteristic \"ghost attractor\" of pain. Down-regulation by NAc activation exerts force towards the attractor of internal context, leading to the brain less frequent \"visiting\" pain-associated states.\nE Visualizing meta-analytic activation maps (see Supplementary Table %s for details) on the fcHNN projection captures intimate relations between the corresponding tasks and F serves as a basis for a fcHNN-based theoretical interpretative framework for spontaneous and task-based brain dynamics. In the proposed framework, task-based activity is not a mere response to external stimuli in certain brain locations but a perturbation of the brain's characteristic dynamic trajectories, constrained by the underlying functional connectivity. From this perspective, \"activity maps\" from conventional task-based fMRI analyses capture time-averaged differences in these whole brain dynamics.","urlSource":"figures/task_validity.png","key":"OYMQBtMdHo"},{"type":"caption","children":[{"type":"paragraph","position":{"start":{"line":290,"column":0},"end":{"line":303,"column":0}},"children":[{"type":"captionNumber","kind":"figure","label":"task-validity","identifier":"task-validity","html_id":"task-validity","enumerator":"4","children":[{"type":"text","value":"Figure ","key":"QhAC5tNSLb"},{"type":"text","value":"4","key":"rQ30CrXWjc"},{"type":"text","value":":","key":"xXLYbUp4y4"}],"template":"Figure %s:","key":"tqbwIKo6Ef"},{"type":"strong","children":[{"type":"text","value":"Empirical Hopfield-networks reconstruct real task-based brain activity.","key":"a4ZJD7URKe"}],"key":"LPiedxkUja"},{"type":"text","value":" ","key":"gYC3zBBr7A"},{"type":"break","key":"UpzjeykLOx"},{"type":"text","value":"\n","key":"j8J1X62P2j"},{"type":"strong","children":[{"type":"text","value":"A","key":"i6pUza3u9L"}],"key":"vk2z6ZWy9a"},{"type":"text","value":" Functional MRI time-frames during pain stimulation from ","key":"yrXhc9lXWN"},{"type":"crossReference","children":[{"type":"text","value":"study 4","key":"w2aJRwr7DD"}],"urlSource":"#tab-samples","identifier":"tab-samples","label":"tab-samples","kind":"table","template":"Table %s","enumerator":"1","resolved":true,"html_id":"tab-samples","remote":true,"url":"/methods","dataUrl":"/methods.json","key":"zbWEDRlOR6"},{"type":"text","value":" (second ","key":"xPSKK5xrPB"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"gQuZ6S76H4"}],"key":"ASiDNUwFmo"},{"type":"text","value":" projection plot)\nand self-regulation (third and fourth) are distributed differently on the ","key":"e8ADTLzuvP"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"NWS04Ffm5L"}],"key":"sF5pTjjGA0"},{"type":"text","value":" projection than brain states\nduring rest (first projection, permutation test, p<0.001 for all). Energies, as defined by the Hopfield model, are also\nsignificantly different between rest and the pain conditions (permutation test, p<0.001), with higher energies during\npain stimulation. Triangles denote participant-level mean activations in the various blocks (corrected for\nhemodynamics). Small circle plots show the directions of the change for each individual (points) as well as the mean direction\nacross participants (arrow), as compared to the reference state (downregulation for the last circle plot, rest for all\nother circle plots).\n","key":"UeFCVfxZMp"},{"type":"strong","children":[{"type":"text","value":"B","key":"r0XMpnrMx8"}],"key":"QazzlSiROl"},{"type":"text","value":" Flow-analysis (difference in the average timeframe-to-timeframe transition direction) reveals a non-linear difference in brain dynamics during pain and rest (left). When introducing weak pain-related signal in the ","key":"rVxpzCdTXC"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"SJ8HSGWFwD"}],"key":"fvVlzBg516"},{"type":"text","value":" model during stochastic relaxation, it accurately reproduces these non-linear flow differences (right).\n","key":"DYCXxzxGGx"},{"type":"strong","children":[{"type":"text","value":"C","key":"fyYXXhmsRF"}],"key":"DRKX5L7f12"},{"type":"text","value":" Simulating activity in the Nucleus Accumbens (NAc) (the region showing significant activity differences in ","key":"WZjwuUejSV"},{"type":"cite","label":"woo2015distinct","identifier":"woo2015distinct","children":[{"type":"text","value":"Woo ","key":"c9gpKFKLZ3"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"yFpgfvvtJD"}],"key":"bl8MZI14di"},{"type":"text","value":", 2015","key":"UuTliWEAK9"}],"key":"rr7xRV9ALi"},{"type":"text","value":") reconstructs the observed non-linear flow difference between up- and downregulation (left).\n","key":"Wm6oR6XpCR"},{"type":"strong","children":[{"type":"text","value":"D","key":"JsFBElifq0"}],"key":"tMydJ5iKqa"},{"type":"text","value":" Schematic representation of brain dynamics during pain and its up- and downregulation, visualized on the ","key":"qWBy325las"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"yH8mHJd7pB"}],"key":"mMKNruvl71"},{"type":"text","value":" projection. In the proposed framework, pain does not simply elicit a direct response in certain regions, but instead, shifts spontaneous brain dynamics towards the \"action\" attractor, converging to a characteristic \"ghost attractor\" of pain. Down-regulation by NAc activation exerts force towards the attractor of internal context, leading to the brain less frequent \"visiting\" pain-associated states.\n","key":"AAfNq0dwxQ"},{"type":"strong","children":[{"type":"text","value":"E","key":"iezJoTegyq"}],"key":"LK8Jsg8JBE"},{"type":"text","value":" Visualizing meta-analytic activation maps (see ","key":"jNwnxBKeip"},{"type":"crossReference","kind":"table","identifier":"si-tab-neurosynth","label":"si-tab-neurosynth","children":[{"type":"text","value":"Supplementary Table ","key":"FJQx4tcORu"},{"type":"text","value":"1","key":"JznWHS47gy"}],"template":"Table %s","enumerator":"1","resolved":true,"html_id":"si-tab-neurosynth","remote":true,"url":"/supplement","dataUrl":"/supplement.json","key":"piJf4qwNNt"},{"type":"text","value":" for details) on the ","key":"IOKA4PQfUo"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"sUDU3vUw5W"}],"key":"sp3XxzvwMY"},{"type":"text","value":" projection captures intimate relations between the corresponding tasks and ","key":"VtddsLpeJK"},{"type":"strong","children":[{"type":"text","value":"F","key":"ZOrh4dgVZg"}],"key":"ZprOi75FMH"},{"type":"text","value":" serves as a basis for a ","key":"snmCs9YJu2"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"IWjolzGZQu"}],"key":"ztRoAOHk0u"},{"type":"text","value":"-based theoretical interpretative framework for spontaneous and task-based brain dynamics. In the proposed framework, task-based activity is not a mere response to external stimuli in certain brain locations but a perturbation of the brain's characteristic dynamic trajectories, constrained by the underlying functional connectivity. From this perspective, \"activity maps\" from conventional task-based ","key":"Wtm52cvRmS"},{"type":"abbreviation","title":"functional Magnetic Resonance Imaging","children":[{"type":"text","value":"fMRI","key":"JxTk95GPkw"}],"key":"zSNDEG6HGC"},{"type":"text","value":" analyses capture time-averaged differences in these whole brain dynamics.","key":"ndcXuXerHo"}],"key":"BrQATvhYg6"}],"key":"A55FDEAZ3W"}],"enumerator":"4","html_id":"task-validity","key":"WePzqBn5Us"},{"type":"paragraph","position":{"start":{"line":305,"column":0},"end":{"line":307,"column":0}},"children":[{"type":"text","value":"Next, we conducted a \"flow analysis\" on the ","key":"iZw3kgXwCI"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"NqO3EWcB6a"}],"key":"vuFl3SKH5p"},{"type":"text","value":" projection, quantifying how the average timeframe-to-timeframe transition direction differs on the ","key":"itabIYN6Xi"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"XdEAMaEjO4"}],"key":"yD33mz8Xqt"},{"type":"text","value":" projection between conditions (see ","key":"Zt9Ars2pwH"},{"type":"crossReference","children":[{"type":"text","value":"Methods","key":"UlyIPeO8Mr"}],"urlSource":"#evaluation-task-based-dynamics","identifier":"evaluation-task-based-dynamics","label":"evaluation-task-based-dynamics","kind":"heading","template":"{name}","resolved":true,"html_id":"evaluation-task-based-dynamics","remote":true,"url":"/methods","dataUrl":"/methods.json","key":"M4ewIqjVkz"},{"type":"text","value":").\nThis analysis unveiled that during pain (","key":"NxWbCD4pLF"},{"type":"crossReference","kind":"figure","identifier":"task-validity","label":"task-validity","children":[{"type":"text","value":"Figure ","key":"xCsU8BVZCV"},{"type":"text","value":"4","key":"GNsvDIXRsM"}],"template":"Figure %s","enumerator":"4","resolved":true,"html_id":"task-validity","key":"IwSPNBm18t"},{"type":"text","value":"B, left side), brain activity tends to gravitate towards a distinct point on the projection on the boundary the basins of the internal and action attractors, which we term the \"ghost attractor\" of pain (similar to ","key":"T2qdFxiof4"},{"type":"cite","label":"vohryzek2020ghost","identifier":"vohryzek2020ghost","children":[{"type":"text","value":"Vohryzek ","key":"rkqZ0ZQ8Yl"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"Vi63qaYa0l"}],"key":"PSQ3EJpkZw"},{"type":"text","value":", 2020","key":"Uq55BrbCrt"}],"key":"n1pP141r9Z"},{"type":"text","value":"). In case of downregulation (as compared to upregulation), brain activity is pulled away from the pain-related \"ghost attractor\" (","key":"Eb8VThK5vz"},{"type":"crossReference","kind":"figure","identifier":"task-validity","label":"task-validity","children":[{"type":"text","value":"Figure ","key":"G8kPiZnQov"},{"type":"text","value":"4","key":"HAvs4oXCna"}],"template":"Figure %s","enumerator":"4","resolved":true,"html_id":"task-validity","key":"VVIU689hqo"},{"type":"text","value":"C, left side), towards the attractor of internal context.","key":"a1T0lMVUkb"}],"key":"nizrtgNyLa"},{"type":"paragraph","position":{"start":{"line":308,"column":0},"end":{"line":309,"column":0}},"children":[{"type":"text","value":"Our ","key":"k9lh7oJg26"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"QfoJeKvvCk"}],"key":"oTAPcgN4aS"},{"type":"text","value":" was able to accurately reconstruct these non-linear dynamics by adding a small amount of realistic \"control signal\" (similarly to network control theory ","key":"WQqTr9Oycw"},{"type":"citeGroup","kind":"narrative","children":[{"type":"cite","label":"liu2011controllability","identifier":"liu2011controllability","children":[{"type":"text","value":"Liu ","key":"qaVypkaQQv"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"Kqpf1Pzlgr"}],"key":"FBEPeiPjlD"},{"type":"text","value":" (2011)","key":"YcfysNCbWv"}],"key":"Dn5Z35VwmS"},{"type":"cite","label":"gu2015controllability","identifier":"gu2015controllability","children":[{"type":"text","value":"Gu ","key":"OgqMTnPeJj"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"w9848jpm2O"}],"key":"Y69MllG9IJ"},{"type":"text","value":" (2015)","key":"YkuCSpkYxf"}],"key":"d5svu7FMC9"}],"key":"E3zPNbqT6V"},{"type":"text","value":"). To simulate the alterations in brain dynamics during pain stimulation, we acquired a meta-analytic pain activation map ","key":"KiHoWJfRm9"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"zunhammer2021meta","identifier":"zunhammer2021meta","children":[{"type":"text","value":"Zunhammer ","key":"S7fPN9UHpJ"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"wzWyvPGGHx"}],"key":"XKx81LjS4J"},{"type":"text","value":", 2021","key":"lfZCnAUbeh"}],"key":"ma5tM6FUeU"}],"key":"dA1n8HgiZV"},{"type":"text","value":" (n=603) and incorporated it as a control signal added to each iteration of the stochastic relaxation procedure. The ghost attractor found in the empirical data was present across a relatively wide range of signal-to-noise (SNR) values (","key":"qwHMwNx2YW"},{"type":"crossReference","kind":"figure","identifier":"si_pain_ghost_attractor_sim","label":"si_pain_ghost_attractor_sim","children":[{"type":"text","value":"Supplementary Figure ","key":"OUbrxMj9hV"},{"type":"text","value":"5","key":"BgEm6a7ie8"}],"template":"Figure %s","enumerator":"5","resolved":true,"html_id":"si-pain-ghost-attractor-sim","remote":true,"url":"/supplement","dataUrl":"/supplement.json","key":"DMzRerpjh3"},{"type":"text","value":"). Results with SNR=0.005 are presented on ","key":"hgcFYcc44Y"},{"type":"crossReference","kind":"figure","identifier":"task-validity","label":"task-validity","children":[{"type":"text","value":"Figure ","key":"A1UIbO0fdj"},{"type":"text","value":"4","key":"oamGoDKNBH"}],"template":"Figure %s","enumerator":"4","resolved":true,"html_id":"task-validity","key":"itXnvL19x0"},{"type":"text","value":"B, right side (Pearson's r = 0.46, p=0.005 based on randomizing conditions on a per-participant basis).","key":"kbmLWMrptf"}],"key":"ADevB8g600"},{"type":"paragraph","position":{"start":{"line":310,"column":0},"end":{"line":313,"column":0}},"children":[{"type":"text","value":"The same model was also able to reconstruct the observed non-linear differences in brain dynamics between the up- and downregulation conditions (Pearson's r = 0.62, p=0.023) without any further optimization (SNR=0.005,\n","key":"MTeWSDVepu"},{"type":"crossReference","kind":"figure","identifier":"task-validity","label":"task-validity","children":[{"type":"text","value":"Figure ","key":"rkDuso4gdE"},{"type":"text","value":"4","key":"Zjr9Ny1BSU"}],"template":"Figure %s","enumerator":"4","resolved":true,"html_id":"task-validity","key":"NehWKrEYhY"},{"type":"text","value":"C, right side). The only change we made to the model was the addition (downregulation) or\nsubtraction (upregulation) of control signal in the NAc (the region in which ","key":"IPpu4ErBi1"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"woo2015distinct","identifier":"woo2015distinct","children":[{"type":"text","value":"Woo ","key":"pbpUxMRdUV"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"o9KaXemqX2"}],"key":"D1T0xJvvIq"},{"type":"text","value":", 2015","key":"J23C2l8IJF"}],"key":"bqDbFDenDa"}],"key":"kH4fnM0hoF"},{"type":"text","value":" observed significant changes between up- and downregulation), introducing a signal difference of ","key":"iGT0OcBgU5"},{"type":"inlineMath","value":"\\Delta","html":"Δ\\DeltaΔ","key":"CD4z2tGy94"},{"type":"text","value":"SNR=0.005 (the same value we found optimal in the pain-analysis). Results were reproducible with lower NAc SNRs, too (","key":"nQijJuzfYG"},{"type":"crossReference","kind":"figure","identifier":"si_downreg_trajectory_sim","label":"si_downreg_trajectory_sim","children":[{"type":"text","value":"Supplementary Figure ","key":"eWcYFGO4j5"},{"type":"text","value":"6","key":"tqXPp8vzUh"}],"template":"Figure %s","enumerator":"6","resolved":true,"html_id":"si-downreg-trajectory-sim","remote":true,"url":"/supplement","dataUrl":"/supplement.json","key":"nkvpbWHj6l"},{"type":"text","value":").","key":"ZtlLK73YjV"}],"key":"bUis4oGa73"},{"type":"paragraph","position":{"start":{"line":314,"column":0},"end":{"line":316,"column":0}},"children":[{"type":"text","value":"To provide a comprehensive picture on how tasks and stimuli other then pain map onto the ","key":"Qu5HvlSBPT"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"cMqvEdcdsG"}],"key":"YIPL5Sydw5"},{"type":"text","value":" projection, we obtained various task-based meta-analytic activation maps from Neurosynth (see ","key":"GTwl553uCa"},{"type":"crossReference","children":[{"type":"text","value":"Methods","key":"PvE0JMdcjK"}],"urlSource":"#evaluation-task-based-dynamics","identifier":"evaluation-task-based-dynamics","label":"evaluation-task-based-dynamics","kind":"heading","template":"{name}","resolved":true,"html_id":"evaluation-task-based-dynamics","remote":true,"url":"/methods","dataUrl":"/methods.json","key":"f8YGN5QRsC"},{"type":"text","value":") and plotted them on the ","key":"vYZAPKaBhe"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"NbjXt0nE81"}],"key":"b3rTJwQKj9"},{"type":"text","value":" projection (","key":"w74tBkdsLW"},{"type":"crossReference","kind":"figure","identifier":"task-validity","label":"task-validity","children":[{"type":"text","value":"Figure ","key":"e4LX3JaEyZ"},{"type":"text","value":"4","key":"DT487Bdkw2"}],"template":"Figure %s","enumerator":"4","resolved":true,"html_id":"task-validity","key":"eKthD34PvO"},{"type":"text","value":"E). This analysis reinforced and extended our interpretation of the four investigated attractor states and shed more light on how various functions are mapped on the axes of internal vs. external context and perception vs. action.\nIn the coordinate system of the ","key":"jxDMrbXLPE"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"VFJtsgMg8Z"}],"key":"rwik0XMoen"},{"type":"text","value":" projection, visual processing is labeled \"external-perception\", sensory-motor processes \"external-active\", language, verbal cognition and working memory is labelled \"internal-active\" and long-term memory as well as social and autobiographic schemata fall into the \"internal-perception\" regime (","key":"U6e5oZCqL7"},{"type":"crossReference","kind":"figure","identifier":"task-validity","label":"task-validity","children":[{"type":"text","value":"Figure ","key":"bangmh1P1c"},{"type":"text","value":"4","key":"N6uloqamNI"}],"template":"Figure %s","enumerator":"4","resolved":true,"html_id":"task-validity","key":"gJvnOHwHrC"},{"type":"text","value":"F).","key":"jRc1qJLrvg"}],"key":"RBwd6thtC3"},{"type":"heading","depth":3,"position":{"start":{"line":317,"column":0},"end":{"line":318,"column":0}},"children":[{"type":"text","value":"Clinical relevance","key":"i4v7hze2FQ"}],"identifier":"clinical-relevance","label":"Clinical relevance","html_id":"clinical-relevance","implicit":true,"key":"pXjaB0gBdg"},{"type":"paragraph","position":{"start":{"line":319,"column":0},"end":{"line":323,"column":0}},"children":[{"type":"text","value":"We obtained data from n=172 autism spectrum disorder (","key":"HfeV6VzAyl"},{"type":"abbreviation","title":"Autism Spectrum Disorder","children":[{"type":"text","value":"ASD","key":"cpNtceur95"}],"key":"Q2Bcv03RTN"},{"type":"text","value":") and typically developing control (TDC) individuals, acquired at the New York University Langone Medical Center, New York, NY, USA (NYU) and generously shared in the Autism Brain Imaging Data Exchange dataset (","key":"VH3rWbSW51"},{"type":"crossReference","children":[{"type":"text","value":"study 7","key":"OwX7Rzw8wc"}],"urlSource":"#tab-samples","identifier":"tab-samples","label":"tab-samples","kind":"table","template":"Table %s","enumerator":"1","resolved":true,"html_id":"tab-samples","remote":true,"url":"/methods","dataUrl":"/methods.json","key":"qkHqTLDcHG"},{"type":"text","value":": ","key":"xhp0peYtgJ"},{"type":"abbreviation","title":"Autism Brain Imaging Data Exchange","children":[{"type":"text","value":"ABIDE","key":"GYaDluMrQK"}],"key":"Uaofr8v7pd"},{"type":"text","value":", ","key":"X3KNwXWxgQ"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"di2014autism","identifier":"di2014autism","children":[{"type":"text","value":"Di Martino ","key":"KGZHXu4sxZ"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"elS2LKv1Ou"}],"key":"laVKK6jhB5"},{"type":"text","value":", 2014","key":"mOwwYhbxy4"}],"key":"JAQkFsx4XC"}],"key":"PJkgjAMb8x"},{"type":"text","value":".\nAfter excluding high-motion cases (see ","key":"Dk3padDT2q"},{"type":"crossReference","children":[{"type":"text","value":"Methods","key":"cFbX7fosYE"}],"urlSource":"#clinical-data","identifier":"clinical-data","label":"clinical-data","kind":"heading","template":"{name}","resolved":true,"html_id":"clinical-data","remote":true,"url":"/methods","dataUrl":"/methods.json","key":"RgvPSDafIz"},{"type":"text","value":"), we visualized the distribution of time-frames on the ","key":"YukdbBNi9F"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"jl5XfEc1lS"}],"key":"MwX6hs24ND"},{"type":"text","value":"-projection separately for the ","key":"rRKPgCmhO5"},{"type":"abbreviation","title":"Autism Spectrum Disorder","children":[{"type":"text","value":"ASD","key":"qqpZDBLuNx"}],"key":"uiOFXRwEaQ"},{"type":"text","value":" and TDC groups (","key":"CM3QR1r8Cs"},{"type":"crossReference","kind":"figure","identifier":"clinical-validity","label":"clinical-validity","children":[{"type":"text","value":"Figure ","key":"c2CTMkkQiW"},{"type":"text","value":"5","key":"wXxe5tn9zt"}],"template":"Figure %s","enumerator":"5","resolved":true,"html_id":"clinical-validity","key":"iBlXwNtPtQ"},{"type":"text","value":"A).\nFirst, we assigned all timeframes to one of the 4 attractor states with the ","key":"qQzNle5O9u"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"thGkhpcGLe"}],"key":"FaOkDxw1H6"},{"type":"text","value":" from study 1 and found several significant differences in the mean activity on the attractor basins (see ","key":"z7sNqnzoDK"},{"type":"crossReference","children":[{"type":"text","value":"Methods","key":"Bl9XDgzVR0"}],"urlSource":"#clinical-data","identifier":"clinical-data","label":"clinical-data","kind":"heading","template":"{name}","resolved":true,"html_id":"clinical-data","remote":true,"url":"/methods","dataUrl":"/methods.json","key":"ecas6oEJlB"},{"type":"text","value":") of the ","key":"gkU495S9By"},{"type":"abbreviation","title":"Autism Spectrum Disorder","children":[{"type":"text","value":"ASD","key":"XEvfkfzx8O"}],"key":"TwfkwszVho"},{"type":"text","value":" group as compared to the respective controls (","key":"X2BNxveku8"},{"type":"crossReference","kind":"figure","identifier":"clinical-validity","label":"clinical-validity","children":[{"type":"text","value":"Figure ","key":"ustgF2niVR"},{"type":"text","value":"5","key":"ugSa3pqxo6"}],"template":"Figure %s","enumerator":"5","resolved":true,"html_id":"clinical-validity","key":"Is2dg4EY3x"},{"type":"text","value":"B).\nStrongest differences were found on the \"action-perception\" axis (","key":"Y8o9FgdFD0"},{"type":"crossReference","kind":"table","identifier":"tab-clinical-results","label":"tab-clinical-results","children":[{"type":"text","value":"Table ","key":"trqimyKDks"},{"type":"text","value":"1","key":"eSYBDy4djc"}],"template":"Table %s","enumerator":"1","resolved":true,"html_id":"tab-clinical-results","key":"IQMEoh0zYv"},{"type":"text","value":"), with increased activity of the sensory-motor and mid","key":"tfa0ljirpr"},{"type":"abbreviation","title":"dorsolateral","children":[{"type":"text","value":"dl","key":"ZfUVLKJQ3K"}],"key":"fzVozEPswz"},{"type":"text","value":"e cingular cortices during \"action-execution\" related states and increased visual and decreased sensory and auditory activity during \"perception\" states, likely reflecting the widely acknowledged, yet poorly understood, perceptual atypicalities in ","key":"uBpxn9ZcN2"},{"type":"abbreviation","title":"Autism Spectrum Disorder","children":[{"type":"text","value":"ASD","key":"ZNZFemUaRj"}],"key":"fDhV5nsiQM"},{"type":"text","value":" ","key":"WnOqnE11uU"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"hadad2019perception","identifier":"hadad2019perception","children":[{"type":"text","value":"Hadad & Schwartz, 2019","key":"gWZTkd4Ltv"}],"key":"cPNKIQdGrX"}],"key":"L4r3dBEDnD"},{"type":"text","value":". ","key":"uplJQEdZMh"},{"type":"abbreviation","title":"Autism Spectrum Disorder","children":[{"type":"text","value":"ASD","key":"EayBBcEXeq"}],"key":"UG22v0BW3r"},{"type":"text","value":" related changes in the internal-external axis were characterized by more involvement of the posterior cingulate, the precuneus, the nucleus accumbens, the dorsolateral prefrontal cortex (","key":"ZkZTjFC3DJ"},{"type":"abbreviation","title":"dorsolateral","children":[{"type":"text","value":"dl","key":"xZpDQ0wloy"}],"key":"HJdvkpRkjq"},{"type":"abbreviation","title":"Prefrontal Cortex","children":[{"type":"text","value":"PFC","key":"tLwSrfGdil"}],"key":"eVmH0OxKAq"},{"type":"text","value":"), the cerebellum (Crus II, lobule VII) and inferior temporal regions during activity of the internalizing subsystem (","key":"deonlyLeGd"},{"type":"crossReference","kind":"table","identifier":"tab-clinical-results","label":"tab-clinical-results","children":[{"type":"text","value":"Table ","key":"G7zy5DMLxY"},{"type":"text","value":"1","key":"tUgUPX1MFT"}],"template":"Table %s","enumerator":"1","resolved":true,"html_id":"tab-clinical-results","key":"P9SEy9jK9v"},{"type":"text","value":"). While similar, default mode network (DMN)-related changes have often been attributed to an atypical integration of information about the “self” and the “other” ","key":"HLwPqp3dS8"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"padmanabhan2017default","identifier":"padmanabhan2017default","children":[{"type":"text","value":"Padmanabhan ","key":"ICPaRIJ79r"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"XLHq9Jqlph"}],"key":"BmzUHIjpg7"},{"type":"text","value":", 2017","key":"wMQXLlN3O6"}],"key":"Y4L7Lo9VWX"}],"key":"UnzudkqZgA"},{"type":"text","value":", a more detailed ","key":"YOZBR8VbLG"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"dThTRp07uf"}],"key":"ARBL8cu9bC"},{"type":"text","value":"-analysis may help to further disentangle the specific nature of these changes.","key":"rcClrIh8Ys"}],"key":"oAKCOT1Uia"},{"type":"container","kind":"figure","identifier":"clinical-validity","label":"clinical-validity","children":[{"type":"image","url":"/state_analysis-ee5a9d498a950579340cc02d477e17dc.svg","alt":"Connectome-based Hopfield analysis of autism spectrum disorder.

\nA The distribution of time-frames on the fcHNN-projection separately for ASD patients and typically developing control (TDC) participants.

\nB We quantified attractor state activations in the Autism Brain Imaging Data Exchange datasets (study 7) as the\nindividual-level mean activation of all time-frames belonging to the same attractor state. This analysis captured alterations similar to those previously associated to ASD-related perceptual atypicalities (visual, auditory and somatosensory cortices) as well as atypical integration of information about the “self” and the “other” (default mode network regions). All results are corrected for multiple comparisons across brain regions and attractor states (122*4 comparisons) with Bonferroni-correction. See and Supplementary Figure %s for detailed results.

\nC The comparison of data generated by fcHNNs initialized with ASD and TDC connectomes, respectively, revealed a characteristic pattern of differences in the system's dynamics, with increased pull towards (and potentially a higher separation between) the action and perception attractors attractors and a lower tendency of trajectories going towards the internal and external attractors.

\nAbbreviations: MCC: middle cingulate cortex, ACC: anterior cingulate cortex, pg: perigenual, PFC: prefrontal cortex, dm: dorsomedial, dl: dorsolateral, STG: superior temporal gyrus, ITG: inferior temporal gyrus, Caud/Acc: caudate-accumbens, SM: sensorimotor, V1: primary visual, A1: primary auditory, SMA: supplementary motor cortex, ASD: autism spectrum disorder, TDC: typically developing control.","urlSource":"figures/state_analysis.svg","key":"zmCugcrVem"},{"type":"caption","children":[{"type":"paragraph","position":{"start":{"line":326,"column":0},"end":{"line":332,"column":0}},"children":[{"type":"captionNumber","kind":"figure","label":"clinical-validity","identifier":"clinical-validity","html_id":"clinical-validity","enumerator":"5","children":[{"type":"text","value":"Figure ","key":"fIa08pROsE"},{"type":"text","value":"5","key":"yJvkMCVJaq"},{"type":"text","value":":","key":"Lcf3FnuLU4"}],"template":"Figure %s:","key":"leU6CRE8Ku"},{"type":"strong","children":[{"type":"text","value":"Connectome-based Hopfield analysis of autism spectrum disorder.","key":"Zs4OMJeP6a"}],"key":"ZdVmqpw0yZ"},{"type":"text","value":" ","key":"IoZN5rGRQC"},{"type":"break","key":"RNALzcfcEJ"},{"type":"text","value":"\n","key":"HM1mwlfntY"},{"type":"strong","children":[{"type":"text","value":"A","key":"uUzcCG7V8w"}],"key":"vC6GTtXw2S"},{"type":"text","value":" The distribution of time-frames on the ","key":"ldvy0RSmgo"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"Q5I33nzovF"}],"key":"KbrGVrDR2r"},{"type":"text","value":"-projection separately for ","key":"EU7ndRFyqH"},{"type":"abbreviation","title":"Autism Spectrum Disorder","children":[{"type":"text","value":"ASD","key":"KtcQTo1Zrl"}],"key":"DSwVjZraVu"},{"type":"text","value":" patients and typically developing control (TDC) participants. ","key":"mRRL639EEA"},{"type":"break","key":"Y8uonhH9hW"},{"type":"text","value":"\n","key":"GXkMTX3vw1"},{"type":"strong","children":[{"type":"text","value":"B","key":"Z5I77XXZwb"}],"key":"zMLIeR5gd1"},{"type":"text","value":" We quantified attractor state activations in the Autism Brain Imaging Data Exchange datasets (","key":"pNhO8bgVAC"},{"type":"crossReference","children":[{"type":"text","value":"study 7","key":"GHaf62b4Ax"}],"urlSource":"#tab-samples","identifier":"tab-samples","label":"tab-samples","kind":"table","template":"Table %s","enumerator":"1","resolved":true,"html_id":"tab-samples","remote":true,"url":"/methods","dataUrl":"/methods.json","key":"ntiMjXq7yb"},{"type":"text","value":") as the\nindividual-level mean activation of all time-frames belonging to the same attractor state. This analysis captured alterations similar to those previously associated to ","key":"QANxgb3Dk0"},{"type":"abbreviation","title":"Autism Spectrum Disorder","children":[{"type":"text","value":"ASD","key":"qPSIwiI55L"}],"key":"dIdg9TKwjE"},{"type":"text","value":"-related perceptual atypicalities (visual, auditory and somatosensory cortices) as well as atypical integration of information about the “self” and the “other” (default mode network regions). All results are corrected for multiple comparisons across brain regions and attractor states (122*4 comparisons) with Bonferroni-correction. See ","key":"PU9HBbGO6N"},{"type":"crossReference","kind":"table","identifier":"tab-clinical-results","label":"tab-clinical-results","children":[{"type":"text","value":"Table ","key":"LoFHihYQxr"},{"type":"text","value":"1","key":"hYhH0Wc10j"}],"template":"Table %s","enumerator":"1","resolved":true,"html_id":"tab-clinical-results","key":"kDbZfDEbBE"},{"type":"text","value":" and ","key":"byhMBvGaJy"},{"type":"crossReference","kind":"figure","identifier":"si_clinical_results_table","label":"si_clinical_results_table","children":[{"type":"text","value":"Supplementary Figure ","key":"V6c9HNBI45"},{"type":"text","value":"8","key":"srMxwDP3Mu"}],"template":"Figure %s","enumerator":"8","resolved":true,"html_id":"si-clinical-results-table","remote":true,"url":"/supplement","dataUrl":"/supplement.json","key":"xcbG5PCWew"},{"type":"text","value":" for detailed results. ","key":"Dts69vk2eO"},{"type":"break","key":"Y90k49yBlG"},{"type":"text","value":"\n","key":"lOLGGQR33g"},{"type":"strong","children":[{"type":"text","value":"C","key":"O03WTuX86y"}],"key":"Ce25pE34jc"},{"type":"text","value":" The comparison of data generated by ","key":"JO8ex79cQQ"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"ZDp8PYW9df"}],"key":"UFiKMVyMOm"},{"type":"text","value":"s initialized with ","key":"aVEUDxyDOb"},{"type":"abbreviation","title":"Autism Spectrum Disorder","children":[{"type":"text","value":"ASD","key":"nvpFiptQJX"}],"key":"sP0VKN4Dvs"},{"type":"text","value":" and TDC connectomes, respectively, revealed a characteristic pattern of differences in the system's dynamics, with increased pull towards (and potentially a higher separation between) the action and perception attractors attractors and a lower tendency of trajectories going towards the internal and external attractors. 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We observed a highly similar pattern in the real data (Pearson's correlation: 0.66), statistically significant after permutation testing (shuffling the group assignment, p=0.009).","key":"IsM9pgrM5r"}],"key":"FVTi6NYUzx"},{"type":"heading","depth":2,"position":{"start":{"line":386,"column":0},"end":{"line":387,"column":0}},"children":[{"type":"text","value":"Discussion","key":"Y6DgC0cfww"}],"identifier":"discussion","label":"Discussion","html_id":"discussion","implicit":true,"key":"V3oZ2cLKjr"},{"type":"paragraph","position":{"start":{"line":388,"column":0},"end":{"line":389,"column":0}},"children":[{"type":"text","value":"In this study, we have introduced and validated a simple yet robust computational generative framework that elucidates how activity propagation within the functional connectome orchestrates large-scale brain dynamics, leading to the spontaneous emergence of brain states and characteristic dynamic responses to perturbations.","key":"jDEwPmxEnY"}],"key":"yDPWgJkv0m"},{"type":"paragraph","position":{"start":{"line":390,"column":0},"end":{"line":392,"column":0}},"children":[{"type":"text","value":"The construct validity of our model is rooted in the activity flow principle, first introduced by ","key":"quC6g1vga1"},{"type":"citeGroup","kind":"narrative","children":[{"type":"cite","label":"cole2016activity","identifier":"cole2016activity","children":[{"type":"text","value":"Cole ","key":"h2VvyYcagJ"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"GkwP2GMewo"}],"key":"MzeBqSsPSX"},{"type":"text","value":" (2016)","key":"CwdgVVf7t8"}],"key":"sIB6z9kJiT"}],"key":"u12fQ5Q4aA"},{"type":"text","value":". The activity flow principle states that activity in a brain region can be predicted by a weighted combination of the activity of all other regions, where the weights are set to the functional connectivity of those regions to the held-out region. This principle has been shown to hold across a wide range of experimental and clinical conditions ","key":"No3u7ekq6r"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"cole2016activity","identifier":"cole2016activity","children":[{"type":"text","value":"Cole ","key":"raEqRaMT8o"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"exDFjCGpfM"}],"key":"DVgB4knbbU"},{"type":"text","value":", 2016","key":"PHZ2tVuUwM"}],"key":"fNKYqTHNp3"},{"type":"cite","label":"ito2017cognitive","identifier":"ito2017cognitive","children":[{"type":"text","value":"Ito ","key":"vAob2PaSSs"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"olshqFxait"}],"key":"cnZeJapG8E"},{"type":"text","value":", 2017","key":"AIi9FdUMgM"}],"key":"dLNyb80C8G"},{"type":"cite","label":"mill2022network","identifier":"mill2022network","children":[{"type":"text","value":"Mill ","key":"AB6shsCuR6"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"ZEedjdRILF"}],"key":"ab7ZLwmz1F"},{"type":"text","value":", 2022","key":"VbuhqDDWOP"}],"key":"eSLvnsQPKT"},{"type":"cite","label":"hearne2021activity","identifier":"hearne2021activity","children":[{"type":"text","value":"Hearne ","key":"DvPKAOu8XZ"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"Jz86vtXuxF"}],"key":"pBK4p8yd5T"},{"type":"text","value":", 2021","key":"zR7Djylc6y"}],"key":"qxL40hHnTm"},{"type":"cite","label":"chen2018human","identifier":"chen2018human","children":[{"type":"text","value":"Chen ","key":"JqcxQ434uD"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"bWMcMrZ3wa"}],"key":"k8qJ9CGxwO"},{"type":"text","value":", 2018","key":"PDocfm2Csu"}],"key":"P03ZHBteTG"}],"key":"uHR77PEIqW"},{"type":"text","value":".\nThe proposed approach is based on the intuition that the repeated, iterative application of the activity flow equation in a system exhibits close analogies with a type of recurrent artificial neural networks, known as Hopfield networks ","key":"VGE6GK4Mkg"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"hopfield1982neural","identifier":"hopfield1982neural","children":[{"type":"text","value":"Hopfield, 1982","key":"T3ZfOgBo4l"}],"key":"ReiZMCT71l"}],"key":"x43rqwh6bp"},{"type":"text","value":".","key":"si2ov86Ht3"}],"key":"a8VsXYG5Ld"},{"type":"paragraph","position":{"start":{"line":393,"column":0},"end":{"line":395,"column":0}},"children":[{"type":"text","value":"Hopfield networks have been widely acknowledged for their relevance for brain function, including the ability to store and recall memories ","key":"EVJZIgeb0R"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"hopfield1982neural","identifier":"hopfield1982neural","children":[{"type":"text","value":"Hopfield, 1982","key":"a9CgEnFZU6"}],"key":"DKk7WkYepE"}],"key":"dvq6B2eDTr"},{"type":"text","value":", self-repair ","key":"PJ1PL094QN"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"murre2003selfreparing","identifier":"murre2003selfreparing","children":[{"type":"text","value":"Murre ","key":"daI46Spfp7"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"sB96EfugOj"}],"key":"iQZq0SI0UN"},{"type":"text","value":", 2003","key":"Ds4f5HM1Ss"}],"key":"n70Bi2LF7w"}],"key":"SfbMW0G8TR"},{"type":"text","value":",\na staggering robustness to noisy or corrupted inputs ","key":"jQB3kRCmId"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"hertz1991introduction","identifier":"hertz1991introduction","children":[{"type":"text","value":"Hertz ","key":"cr9dtbjkre"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"cwrmmNrFAP"}],"key":"XPyqvz1q59"},{"type":"text","value":", 1991","key":"rsXAqOu7Ic"}],"key":"Du4w1bIiSO"}],"key":"hj9Exw4UKw"},{"type":"text","value":" (see also ","key":"IXqZlnFkwf"},{"type":"crossReference","kind":"figure","identifier":"si_noise_robustness_weights","label":"si_noise_robustness_weights","children":[{"type":"text","value":"Supplementary Figure ","key":"HCiRbaGdPx"},{"type":"text","value":"7","key":"LhSiPTU6tP"}],"template":"Figure %s","enumerator":"7","resolved":true,"html_id":"si-noise-robustness-weights","remote":true,"url":"/supplement","dataUrl":"/supplement.json","key":"zAh3sfsyt3"},{"type":"text","value":") and the ability to produce multistable dynamics organized by the \"gravitational pull\" of a finite number of attractor states ","key":"jeVPvol14P"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"khona2022attractor","identifier":"khona2022attractor","children":[{"type":"text","value":"Khona & Fiete, 2022","key":"i6uU07Tvwp"}],"key":"PXs6sfB7Rx"}],"key":"NvhsMHz9dp"},{"type":"text","value":". While many of such properties of Hopfield networks have previously been proposed as a model for micro-scale neural systems (see ","key":"Vk1bmZFX1V"},{"type":"cite","label":"khona2022attractor","identifier":"khona2022attractor","children":[{"type":"text","value":"Khona & Fiete, 2022","key":"kfO5ZgdWWV"}],"key":"cKJMoY5y77"},{"type":"text","value":" for a review), the proposed link between macro-scale activity propagation and Hopfield networks allows transferring the vast body of knowledge on Hopfield networks to the study of large-scale brain dynamics.","key":"ikBFnZjKfO"}],"key":"tOUl6Yujvg"},{"type":"paragraph","position":{"start":{"line":396,"column":0},"end":{"line":398,"column":0}},"children":[{"type":"text","value":"Integrating Cole's activity flow principle with the ","key":"ki7uGZBZRQ"},{"type":"abbreviation","title":"Hopfield Neural Network","children":[{"type":"text","value":"HNN","key":"FDficpsBk1"}],"key":"ZV37D5PKSl"},{"type":"text","value":" architecture mandates the 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","key":"hokozVIaq2"},{"type":"crossReference","kind":"figure","identifier":"si_state_occupancy_null_models","label":"si_state_occupancy_null_models","children":[{"type":"text","value":"4","key":"OlroKMHVXK"}],"template":"Figure %s","enumerator":"4","resolved":true,"html_id":"si-state-occupancy-null-models","remote":true,"url":"/supplement","dataUrl":"/supplement.json","key":"vyy0YRbr1A"},{"type":"text","value":", ","key":"jgIbNhkZCr"},{"type":"crossReference","kind":"figure","identifier":"si_pain_ghost_attractor_sim","label":"si_pain_ghost_attractor_sim","children":[{"type":"text","value":"5","key":"jzkUbxygWo"}],"template":"Figure %s","enumerator":"5","resolved":true,"html_id":"si-pain-ghost-attractor-sim","remote":true,"url":"/supplement","dataUrl":"/supplement.json","key":"FjV5NPcgml"},{"type":"text","value":", ","key":"xrm149tmbd"},{"type":"crossReference","kind":"figure","identifier":"si_downreg_trajectory_sim","label":"si_downreg_trajectory_sim","children":[{"type":"text","value":"6","key":"SHczBP6tRr"}],"template":"Figure %s","enumerator":"6","resolved":true,"html_id":"si-downreg-trajectory-sim","remote":true,"url":"/supplement","dataUrl":"/supplement.json","key":"pRPUQngxcT"},{"type":"text","value":")). To underscore the potency of this simplicity and stability, in the present work, we avoided any unnecessary parameter optimization. It is likely, however, that extensive parameter optimization could further improve the performance of the model.","key":"GOsUyCHSRD"}],"key":"fC9jCcUiSI"},{"type":"paragraph","position":{"start":{"line":401,"column":0},"end":{"line":402,"column":0}},"children":[{"type":"text","value":"Another advantage of ","key":"zbRemnyG0R"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"xDFSuLE73Y"}],"key":"siCmcePyCW"},{"type":"text","value":"s over more detailed models is that ","key":"CQx3LuyJ1x"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"zDhDH7hED2"}],"key":"ZLiXM81vFg"},{"type":"text","value":"s establish a simple and easily interpretable link between two highly prevalent metrics of brain function: functional connectivity and brain activity. This connection is not solely phenomenological, but also mathematical, facilitating the exploration and prediction of alterations in the system's dynamics in response to perturbations affecting both activity and connectivity.","key":"GgRny8Pfws"}],"key":"c6KOKb3KYL"},{"type":"paragraph","position":{"start":{"line":403,"column":0},"end":{"line":404,"column":0}},"children":[{"type":"text","value":"The proposed model also exhibits several advantages over linear network control theory-based ","key":"tYcmqF90Pi"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"gu2015controllability","identifier":"gu2015controllability","children":[{"type":"text","value":"Gu ","key":"Db5NN0xIYo"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"UmE2ypRstu"}],"key":"Azt906UOLy"},{"type":"text","value":", 2015","key":"QLgV6o6A4J"}],"key":"Do8BBFl5du"}],"key":"gVRKYX268I"},{"type":"text","value":" approaches. First, the ","key":"YbaNYbL9eR"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"fy9QOCW5aW"}],"key":"nJaR3HqW45"},{"type":"text","value":" approach works with direct activity flow estimates and does not require knowledge about the structural-functional coupling in the brain. Second, the ","key":"qqlStg2Mch"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"tazPp8Rrur"}],"key":"iC75hIL7Tt"},{"type":"text","value":" approach is based on a non-linear ","key":"Ox8Xti4oKb"},{"type":"abbreviation","title":"Artificial Neural Network","children":[{"type":"text","value":"ANN","key":"YSDFiieJTG"}],"key":"hoEYNSg0VN"},{"type":"text","value":" architecture, thus, similarly to neuroconnectionist approaches, allows leveraging on knowledge about the ","key":"QnNC6VxUeN"},{"type":"abbreviation","title":"Artificial Neural Network","children":[{"type":"text","value":"ANN","key":"lJo78qKs5x"}],"key":"PlLiOsuRRT"},{"type":"text","value":" architecture itself. Specifically, the ","key":"G0vY1so10B"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"t1eFQ51QeJ"}],"key":"bjSRs0pSM7"},{"type":"text","value":"s provide a mechanistic account for the emergence of large-scale canonical brain networks ","key":"yjXxLK1OYn"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"zalesky2014time","identifier":"zalesky2014time","children":[{"type":"text","value":"Zalesky ","key":"FGMS9ATT3l"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"tU66ielpy9"}],"key":"hfBxo9tu0A"},{"type":"text","value":", 2014","key":"bTd76EU5Gs"}],"key":"SpwbX3S926"}],"key":"JUjCosGcNX"},{"type":"text","value":" and brain states or the presence of \"ghost attractors\" ","key":"ybioMRwn6K"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"deco2012ongoing","identifier":"deco2012ongoing","children":[{"type":"text","value":"Deco & Jirsa, 2012","key":"JIjNDdgs3j"}],"key":"hPXb6yA6tS"},{"type":"cite","label":"vohryzek2020ghost","identifier":"vohryzek2020ghost","children":[{"type":"text","value":"Vohryzek ","key":"QWBotp0Nob"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"fIHtPAblkO"}],"key":"VNCGrXI1Kg"},{"type":"text","value":", 2020","key":"CJhr7WzwyK"}],"key":"LaheG3gNIh"}],"key":"DPrduKbwMP"},{"type":"text","value":", via the key concept in the Hopfield network framework, the attractor states.","key":"JMWxjLkrg2"}],"key":"Oi7ingALol"},{"type":"paragraph","position":{"start":{"line":405,"column":0},"end":{"line":406,"column":0}},"children":[{"type":"text","value":"In comparison to conventional neuroconnectionist approaches, ","key":"IFJaZ27YjO"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"TjVww7eXn1"}],"key":"aGKW0zTtEn"},{"type":"text","value":"s do not need to be trained to solve tasks and thus allow for the exploration of spontaneous brain dynamics. However, it is worth mentioning that, like any other ","key":"Iu1NSZe3G0"},{"type":"abbreviation","title":"Artificial Neural Network","children":[{"type":"text","value":"ANN","key":"MWnFa1nTJo"}],"key":"uF2HzZpvvd"},{"type":"text","value":"s, ","key":"h8fwIhmm28"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"IiARXs1R4P"}],"key":"tQVVvvkHyt"},{"type":"text","value":"s can also be further trained via established ","key":"XyuZEpcrK8"},{"type":"abbreviation","title":"Artificial Neural Network","children":[{"type":"text","value":"ANN","key":"PgXqjhJRzw"}],"key":"CCi8ZOlRfC"},{"type":"text","value":" training techniques (e.g. via the Hebbian learning rule) to \"solve\" various tasks or to match altered dynamics during development or in clinical populations. In this interesting future direction, the training procedure itself becomes part of the model, providing testable hypotheses about the formation, and various malformations, of brain dynamics.","key":"cWsTfua9QJ"}],"key":"rFJz0ax42z"},{"type":"paragraph","position":{"start":{"line":407,"column":0},"end":{"line":410,"column":0}},"children":[{"type":"text","value":"Given its simplicity, it is remarkable, if not surprising, how accurately the ","key":"ssiMDcg9AX"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"GaSSq4VInU"}],"key":"b9ZfSCzxlr"},{"type":"text","value":" model is able to reconstruct and predict brain dynamics under a wide range of conditions. Particularly interesting is the result that the two-dimensional ","key":"Nnl6u0gwTi"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"wCPTFcuoNS"}],"key":"hKLxDfxJdG"},{"type":"text","value":" projection can explain more variance in real resting state ","key":"BXRnMLz1Yb"},{"type":"abbreviation","title":"functional Magnetic Resonance Imaging","children":[{"type":"text","value":"fMRI","key":"gSGkC70Rkp"}],"key":"nUbLuddcHr"},{"type":"text","value":" data than the first two principal components derived from the data itself.\nA plausible explanation for the remarkable reconstruction performance is that, trough their known noise tolerance, ","key":"vuhfmX75Om"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"kV1kjskwX5"}],"key":"DmBLNOEkhs"},{"type":"text","value":"s are able to capture essential principles of the underlying dynamic processes even if our empirical measurements are corrupted by noise and low sampling rate.\nIndeed, ","key":"dZYaXv5wqf"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"iJncA8PFW3"}],"key":"lANLQX4F7o"},{"type":"text","value":" attractor states were found to be robust to noisy weights (","key":"ePROh4wMEH"},{"type":"crossReference","kind":"figure","identifier":"si_noise_robustness_weights","label":"si_noise_robustness_weights","children":[{"type":"text","value":"Supplementary Figure ","key":"CocpCxnjgk"},{"type":"text","value":"7","key":"VPDkXe0Seh"}],"template":"Figure %s","enumerator":"7","resolved":true,"html_id":"si-noise-robustness-weights","remote":true,"url":"/supplement","dataUrl":"/supplement.json","key":"c6OepwqEwg"},{"type":"text","value":") and highly replicable across datasets acquired at different sites, with different scanners and imaging sequences (study 2 and 3). The observed level of replicability allowed us to re-use the ","key":"qvG3kKgYJz"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"klEExy3gyh"}],"key":"iskhhcPDWQ"},{"type":"text","value":" model constructed with the connectome of study 1 for all subsequent analyses, without any further fine-tuning or study-specific parameter optimization.","key":"iCXQL8Bmhd"}],"key":"FMI7kMQW4g"},{"type":"paragraph","position":{"start":{"line":411,"column":0},"end":{"line":413,"column":0}},"children":[{"type":"text","value":"Conceptually, the notion of a global attractor model of the brain network is not new ","key":"e6eiJXVSk9"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"deco2012ongoing","identifier":"deco2012ongoing","children":[{"type":"text","value":"Deco & Jirsa, 2012","key":"jqMTr5k04f"}],"key":"TsQp9NfZdI"}],"key":"GUyIkYYeur"},{"type":"text","value":". The present work shows, however, that the brain as an attractor network necessarily 'leaks' its code in form of the partial correlation across the regional timeseries, allowing us to uncover its large-scale attractor states. Moreover, we demonstrate that the brain's attractor states are not solely local minima in the state-space but act as a driving force for the dynamic trajectories of brain activity. Attractor-dynamics may be the main driving factor for the spatial and temporal autocorrelation structure of the brain, recently described to be predictive of network topology in relation to age, subclinical symptoms of dementia, and pharmacological manipulations with serotonergic drugs ","key":"elMnLgSG22"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"shinn2023functional","identifier":"shinn2023functional","children":[{"type":"text","value":"Shinn ","key":"ZTpFTtZ6eU"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"rJot3lVBaK"}],"key":"qrtaDt0qJz"},{"type":"text","value":", 2023","key":"nzzeFHziJT"}],"key":"pu9yJtJ1zA"}],"key":"tsEDYLp6qS"},{"type":"text","value":".\nNevertheless, attractor states should not be confused with the conventional notion of brain states ","key":"YLOgFK4WJP"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"chen2015introducing","identifier":"chen2015introducing","children":[{"type":"text","value":"Chen ","key":"riJTqS6NFW"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"L1bE4sXBe0"}],"key":"eOuYVZeZ1B"},{"type":"text","value":", 2015","key":"ggNvyrrnOf"}],"key":"oYtDszbPe5"}],"key":"Vkj7ooJ0p6"},{"type":"text","value":" and large-scale functional gradients ","key":"zKnTd7abZ3"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"margulies2016situating","identifier":"margulies2016situating","children":[{"type":"text","value":"Margulies ","key":"O3Fi0Y6GRT"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"AX0qXK5Qkc"}],"key":"lDa2qpDltx"},{"type":"text","value":", 2016","key":"hsRKTLnFP5"}],"key":"gYkUxnPwnk"}],"key":"Hc2wIaRsIx"},{"type":"text","value":". In the ","key":"BuZPoIVuTe"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"TD4DxniuQv"}],"key":"l6xfg93ZyW"},{"type":"text","value":" framework, attractor states can rather be conceptualized as \"Platonic idealizations\" of brain activity, that are continuously approximated - but never reached - by the brain, resulting in re-occurring patterns (brain states) and smooth gradual transitions (large-scale gradients).","key":"TeX9Z1gHM9"}],"key":"lH1cTZKYpZ"},{"type":"paragraph","position":{"start":{"line":414,"column":0},"end":{"line":416,"column":0}},"children":[{"type":"text","value":"Relying on previous work, we can establish a relatively straightforward (although somewhat speculative) correspondence between attractor states and brain function, mapping brain activation on the axes of internal vs. external context ","key":"ximX99kwdk"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"golland2008data","identifier":"golland2008data","children":[{"type":"text","value":"Golland ","key":"yYFVTwuUuZ"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"lThuzRWUrn"}],"key":"Hfs3qkU7Zo"},{"type":"text","value":", 2008","key":"H5AjHDdHm4"}],"key":"KlkOytP2Mk"},{"type":"cite","label":"cioli2014differences","identifier":"cioli2014differences","children":[{"type":"text","value":"Cioli ","key":"pSpsf56ry4"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"FiWyeepQfs"}],"key":"jb03Dhuy8z"},{"type":"text","value":", 2014","key":"T0Q0CDRpvu"}],"key":"VJ6CnmBvsS"}],"key":"sbziy8zv0S"},{"type":"text","value":", as well as perception vs. action ","key":"amwxrnUMnl"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"fuster2004upper","identifier":"fuster2004upper","children":[{"type":"text","value":"Fuster, 2004","key":"w5r5szMexQ"}],"key":"aoTse69Hvh"}],"key":"V22v09U1P1"},{"type":"text","value":".\nThis four-attractor architecture exhibits an appealing analogy with Friston's free energy principle ","key":"nyz9zAgKXO"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"friston2006free","identifier":"friston2006free","children":[{"type":"text","value":"Friston ","key":"WKpaFkrfVU"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"re62rP5Aqa"}],"key":"Gd9eIx8avC"},{"type":"text","value":", 2006","key":"KqjpDpbLEp"}],"key":"i2eiGxhatk"}],"key":"uPDZJqedsF"},{"type":"text","value":" that postulate the necessary existence of subsystems for active and perceptual inference as well as a hierarchically organized (i.e. external and internal) subsystems that give rise to consciousness ","key":"lwZWANuHvM"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"ramstead2023inner","identifier":"ramstead2023inner","children":[{"type":"text","value":"Ramstead ","key":"PhCgX8VHIo"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"BhwctP9uNH"}],"key":"qX6etmWIww"},{"type":"text","value":", 2023","key":"bpxjvq8KcY"}],"key":"kqSOCJnKZd"},{"type":"cite","label":"lee2023life","identifier":"lee2023life","children":[{"type":"text","value":"Lee ","key":"OKh9qdTnLU"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"lcGa7guiY6"}],"key":"mz8sN08uUZ"},{"type":"text","value":", 2023","key":"uHeYwFvEuz"}],"key":"hoGbgss2pX"}],"key":"QUiIc65ppe"},{"type":"text","value":".","key":"PMauTeXgxo"}],"key":"wfCgeCuVtu"},{"type":"paragraph","position":{"start":{"line":417,"column":0},"end":{"line":420,"column":0}},"children":[{"type":"text","value":"Both conceptually and in terms of analysis practices, resting and task states are often treated as separate phenomena. However, in the ","key":"vwtDK6GeHf"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"aqpRiMIcgC"}],"key":"Cas8Ta3jqV"},{"type":"text","value":" framework, the differentiation between task and resting states is considered an artificial dichotomy.\nTask-based brain activity in the ","key":"YdDgXTErP4"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"oi8R1rUY6P"}],"key":"O217H8Mv3z"},{"type":"text","value":" framework is not a mere response to external stimuli in certain brain locations but a perturbation of the brain's characteristic dynamic trajectories, with increased preference for certain locations on the energy landscape (\"ghost attractors\").\nIn our analyses, the ","key":"tPv90thPVJ"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"xplnrYpzoJ"}],"key":"R6w7CcWcLm"},{"type":"text","value":" approach capture and predict participant-level activity changes induced by pain and its self-regulation and gave a mechanistic account for how relatively small activity changes in a single region (NAcc) may result in a significantly altered pain experience.","key":"OP5c0gcp1z"}],"key":"utfBUobv4l"},{"type":"paragraph","position":{"start":{"line":421,"column":0},"end":{"line":422,"column":0}},"children":[{"type":"text","value":"Brain dynamics can not only be perturbed by task or other types of experimental or naturalistic interventions, but also by pathological alterations. Here we have demonstrated (study 7) that ","key":"CLdPX31d6x"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"N2dbWqO4kg"}],"key":"iNyBT170tv"},{"type":"text","value":"-based analyses can characterize and predict altered brain dynamics in autism spectrum disorder (","key":"kfJg5J2Ko6"},{"type":"abbreviation","title":"Autism Spectrum Disorder","children":[{"type":"text","value":"ASD","key":"eHJTj7f135"}],"key":"fmMyz920dJ"},{"type":"text","value":"). The observed ","key":"j4pMBQ0wLR"},{"type":"abbreviation","title":"Autism Spectrum Disorder","children":[{"type":"text","value":"ASD","key":"MXJ8aR3fYs"}],"key":"bETUe0iDDa"},{"type":"text","value":"-associated changes in brain dynamics are indicative of a reduced ability to flexibly switch between internal and external modes of processing, corroborating previous findings that in ","key":"PFfO82zjhd"},{"type":"abbreviation","title":"Autism Spectrum Disorder","children":[{"type":"text","value":"ASD","key":"bYcU1F5M8a"}],"key":"uZ0ZKM9QHB"},{"type":"text","value":", sensory-driven connectivity transitions do not converge to transmodal areas ","key":"urmGvqNT0u"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"hong2019atypical","identifier":"hong2019atypical","children":[{"type":"text","value":"Hong ","key":"PFn36MfOop"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"aVr77wS9OR"}],"key":"ePAa9fXV9x"},{"type":"text","value":", 2019","key":"G1YgwuxyuN"}],"key":"QNwWCluYF2"}],"key":"tmlfk9Tyv0"},{"type":"text","value":". Such findings are in line with previous reports of a reduced influence of context on the interpretation of incoming sensory information in ","key":"hvImIxRPMg"},{"type":"abbreviation","title":"Autism Spectrum Disorder","children":[{"type":"text","value":"ASD","key":"O5CTeVfjGQ"}],"key":"QGYMWYWoQx"},{"type":"text","value":" (e.g. the violation of Weber's law) ","key":"MSpF9csMbM"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"hadad2019perception","identifier":"hadad2019perception","children":[{"type":"text","value":"Hadad & Schwartz, 2019","key":"t2Zc3CQrnI"}],"key":"l5oirGH4vZ"}],"key":"vnCdUfaE2N"},{"type":"text","value":".","key":"kOBk1KltBB"}],"key":"PLzQc522zQ"},{"type":"paragraph","position":{"start":{"line":423,"column":0},"end":{"line":424,"column":0}},"children":[{"type":"text","value":"Together, our findings open up a series of exciting opportunities for the better understanding of brain function in health and disease.","key":"loZ91Vitrh"}],"key":"V9CKgceB2q"},{"type":"paragraph","position":{"start":{"line":425,"column":0},"end":{"line":426,"column":0}},"children":[{"type":"text","value":"First, the 2-dimensional ","key":"Ug5gAzOMEw"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"EPKyLn2tKN"}],"key":"uQI66vy3cu"},{"type":"text","value":" projection offers a streamlined framework not only for the visualization, but also for the ","key":"Eo5mwH9CNm"},{"type":"emphasis","children":[{"type":"text","value":"interpretation","key":"vWrPtH54FI"}],"key":"jquWI7dgqq"},{"type":"text","value":", of brain activity patterns, as it conceptualizes changes related to various behavioral or clinical states or traits as a shift in brain dynamics in relation to brain attractor states.","key":"gjLCoDo7A1"}],"key":"tvYB5QoLyp"},{"type":"paragraph","position":{"start":{"line":427,"column":0},"end":{"line":428,"column":0}},"children":[{"type":"text","value":"Second, ","key":"m7Ld7CY7Q5"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"jbDdRHdzk5"}],"key":"GOhKquwQZi"},{"type":"text","value":" analyses may provide insights into the causes of changes in brain dynamics, by for instance, identifying the regions or connections that act as an \"Achilles heel\" in generating such changes. Such analyses could, for instance, aid the differentiation of primary causes and secondary effects of particular activity or connectivity changes in various clinical conditions.","key":"SgWg2uwxdr"}],"key":"Pf1zuS03zf"},{"type":"paragraph","position":{"start":{"line":429,"column":0},"end":{"line":430,"column":0}},"children":[{"type":"text","value":"Third, the ","key":"WREJlu0B1D"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"iKRg7xYvZv"}],"key":"Yk1sL1cqna"},{"type":"text","value":" approach can provide testable predictions about the effects of pharmacological interventions as well as non-invasive brain stimulation (e.g. transcranial magnetic or direct current stimulation, focused ultrasound, etc) and neurofeedback. Obtaining the optimal stimulation or treatment target within the ","key":"tvQDxwZnRz"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"ujpgjKclX6"}],"key":"HNvCcXYRDw"},{"type":"text","value":" framework (e.g. by means of network control theory ","key":"pItTejZNco"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"liu2011controllability","identifier":"liu2011controllability","children":[{"type":"text","value":"Liu ","key":"cLYCcWaTQh"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"tyyKQNDm0C"}],"key":"XhXTlWa3zu"},{"type":"text","value":", 2011","key":"RG6GtkIznX"}],"key":"Fi8KY2X9Ze"}],"key":"WSPDEVIRS3"},{"type":"text","value":") is one of the most promising future directions with the potential to significantly advance the development of novel, personalized treatment approaches.","key":"anE0EA527i"}],"key":"XgtLDCyxiv"},{"type":"paragraph","position":{"start":{"line":431,"column":0},"end":{"line":433,"column":0}},"children":[{"type":"text","value":"In this initial work, we presented the simplest possible implementation of the ","key":"BILlv66Z88"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"J0XW32nVhe"}],"key":"FobNkdEyue"},{"type":"text","value":" concept. It is clear that the presented analyses exploit only a small proportion of the richness of the full state-space dynamics reconstructed by the ","key":"nnNfW3KnQQ"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"N5DDtQzU9E"}],"key":"klyFs8vRYx"},{"type":"text","value":" model.\nThere are many potential ways to further improve the utility of the ","key":"qOznoCJQx9"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"DsbWPYdEr4"}],"key":"Vt3tY4otAZ"},{"type":"text","value":" approach. Increasing the number of reconstructed attractor states (by increasing the temperature parameter), investigating higher-dimensional dynamics, fine-tuning the hyperparameters, testing the effect of different initializations and perturbations are all important direction for future work, with the potential to further improve the model's accuracy and usefulness.","key":"jUvSl951XK"}],"key":"JWziIodZnd"},{"type":"comment","value":"**other potential topics**:\n - is the functional connectome stationary? Why don't we use dynamic connectivity? See arguments by the Cole-group. Also, the fcHNN model can actually probably also reproduce task-based connectivity, when adding a task-related control signal to the stochastic relaxation procedure (as on Fig. 3). Thus it could be a model of how task-based connectivity and dynamic connectivity changes arise from the underlying rs-fMRI connectome. Maybe it could be even better to use \"latent-FC\" a'la McCormick, 2022, [](https://doi.org/10.1162/netn_a_00234))\n - why no HRF modelling (could be a possible extension, but it is also not part of the activity flow approach and we don't reconstruct time series, per-se, but rather activations)\n - the fcHNN model is not a model of brain function, but a model of brain dynamics. It does not strive to explain various brain regions ability to perform certain computations, but the brain's characteristic dynamic \"trajectories\", and how these are perturbed by tasks and other types of interventions.","position":{"start":{"line":434,"column":0},"end":{"line":438,"column":0}},"key":"UQEczCA1e1"},{"type":"heading","depth":2,"position":{"start":{"line":439,"column":0},"end":{"line":440,"column":0}},"children":[{"type":"text","value":"Conclusion","key":"pgkxErVx16"}],"identifier":"conclusion","label":"Conclusion","html_id":"conclusion","implicit":true,"key":"G3oPE5NlFE"},{"type":"paragraph","position":{"start":{"line":441,"column":0},"end":{"line":442,"column":0}},"children":[{"type":"text","value":"To conclude, here we have proposed a lightweight, high-level computational framework that accurately captures and predicts brain dynamics under a wide range of conditions. The framework models large-scale activity flow in the brain with a recurrent artificial neural network architecture that, instead of being trained to solve specific tasks or mimic certain dynamics, is simply initialized with the empirical functional connectome. The framework identifies neurobiologically meaningful attractor states and provides a model for how these restrict brain dynamics. The proposed framework, referred to as the connectome-based Hopfield neural network (","key":"WCdkFHsX0K"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"mXcFXiwFJZ"}],"key":"UOuOgwyLB1"},{"type":"text","value":") model, can accurately reconstruct and predict brain dynamics under a wide range of conditions, including resting state, task-induced activity changes, as well as brain disorders. Fc","key":"CrcjPDm7mp"},{"type":"abbreviation","title":"Hopfield Neural Network","children":[{"type":"text","value":"HNN","key":"KPed10kx4J"}],"key":"atI3UUHWrE"},{"type":"text","value":"s establish a conceptual link between connectivity and activity and offer a simple, robust, and highly interpretable computational alternative to conventional descriptive approaches to investigating brain function. The generative nature of the proposed model opens up a series of exciting opportunities for future research, including predicting the effect, and understanding the mechanistic bases, of various interventions; thereby paving the way for designing novel treatment approaches.","key":"elgDEI0IMf"}],"key":"tyRdxF1Qtj"}],"key":"OIGrbqAdOp"},{"type":"block","position":{"start":{"line":444,"column":0},"end":{"line":445,"column":0}},"children":[{"type":"heading","depth":2,"position":{"start":{"line":445,"column":0},"end":{"line":446,"column":0}},"children":[{"type":"text","value":"Acknowledgements","key":"ZpMiwmHwZB"}],"identifier":"acknowledgements","label":"Acknowledgements","html_id":"acknowledgements","implicit":true,"key":"KHpFh4xKyc"},{"type":"paragraph","position":{"start":{"line":447,"column":0},"end":{"line":448,"column":0}},"children":[{"type":"text","value":"The work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation; projects ‘TRR289 - Treatment Expectation’, ID 422744262 and ‘SFB1280 - Extinction Learning’, ID 316803389) and by IBS-R015-D1 (Institute for Basic Science; C.W.-W.).","key":"iYPTp3lZsG"}],"key":"k9HHWdqeIp"}],"data":{"part":"acknowledgements"},"key":"O9EYwVVRMd"},{"type":"block","position":{"start":{"line":448,"column":0},"end":{"line":449,"column":0}},"key":"GjnB2IbVZ0"},{"type":"block","position":{"start":{"line":450,"column":0},"end":{"line":451,"column":0}},"children":[{"type":"heading","depth":2,"position":{"start":{"line":452,"column":0},"end":{"line":453,"column":0}},"children":[{"type":"text","value":"Analysis source code","key":"qDYiOLDXza"}],"identifier":"analysis-source-code","label":"Analysis source code","html_id":"analysis-source-code","implicit":true,"key":"v9tOfsNCux"},{"type":"paragraph","position":{"start":{"line":453,"column":0},"end":{"line":454,"column":0}},"children":[{"type":"link","url":"https://github.com/pni-lab/connattractor","children":[{"type":"text","value":"https://​​github​​.com​​/pni​​-lab​​/connattractor","key":"R3rYIrADPr"}],"urlSource":"https://github.com/pni-lab/connattractor","error":true,"key":"vYOFMselQn"}],"key":"ruHZOvHADZ"},{"type":"heading","depth":2,"position":{"start":{"line":455,"column":0},"end":{"line":456,"column":0}},"children":[{"type":"text","value":"Project website","key":"ewqlyW0Znu"}],"identifier":"project-website","label":"Project website","html_id":"project-website","implicit":true,"key":"QCNnwkDuL3"},{"type":"paragraph","position":{"start":{"line":456,"column":0},"end":{"line":457,"column":0}},"children":[{"type":"link","url":"https://pni-lab.github.io/connattractor/","children":[{"type":"text","value":"https://​​pni​​-lab​​.github​​.io​​/connattractor​​/","key":"cFqjkaduzf"}],"urlSource":"https://pni-lab.github.io/connattractor/","key":"svIiutKqLd"}],"key":"EWxv6PPc6K"},{"type":"heading","depth":2,"position":{"start":{"line":458,"column":0},"end":{"line":459,"column":0}},"children":[{"type":"text","value":"Data availability","key":"Lv5QoedpWn"}],"identifier":"data-availability","label":"Data availability","html_id":"data-availability","implicit":true,"key":"elijrIV0BM"},{"type":"paragraph","position":{"start":{"line":459,"column":0},"end":{"line":460,"column":0}},"children":[{"type":"text","value":"Study 1,2 and 4 is available at ","key":"yn5oz3byeQ"},{"type":"link","url":"http://openneuro.org","children":[{"type":"text","value":"openneuro.org","key":"mdPeGb8Iyw"}],"urlSource":"http://openneuro.org","key":"lhUbwMmlKD"},{"type":"text","value":" (ds002608, ds002608, ds000140). Data for study 3 is available upon request. Data for study 5-6 is available at the github page of the project: ","key":"GMD4EgcFOw"},{"type":"link","url":"https://github.com/pni-lab/connattractor","children":[{"type":"text","value":"https://​​github​​.com​​/pni​​-lab​​/connattractor","key":"rTbRUebJOR"}],"urlSource":"https://github.com/pni-lab/connattractor","error":true,"key":"W6g8XwVk2p"},{"type":"text","value":". Study 7 is available at https://fcon_1000.projects.nitrc.org/indi/abide/, preprocessed data is available at ","key":"dMLJwUti6b"},{"type":"link","url":"http://preprocessed-connectomes-project.org/","children":[{"type":"text","value":"http://​​preprocessed​​-connectomes​​-project​​.org​​/","key":"BXTa6n8YVd"}],"urlSource":"http://preprocessed-connectomes-project.org/","key":"dBQPrAJTwB"},{"type":"text","value":".","key":"J4zdYSuRDj"}],"key":"R6GO2mhyGZ"}],"data":{"part":"data-availability"},"key":"Ba82iCvlG5"},{"type":"block","position":{"start":{"line":460,"column":0},"end":{"line":461,"column":0}},"key":"Edy4zS0DQ2"}],"key":"dMA3dl4sUL"},"references":{"cite":{"order":["buzsaki2006rhythms","bassett2017network","liu2013time","zalesky2014time","margulies2016situating","huntenburg2018large","greene2023everyone","vidaurre2017brain","smith2012temporally","chen2018human","hutchison2013dynamic","barttfeld2015signature","meer2020movie","breakspear2017dynamic","murray2018biophysical","kriegeskorte2018cognitive","heinz2019towards","schirner2022dynamic","schiff1994controlling","papadopoulos2017development","chiem2021structure","scheid2021time","gu2015controllability","doerig2023neuroconnectionist","richards2019deep","cole2016activity","ito2017cognitive","hopfield1982neural","krotov2023new","koiran1994dynamics","daunizeau2012stochastic","robinson2005multiscale","golland2008data","cioli2014differences","fuster2004upper","woo2015distinct","vohryzek2020ghost","liu2011controllability","zunhammer2021meta","di2014autism","hadad2019perception","padmanabhan2017default","mill2022network","hearne2021activity","murre2003selfreparing","hertz1991introduction","khona2022attractor","cabral2017functional","deco2012ongoing","shinn2023functional","chen2015introducing","friston2006free","ramstead2023inner","lee2023life","hong2019atypical"],"data":{"buzsaki2006rhythms":{"number":1,"html":"Buzsaki, G. 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The challenge for such models lies in modelling the relation between the structural wiring of the brain and functional connectivity.\nThe \"neuroconnectionist\" approach, on the other hand, ","key":"w6xAY0WmS9"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"doerig2023neuroconnectionist","identifier":"doerig2023neuroconnectionist","children":[{"type":"text","value":"Doerig ","key":"XambBbQse2"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"wMj9uJNLRY"}],"key":"Gh3w3yiTxs"},{"type":"text","value":", 2023","key":"mRhYeqO1mr"}],"key":"je2Gq2dMvh"}],"key":"OsIp4qfrcr"},{"type":"text","value":" aims at \"cognitive/behavioral fidelity\" ","key":"VTG4mVjwxx"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"kriegeskorte2018cognitive","identifier":"kriegeskorte2018cognitive","children":[{"type":"text","value":"Kriegeskorte & Douglas, 2018","key":"X7GXmunoYH"}],"key":"Zj7Pz9UBiw"}],"key":"HXeqkl0gCM"},{"type":"text","value":", by using artificial neural networks (","key":"EtnJS6v4UQ"},{"type":"abbreviation","title":"Artificial Neural Network","children":[{"type":"text","value":"ANN","key":"Ns2rImO9Js"}],"key":"g6DDgWUQsX"},{"type":"text","value":"s) that are trained to perform various tasks, as brain models. However, the need to train ","key":"JgFWTo2zuK"},{"type":"abbreviation","title":"Artificial Neural Network","children":[{"type":"text","value":"ANN","key":"VQLC4NrFX1"}],"key":"BSwHNWl8yc"},{"type":"text","value":"s for specific tasks inherently limits their ability to explain task-independent, spontaneous neural dynamics ","key":"AfZEkL8ZW4"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"richards2019deep","identifier":"richards2019deep","children":[{"type":"text","value":"Richards ","key":"SK3BjBYeNC"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"mP8Zm9U7nr"}],"key":"XVUq5Keksv"},{"type":"text","value":", 2019","key":"ivkwE9Tzwc"}],"key":"jf0vNRQJ4T"}],"key":"WcZHDgm4V5"},{"type":"text","value":".","key":"xQAyUIqlp7"}],"key":"kNY8N4OpTZ"},{"type":"paragraph","position":{"start":{"line":143,"column":0},"end":{"line":146,"column":0}},"children":[{"type":"text","value":"Here we propose a novel approach that combines the advantages of large-scale network models and neuroconnectionism, to investigate brain dynamics.\nSimilar to neuroconnectionism, we utilize an ","key":"kwMSJ74ID0"},{"type":"abbreviation","title":"Artificial Neural Network","children":[{"type":"text","value":"ANN","key":"OxDZoeJv0F"}],"key":"Q9dWTacQMd"},{"type":"text","value":" as a high-level computational model of the brain.\nHowever, our model is not explicitly trained for a specific task. Instead, we set its weights empirically, with data based on the \"activity flow\" ","key":"qPX1Q4mIFG"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"cole2016activity","identifier":"cole2016activity","children":[{"type":"text","value":"Cole ","key":"j6gaeMIxwK"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"LrGIZi1fZn"}],"key":"mvuQ3sfV6u"},{"type":"text","value":", 2016","key":"fNCwtrciZT"}],"key":"AuFd6VkUbX"},{"type":"cite","label":"ito2017cognitive","identifier":"ito2017cognitive","children":[{"type":"text","value":"Ito ","key":"xlyuImjhVy"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"VO2JHJj4Jx"}],"key":"Pn1iUynlM9"},{"type":"text","value":", 2017","key":"YyrhziAEf1"}],"key":"HfPRPyUzgB"}],"key":"uNP64Ut6rP"},{"type":"text","value":" across regions within the functional brain connectome, as measured with functional magnetic resonance imaging (","key":"a3ob0OJCXF"},{"type":"abbreviation","title":"functional Magnetic Resonance Imaging","children":[{"type":"text","value":"fMRI","key":"lmjENTCe7S"}],"key":"yJHKMbUENX"},{"type":"text","value":", ","key":"naC4VPscTD"},{"type":"crossReference","kind":"figure","identifier":"concept","label":"concept","children":[{"type":"text","value":"Figure ","key":"FgrNCLy3O6"},{"type":"text","value":"1","key":"poPApZPflV"}],"template":"Figure %s","enumerator":"1","resolved":true,"html_id":"concept","key":"QxOlRXQnil"},{"type":"text","value":"B).","key":"ml0197doMu"}],"key":"mu0IiYdJPx"},{"type":"paragraph","position":{"start":{"line":147,"column":0},"end":{"line":151,"column":0}},"children":[{"type":"text","value":"Specifically, we employ a continuous-space Hopfield Neural Network (","key":"gaRw3aB17U"},{"type":"abbreviation","title":"Hopfield Neural Network","children":[{"type":"text","value":"HNN","key":"RIU1GqDSD6"}],"key":"LbCtT26GWz"},{"type":"text","value":") ","key":"l0au2rhBnF"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"hopfield1982neural","identifier":"hopfield1982neural","children":[{"type":"text","value":"Hopfield, 1982","key":"XrcVp1Te15"}],"key":"EaLNbGtBDF"},{"type":"cite","label":"krotov2023new","identifier":"krotov2023new","children":[{"type":"text","value":"Krotov, 2023","key":"oTr0opJhJ1"}],"key":"VqjULs0PTQ"}],"key":"frqckjYlsO"},{"type":"text","value":", with its nodes representing large-scale brain areas, and its weights initialized with the functional connectivity values between these areas.\nBased on the topology of the functional connectome, this architecture establishes an energy level for any arbitrary activation patterns and determines a \"trajectory of least action\" towards one of the finite number of stable patterns, known as ","key":"pz9iJ0cBII"},{"type":"emphasis","children":[{"type":"text","value":"attractor states","key":"JlZtjbyNi8"}],"key":"ODZflkLRyG"},{"type":"text","value":", that minimize this energy.\nIn this simplistic yet powerful framework, brain dynamics can be conceptualized as an intricate, high-dimensional path on the energy landscape (","key":"c7LwrI7eHB"},{"type":"crossReference","kind":"figure","identifier":"concept","label":"concept","children":[{"type":"text","value":"Figure ","key":"iNLuBeRmGi"},{"type":"text","value":"1","key":"RTs5gNb3CO"}],"template":"Figure %s","enumerator":"1","resolved":true,"html_id":"concept","key":"p5Rnd6qqIP"},{"type":"text","value":"C), arising from the activity flow ","key":"roQzmf6fjb"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"cole2016activity","identifier":"cole2016activity","children":[{"type":"text","value":"Cole ","key":"hZLo1pO8zQ"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"ltx7MVez1L"}],"key":"SpfG2yW8c6"},{"type":"text","value":", 2016","key":"vciONKvozX"}],"key":"PWBM6sdYmG"}],"key":"K8562tl9g8"},{"type":"text","value":" within the functional connectome and constrained by the \"gravitational pull\" of the attractor states of the system.\nGiven its generative nature, the proposed model offers testable predictions for the effect of various perturbations and alterations of these dynamics, from task-induced activity, to changes related to brain disorders.","key":"X5TZKigayo"}],"key":"nN4osaPUzW"},{"type":"container","kind":"figure","identifier":"concept","label":"concept","children":[{"type":"image","url":"/concept-fc61e809920bee5c5ec3407459032fc2.png","alt":"Connectome-based Hopfield networks as models of macro-scale brain dynamics.

\nA Hopfield artificial neural networks (HNNs) are a form of recurrent artificial neural networks that serve as content-addressable (\"associative\") memory systems. Hopfield networks can be trained to store a finite number of patterns (e.g. via Hebbian learning a.k.a. \"fire together - wire together\"). During the training procedure, the weights of the HNN are trained so that the stored\npatterns become stable attractor states of the network. Thus, when the trained network is presented partial, noisy or corrupted variations of the stored patterns, it can effectively reconstruct the original pattern via an iterative relaxation procedure that converges to the attractor states.\nB We consider regions of the brain as nodes of a Hopfield network. Instead of training the Hopfield network to\nspecific tasks, we set its weights empirically, with the interregional activity flow estimated via functional\nbrain connectivity. Capitalizing on strong analogies between the relaxation rule of Hopfield networks and the\nactivity flow principle that links activity to connectivity in brain networks, we propose the resulting\nfunctional connectome-based Hopfield neural network (fcHNN) as a computational model for macro-scale brain dynamics.C The proposed computational framework assigns an energy level, an attractor state and a position in a\nlow-dimensional embedding to brain activation patterns. Additionally, it models how the entire state-space of viable activation patterns is restricted by the dynamics of the system and how alterations in activity and/or connectivity modify these dynamics.","urlSource":"figures/concept.png","urlOptimized":"/concept-fc61e809920bee5c5ec3407459032fc2.webp","key":"KvDdd2MZpn"},{"type":"caption","children":[{"type":"paragraph","position":{"start":{"line":154,"column":0},"end":{"line":164,"column":0}},"children":[{"type":"captionNumber","kind":"figure","label":"concept","identifier":"concept","html_id":"concept","enumerator":"1","children":[{"type":"text","value":"Figure ","key":"Wz2W0vTkvT"},{"type":"text","value":"1","key":"zWuqr04pw5"},{"type":"text","value":":","key":"u885QsXBXx"}],"template":"Figure %s:","key":"ZjSFdEHjfR"},{"type":"strong","children":[{"type":"text","value":"Connectome-based Hopfield networks as models of macro-scale brain dynamics.","key":"ZkRTS38EhC"}],"key":"oOSqL8fpu3"},{"type":"text","value":" ","key":"q6FpjsMhFY"},{"type":"break","key":"JLBmqP2tiP"},{"type":"break","key":"T0jvEoS9CB"},{"type":"text","value":"\n","key":"TIutAW8gRn"},{"type":"strong","children":[{"type":"text","value":"A","key":"KXAx2Wdado"}],"key":"Tvk7i77RDM"},{"type":"text","value":" Hopfield artificial neural networks (","key":"CIfCkfJUKh"},{"type":"abbreviation","title":"Hopfield Neural Network","children":[{"type":"text","value":"HNN","key":"sQ5WEDAGPc"}],"key":"HriVhKrU0o"},{"type":"text","value":"s) are a form of recurrent artificial neural networks that serve as content-addressable (\"associative\") memory systems. Hopfield networks can be trained to store a finite number of patterns (e.g. via Hebbian learning a.k.a. \"fire together - wire together\"). During the training procedure, the weights of the ","key":"hTpzcMRARO"},{"type":"abbreviation","title":"Hopfield Neural Network","children":[{"type":"text","value":"HNN","key":"uneFnv3Mxo"}],"key":"Xn31yGBrKr"},{"type":"text","value":" are trained so that the stored\npatterns become stable attractor states of the network. Thus, when the trained network is presented partial, noisy or corrupted variations of the stored patterns, it can effectively reconstruct the original pattern via an iterative relaxation procedure that converges to the attractor states.\n","key":"pYdHYEWTCK"},{"type":"strong","children":[{"type":"text","value":"B","key":"eo1SiMw9aH"}],"key":"lCsp490PhK"},{"type":"text","value":" We consider regions of the brain as nodes of a Hopfield network. Instead of training the Hopfield network to\nspecific tasks, we set its weights empirically, with the interregional activity flow estimated via functional\nbrain connectivity. Capitalizing on strong analogies between the relaxation rule of Hopfield networks and the\nactivity flow principle that links activity to connectivity in brain networks, we propose the resulting\nfunctional connectome-based Hopfield neural network (","key":"dGK74lZ2Mm"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"mxwc6GH5GO"}],"key":"IjHULw9yfW"},{"type":"text","value":") as a computational model for macro-scale brain dynamics.","key":"EyZH6SjsBN"},{"type":"break","key":"GYwMyG3PvJ"},{"type":"strong","children":[{"type":"text","value":"C","key":"gNN7z5E0bG"}],"key":"E8Oj9BADLw"},{"type":"text","value":" The proposed computational framework assigns an energy level, an attractor state and a position in a\nlow-dimensional embedding to brain activation patterns. Additionally, it models how the entire state-space of viable activation patterns is restricted by the dynamics of the system and how alterations in activity and/or connectivity modify these dynamics.","key":"ZsMlShRxaM"}],"key":"XDFTJyBWZl"}],"key":"hElYFROa3q"}],"enumerator":"1","html_id":"concept","key":"aisHTlxcPg"},{"type":"paragraph","position":{"start":{"line":166,"column":0},"end":{"line":168,"column":0}},"children":[{"type":"text","value":"In the present work, we use ","key":"vYC2So8FU2"},{"type":"abbreviation","title":"Hopfield Neural Network","children":[{"type":"text","value":"HNN","key":"QHJDdQ72yo"}],"key":"JsG0Slt56s"},{"type":"text","value":"s to explore the functional connectome's attractor-dynamics with the aid of a streamlined, low-dimensional representation of the energy landscape.\nSubsequently, we use a diverse set of experimental, clinical and meta-analytic studies to evaluate our model's ability to reconstruct various characteristics of resting state brain dynamics, as well as its capacity to detect and explain changes induced by experimental tasks or alterations in brain disorders.","key":"oLbkk31wvi"}],"key":"TPdvc0itTF"},{"type":"heading","depth":2,"position":{"start":{"line":169,"column":0},"end":{"line":170,"column":0}},"children":[{"type":"text","value":"Results","key":"x8M1kP7M9g"}],"identifier":"results","label":"Results","html_id":"results","implicit":true,"key":"ZgA5mzmKOL"},{"type":"heading","depth":3,"position":{"start":{"line":171,"column":0},"end":{"line":172,"column":0}},"children":[{"type":"text","value":"Connectome-based Hopfield network as a model of brain dynamics","key":"x93XQcmI0T"}],"identifier":"connectome-based-hopfield-network-as-a-model-of-brain-dynamics","label":"Connectome-based Hopfield network as a model of brain dynamics","html_id":"connectome-based-hopfield-network-as-a-model-of-brain-dynamics","implicit":true,"key":"IKaWrBojE3"},{"type":"paragraph","position":{"start":{"line":173,"column":0},"end":{"line":180,"column":0}},"children":[{"type":"text","value":"First, we explored the attractor states of the functional connectome in a sample of n=41 healthy young\nparticipants (","key":"ZNgelbaDnL"},{"type":"crossReference","children":[{"type":"text","value":"study 1","key":"DgswDwDAJI"}],"urlSource":"#tab-samples","identifier":"tab-samples","label":"tab-samples","kind":"table","template":"Table %s","enumerator":"1","resolved":true,"html_id":"tab-samples","remote":true,"url":"/methods","dataUrl":"/methods.json","key":"Fzsa749zW0"},{"type":"text","value":", see Methods ","key":"oGV7z6RK6F"},{"type":"crossReference","children":[{"type":"text","value":"Table ","key":"SfM4cvmSds"},{"type":"text","value":"1","key":"h6LGznsKuk"}],"urlSource":"#tab-samples","identifier":"tab-samples","label":"tab-samples","kind":"table","template":"Table %s","enumerator":"1","resolved":true,"html_id":"tab-samples","remote":true,"url":"/methods","dataUrl":"/methods.json","key":"mzmMtwaxCq"},{"type":"text","value":" for details). We estimated interregional activity flow ","key":"oIRaanQFDQ"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"cole2016activity","identifier":"cole2016activity","children":[{"type":"text","value":"Cole ","key":"Ll7lBj0SXS"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"srXqAwFfis"}],"key":"KchU65wpUL"},{"type":"text","value":", 2016","key":"kapACkxbAn"}],"key":"HJE2I0X966"},{"type":"cite","label":"ito2017cognitive","identifier":"ito2017cognitive","children":[{"type":"text","value":"Ito ","key":"cWC71eE8mz"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"mJwGhv2CeU"}],"key":"RvWZo0JTaG"},{"type":"text","value":", 2017","key":"Q6VOB1hnUV"}],"key":"YN7yUfVanf"}],"key":"dtTubSU1gR"},{"type":"text","value":"\nas the study-level average of regularized partial correlations among the resting state ","key":"WkNkLscCgd"},{"type":"abbreviation","title":"functional Magnetic Resonance Imaging","children":[{"type":"text","value":"fMRI","key":"L1VDFdjOtn"}],"key":"YsTcMdd5fz"},{"type":"text","value":" timeseries of m = 122\nfunctionally defined brain regions (see ","key":"vjPBijz2Gn"},{"type":"crossReference","children":[{"type":"text","value":"Methods","key":"j4guVPNrS1"}],"urlSource":"#Functional-connectome","identifier":"functional-connectome","label":"Functional-connectome","kind":"heading","template":"{name}","resolved":true,"html_id":"functional-connectome","remote":true,"url":"/methods","dataUrl":"/methods.json","key":"aR2yjy0TIs"},{"type":"text","value":" for details). We then used the standardized\nfunctional connectome as the ","key":"YIX3Mw1i5C"},{"type":"inlineMath","value":"w_{ij}","html":"wijw_{ij}wij​","key":"OSIEn7ANSB"},{"type":"text","value":" weights of a fully connected recurrent ","key":"KRzA014yIL"},{"type":"abbreviation","title":"Artificial Neural Network","children":[{"type":"text","value":"ANN","key":"tO0AsiAoS2"}],"key":"FPeG7GiAZv"},{"type":"text","value":", specifically a continuous-state Hopfield network ","key":"P8cHkbShgC"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"hopfield1982neural","identifier":"hopfield1982neural","children":[{"type":"text","value":"Hopfield, 1982","key":"RMrwm8MhzT"}],"key":"h7RV48VW40"},{"type":"cite","label":"koiran1994dynamics","identifier":"koiran1994dynamics","children":[{"type":"text","value":"Koiran, 1994","key":"xUdQNEMI7F"}],"key":"wNcNATDKQo"}],"key":"yNz5kpJwHo"},{"type":"text","value":", consisting of ","key":"I2yoRFedNA"},{"type":"inlineMath","value":"m","html":"mmm","key":"Z32f5Mca0R"},{"type":"text","value":" neural units, each having an activity\n","key":"j3geVFp7cE"},{"type":"inlineMath","value":"a_i \\in [-1,1] \\subset \\mathbb{R})","html":"ai∈[−1,1]⊂R)a_i \\in [-1,1] \\subset \\mathbb{R})ai​∈[−1,1]⊂R)","key":"ScpLa0ciCD"},{"type":"text","value":". Hopfield networks can be initialized by an arbitrary activation pattern (consisting of ","key":"O263ptsPYL"},{"type":"inlineMath","value":"m","html":"mmm","key":"jVqBh1oUqg"},{"type":"text","value":" activation values) and iteratively updated (i.e. \"relaxed\") until their energy converges a local minimum, that is, to one of the finite number of attractor states (see ","key":"kmzyx1dz4Y"},{"type":"crossReference","children":[{"type":"text","value":"Methods","key":"gN1z00SBG7"}],"urlSource":"#connectome-based-hopfield-networks","identifier":"connectome-based-hopfield-networks","label":"connectome-based-hopfield-networks","kind":"heading","template":"{name}","resolved":true,"html_id":"connectome-based-hopfield-networks","remote":true,"url":"/methods","dataUrl":"/methods.json","key":"r71tk2Az4S"},{"type":"text","value":"). The relaxation procedure is based on a simple rule; in each iteration, the activity of a region is constructed as the weighted average of the activities of all other regions, with weights defined by the connectivity between them. The average is then transformed by a sigmoidal activation function, to keep it in the desired [-1,1] interval.\nThis can be expressed by the following equation:","key":"hvHpCMCKVH"}],"key":"ICUVnLMUbc"},{"type":"math","identifier":"hopfield-update","label":"hopfield-update","value":"\\dot{a}_i = S(\\beta \\sum_{j=1}^m w_{ij}a_j - b_i)","html":"a˙i=S(β∑j=1mwijaj−bi)\\dot{a}_i = S(\\beta \\sum_{j=1}^m w_{ij}a_j - b_i)a˙i​=S(βj=1∑m​wij​aj​−bi​)","enumerator":"1","html_id":"hopfield-update","key":"NLf8tBmhog"},{"type":"paragraph","position":{"start":{"line":186,"column":0},"end":{"line":189,"column":0}},"children":[{"type":"text","value":"where ","key":"xFdMMLhUC3"},{"type":"inlineMath","value":"\\dot{a}_i","html":"a˙i\\dot{a}_ia˙i​","key":"iCZRBWRW8T"},{"type":"text","value":" is the activity of neural unit ","key":"ApI1LR04Gv"},{"type":"inlineMath","value":"i","html":"iii","key":"gv2CUJJeg0"},{"type":"text","value":" in the next iteration and ","key":"c7WGDGCQ22"},{"type":"inlineMath","value":"S(a_j)","html":"S(aj)S(a_j)S(aj​)","key":"LIXEw7gOve"},{"type":"text","value":" is the sigmoidal activation\nfunction (","key":"KXoCPUcwmx"},{"type":"inlineMath","value":"S(a) = tanh(a)","html":"S(a)=tanh(a)S(a) = tanh(a)S(a)=tanh(a)","key":"gW0mnLYngb"},{"type":"text","value":" in our implementation) and ","key":"YSzwD73vLE"},{"type":"inlineMath","value":"b_i","html":"bib_ibi​","key":"jDp5brdLLj"},{"type":"text","value":" is the bias of unit ","key":"FtOZXEkziV"},{"type":"inlineMath","value":"i","html":"iii","key":"HJymK4EpsV"},{"type":"text","value":" and ","key":"fNcg7h4KY1"},{"type":"inlineMath","value":"\\beta","html":"β\\betaβ","key":"pxjFJr4RrG"},{"type":"text","value":" is the so-called temperature parameter. For the sake of simplicity, we set ","key":"W1zlIeQRv2"},{"type":"inlineMath","value":"b_i=0","html":"bi=0b_i=0bi​=0","key":"VKhZsAzmpu"},{"type":"text","value":" in all our experiments. We refer to this architecture as a functional connectivity-based Hopfield Neural Network (","key":"VJVSIjqaDb"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"pAkKwynW3I"}],"key":"gJA37hs0si"},{"type":"text","value":").\nThe relaxation of a ","key":"N7FCyGAL2S"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"Vwg8IvXQr1"}],"key":"gIiUDzoNxf"},{"type":"text","value":" model can be conceptualized as the repeated application of the activity flow principle ","key":"SceM5jGXUb"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"cole2016activity","identifier":"cole2016activity","children":[{"type":"text","value":"Cole ","key":"DI3o8wIabz"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"wOUi4Mcu5G"}],"key":"BtrCY6fs6d"},{"type":"text","value":", 2016","key":"yhrJRvRRh3"}],"key":"Ftj2Jsb9oU"},{"type":"cite","label":"ito2017cognitive","identifier":"ito2017cognitive","children":[{"type":"text","value":"Ito ","key":"iJ6OCKy5Zn"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"mKbJPuuBJU"}],"key":"thTDQVRjPs"},{"type":"text","value":", 2017","key":"W6LU2a6Xc6"}],"key":"o2YjnDAEmH"}],"key":"ppc23rCerX"},{"type":"text","value":" , simultaneously for all regions: ","key":"HDtVUMHzie"},{"type":"inlineMath","value":"\\dot{a}_i = \\sum_{j=1}^m w_{ij}a_j","html":"a˙i=∑j=1mwijaj\\dot{a}_i = \\sum_{j=1}^m w_{ij}a_ja˙i​=∑j=1m​wij​aj​","key":"kw9dP1kCtA"},{"type":"text","value":". The update rule also exhibits analogies with network control theory ","key":"TporMeyHFo"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"gu2015controllability","identifier":"gu2015controllability","children":[{"type":"text","value":"Gu ","key":"eNZ7UdRsN0"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"Y2aFAVLvbf"}],"key":"WE6RwhH7yG"},{"type":"text","value":", 2015","key":"k4Z7DgYpZJ"}],"key":"p9LtFWLGVW"}],"key":"uClxscNFG3"},{"type":"text","value":" and the inner workings of neural mass models, as applied e.g. in dynamic causal modeling ","key":"dlIx7glUzl"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"daunizeau2012stochastic","identifier":"daunizeau2012stochastic","children":[{"type":"text","value":"Daunizeau ","key":"jaaQB1E0z4"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"pK4RhLLzgg"}],"key":"mD4jV8hefT"},{"type":"text","value":", 2012","key":"agwcW8huxX"}],"key":"vycsAcesyn"}],"key":"JV8NT42rGB"},{"type":"text","value":".","key":"AaiayUm6aw"}],"key":"OwCP4DxYEq"},{"type":"paragraph","position":{"start":{"line":190,"column":0},"end":{"line":193,"column":0}},"children":[{"type":"text","value":"Hopfield networks assign an energy value to each possible activity configuration ","key":"K5TDo00IOg"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"hopfield1982neural","identifier":"hopfield1982neural","children":[{"type":"text","value":"Hopfield, 1982","key":"vOMntx8Tw7"}],"key":"HRMQ2wSCWK"},{"type":"cite","label":"koiran1994dynamics","identifier":"koiran1994dynamics","children":[{"type":"text","value":"Koiran, 1994","key":"sz0stUTgeZ"}],"key":"WsgkwcJLgN"}],"key":"F0IFsbPfJ8"},{"type":"text","value":", which decreases during the relaxation procedure until reaching an equilibrium state with minimal energy (","key":"aCBUI7LdeU"},{"type":"crossReference","kind":"figure","identifier":"attractors","label":"attractors","children":[{"type":"text","value":"Figure ","key":"bDMxlavOSR"},{"type":"text","value":"2","key":"M1aWmc7pAQ"}],"template":"Figure %s","enumerator":"2","resolved":true,"html_id":"attractors","key":"I0S2DXub6A"},{"type":"text","value":"A, top panel).\nWe used a large number of random initializations to obtain all possible attractor states of the connectome-based\nHopfield network in study 1 (","key":"KAH8C7uQTv"},{"type":"crossReference","kind":"figure","identifier":"attractors","label":"attractors","children":[{"type":"text","value":"Figure ","key":"c3KX7AF9CU"},{"type":"text","value":"2","key":"Uz9E9HZ4JQ"}],"template":"Figure %s","enumerator":"2","resolved":true,"html_id":"attractors","key":"CzvE7NbiWt"},{"type":"text","value":"A, bottom panel).","key":"BIKBiEfxxU"}],"key":"RmrOTDBYxb"},{"type":"container","kind":"figure","identifier":"attractors","label":"attractors","children":[{"type":"image","url":"/embedding_method-78f8071a1c322ba464e9c83b5777e057.png","alt":"Attractor states and state-space dynamics of connectome-based Hopfield networks

\nA Top: During so-called relaxation procedure, activities in the nodes of an fcHNN model are iteratively updated based on the activity of all other regions and the connectivity between them. The energy of a\nconnectome-based Hopfield network decreases during the relaxation procedure until reaching an equilibrium state with\nminimal energy, i.e. an attractor state. Bottom: Four attractor states of the fcHNN derived from the\ngroup-level functional connectivity matrix from study 1 (n=44).\nB Top: In presence of weak noise (stochastic update), the system\ndoes not converge to equilibrium anymore. Instead, activity transverses on the state landscape in a way\nrestricted by the topology of the connectome and the \"gravitational pull\" of the attractor states. Bottom: We sample\nthe \"state landscape\" by running the stochastic relaxation procedure for an extended amount of time (e.g. 100.000 consecutive\nstochastic updates), each point representing an activation configuration or state. To construct a\nlow-dimensional representation of the state space, we take the first two principal components of the simulated activity\npatterns. The first two principal components explain approximately 58-85% of the variance of state energy (depending\non the noise parameter \\sigma, see Supplementary Figure %s).\nC We map all states of the state space sample to their corresponding attractor state, with the conventional\nHopfield relaxation procedure (A). The four attractor states are also visualized in their corresponding position on the\nPCA-based projection. The first two principal components yield a clear separation of the attractive state basins\n(cross-validated classification accuracy: 95.5%, Supplementary Figure %s). We refer to the resulting visualization\nas the fcHNN projection and use it to visualize fcHNN-derived and empirical brain dynamics throughout the rest of\nthe manuscript.\nE At its simplest form, the fcHNN framework entails only two free hyperparameters: the temperature parameter\n\\beta (left) that controls the number of attractor states and the noise parameter of the stochastic relaxation\n\\sigma. To avoid overfitting these parameters to the empirical data, we set \\beta=0.04 and \\sigma=0.37 for the\nrest of the paper (dotted boxes).","urlSource":"figures/embedding_method.png","urlOptimized":"/embedding_method-78f8071a1c322ba464e9c83b5777e057.webp","key":"fRugCce2aW"},{"type":"caption","children":[{"type":"paragraph","position":{"start":{"line":196,"column":0},"end":{"line":219,"column":0}},"children":[{"type":"captionNumber","kind":"figure","label":"attractors","identifier":"attractors","html_id":"attractors","enumerator":"2","children":[{"type":"text","value":"Figure ","key":"xweMrxmCRS"},{"type":"text","value":"2","key":"ySi4WKgeMS"},{"type":"text","value":":","key":"o5D4rLoAfY"}],"template":"Figure %s:","key":"QX1E4UNm78"},{"type":"strong","children":[{"type":"text","value":"Attractor states and state-space dynamics of connectome-based Hopfield networks","key":"aC4kUgNJK2"}],"key":"SskashBdSl"},{"type":"text","value":" ","key":"iyqP0acYgv"},{"type":"break","key":"zdtz2lfQNG"},{"type":"break","key":"XEc1RDAv77"},{"type":"text","value":"\n","key":"iBOPtie8wO"},{"type":"strong","children":[{"type":"text","value":"A","key":"JIo4pPqqz2"}],"key":"H8POLQdbtl"},{"type":"text","value":" Top: During so-called relaxation procedure, activities in the nodes of an ","key":"OYtSHN6C6G"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"ifrhmtiHQV"}],"key":"Y9PaxjxUbb"},{"type":"text","value":" model are iteratively updated based on the activity of all other regions and the connectivity between them. The energy of a\nconnectome-based Hopfield network decreases during the relaxation procedure until reaching an equilibrium state with\nminimal energy, i.e. an attractor state. Bottom: Four attractor states of the ","key":"yBY6PGvOrf"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"lTvITKhdz6"}],"key":"oEv0Xe90Zr"},{"type":"text","value":" derived from the\ngroup-level functional connectivity matrix from ","key":"gd1nAt0S46"},{"type":"crossReference","children":[{"type":"text","value":"study 1","key":"YRqRGFFcP0"}],"urlSource":"#tab-samples","identifier":"tab-samples","label":"tab-samples","kind":"table","template":"Table %s","enumerator":"1","resolved":true,"html_id":"tab-samples","remote":true,"url":"/methods","dataUrl":"/methods.json","key":"B44Rahh6vc"},{"type":"text","value":" (n=44).\n","key":"HKLvXRkTT2"},{"type":"strong","children":[{"type":"text","value":"B","key":"AviXxvJQyn"}],"key":"xkFhXCeOlB"},{"type":"text","value":" Top: In presence of weak noise (stochastic update), the system\ndoes not converge to equilibrium anymore. Instead, activity transverses on the state landscape in a way\nrestricted by the topology of the connectome and the \"gravitational pull\" of the attractor states. Bottom: We sample\nthe \"state landscape\" by running the stochastic relaxation procedure for an extended amount of time (e.g. 100.000 consecutive\nstochastic updates), each point representing an activation configuration or state. To construct a\nlow-dimensional representation of the state space, we take the first two principal components of the simulated activity\npatterns. The first two principal components explain approximately 58-85% of the variance of state energy (depending\non the noise parameter ","key":"P0DBulx3do"},{"type":"inlineMath","value":"\\sigma","html":"σ\\sigmaσ","key":"xtebldvOJ3"},{"type":"text","value":", see ","key":"u5IEKAdjHa"},{"type":"crossReference","kind":"figure","identifier":"si_expl_variance_energy","label":"si_expl_variance_energy","children":[{"type":"text","value":"Supplementary Figure ","key":"Tcjz6pwQAr"},{"type":"text","value":"1","key":"M1PNRrj8eQ"}],"template":"Figure %s","enumerator":"1","resolved":true,"html_id":"si-expl-variance-energy","remote":true,"url":"/supplement","dataUrl":"/supplement.json","key":"at6WohnJNs"},{"type":"text","value":").\n","key":"MX7mqzxVz7"},{"type":"strong","children":[{"type":"text","value":"C","key":"cWAu5paujy"}],"key":"Sz4iqjTFZg"},{"type":"text","value":" We map all states of the state space sample to their corresponding attractor state, with the conventional\nHopfield relaxation procedure (A). The four attractor states are also visualized in their corresponding position on the\n","key":"IZCApd37tU"},{"type":"abbreviation","title":"Principal Component","children":[{"type":"text","value":"PC","key":"T3Kq23p2i0"}],"key":"w8lPKMdgin"},{"type":"text","value":"A-based projection. The first two principal components yield a clear separation of the attractive state basins\n(cross-validated classification accuracy: 95.5%, ","key":"dngr894yJ1"},{"type":"crossReference","kind":"figure","identifier":"si_classification_acc_state_basins","label":"si_classification_acc_state_basins","children":[{"type":"text","value":"Supplementary Figure ","key":"Mkazck5MNd"},{"type":"text","value":"2","key":"ndMragFWoI"}],"template":"Figure %s","enumerator":"2","resolved":true,"html_id":"si-classification-acc-state-basins","remote":true,"url":"/supplement","dataUrl":"/supplement.json","key":"CHeEJdrXQ2"},{"type":"text","value":"). We refer to the resulting visualization\nas the ","key":"AQf5ruV5qO"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"WmM33FjuLc"}],"key":"p7QMgHtwdu"},{"type":"text","value":" projection and use it to visualize ","key":"X1XjZnS6Xi"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"qyb3tYs6zr"}],"key":"nJQ0ErbKm2"},{"type":"text","value":"-derived and empirical brain dynamics throughout the rest of\nthe manuscript.\n","key":"w0fyPWQmAj"},{"type":"strong","children":[{"type":"text","value":"E","key":"vukdDcJPca"}],"key":"eeKlL54ZtQ"},{"type":"text","value":" At its simplest form, the ","key":"Xfl0GBW7SY"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"KRZ6xgBk4G"}],"key":"MHolRYPAHM"},{"type":"text","value":" framework entails only two free hyperparameters: the temperature parameter\n","key":"MUFGgnZb82"},{"type":"inlineMath","value":"\\beta","html":"β\\betaβ","key":"xz8UjdxjIY"},{"type":"text","value":" (left) that controls the number of attractor states and the noise parameter of the stochastic relaxation\n","key":"syoVflyPae"},{"type":"inlineMath","value":"\\sigma","html":"σ\\sigmaσ","key":"ElZ9fXBJq3"},{"type":"text","value":". To avoid overfitting these parameters to the empirical data, we set ","key":"S17KJp8gf5"},{"type":"inlineMath","value":"\\beta=0.04","html":"β=0.04\\beta=0.04β=0.04","key":"d54W5ax5Wf"},{"type":"text","value":" and ","key":"I98yEjOUsU"},{"type":"inlineMath","value":"\\sigma=0.37","html":"σ=0.37\\sigma=0.37σ=0.37","key":"rXuTQlRoyG"},{"type":"text","value":" for the\nrest of the paper (dotted boxes).","key":"gaOrDeIX6M"}],"key":"Oi63UwSII8"}],"key":"IfWM8qKXjl"}],"enumerator":"2","html_id":"attractors","key":"coOGd6PyvU"},{"type":"paragraph","position":{"start":{"line":221,"column":0},"end":{"line":225,"column":0}},"children":[{"type":"text","value":"Consistent with theoretical expectations, we observed that increasing the temperature parameter ","key":"PRHlb77zA2"},{"type":"inlineMath","value":"\\beta","html":"β\\betaβ","key":"mLS73wMQJf"},{"type":"text","value":" led to an\nincreasing number of attractor states (","key":"StZUoIn2Y4"},{"type":"crossReference","kind":"figure","identifier":"attractors","label":"attractors","children":[{"type":"text","value":"Figure ","key":"ErguPnRZ0F"},{"type":"text","value":"2","key":"SHpbpWEGWe"}],"template":"Figure %s","enumerator":"2","resolved":true,"html_id":"attractors","key":"BB9Ov3Ur5y"},{"type":"text","value":"E, left, ","key":"xovylxhUiY"},{"type":"crossReference","kind":"figure","identifier":"si_att_state_emergence_over_beta","label":"si_att_state_emergence_over_beta","children":[{"type":"text","value":"Supplementary Figure ","key":"VNLfwD95JP"},{"type":"text","value":"3","key":"XdpbX1hvSw"}],"template":"Figure %s","enumerator":"3","resolved":true,"html_id":"si-att-state-emergence-over-beta","remote":true,"url":"/supplement","dataUrl":"/supplement.json","key":"yLNLjReBQL"},{"type":"text","value":"), appearing in symmetric pairs\n(i.e. ","key":"GWImAyM22p"},{"type":"inlineMath","value":"a_i^{(1)} = -a_i^{(2)}","html":"ai(1)=−ai(2)a_i^{(1)} = -a_i^{(2)}ai(1)​=−ai(2)​","key":"h3E2NqTWCA"},{"type":"text","value":"). For simplicity, we set the temperature parameter for the rest of the paper to a value\nresulting in 4 distinct attractor states (","key":"AX7oLbPMFp"},{"type":"inlineMath","value":"\\beta=0.4","html":"β=0.4\\beta=0.4β=0.4","key":"AtTInk5RDv"},{"type":"text","value":").","key":"u6CjajTa5T"}],"key":"tim7QBRn7d"},{"type":"paragraph","position":{"start":{"line":226,"column":0},"end":{"line":229,"column":0}},"children":[{"type":"text","value":"Fc","key":"RJgJukfPfe"},{"type":"abbreviation","title":"Hopfield Neural Network","children":[{"type":"text","value":"HNN","key":"l0eWgQdiTr"}],"key":"ua9xyNtdaM"},{"type":"text","value":"s, without any modifications, always converge to an equilibrium state.\nTo incorporate stochastic fluctuations in neuronal activity ","key":"ZmjoNhQYYf"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"robinson2005multiscale","identifier":"robinson2005multiscale","children":[{"type":"text","value":"Robinson ","key":"Xmw6lhfy0d"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"Z1NRUGvQ4g"}],"key":"kgTQdzfDHd"},{"type":"text","value":", 2005","key":"tKP35dcdEK"}],"key":"LjnJBoaEZA"}],"key":"P7ii2p9le4"},{"type":"text","value":", we introduced weak\nGaussian noise to the ","key":"nFRkxBFBRZ"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"C8r9srQWdN"}],"key":"yBQdFqzckw"},{"type":"text","value":" relaxation procedure. This procedure, referred to as stochastic relaxation, prevents the system from reaching equilibrium and, somewhat similarly to stochastic DCM ","key":"QYvSywLXha"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"daunizeau2012stochastic","identifier":"daunizeau2012stochastic","children":[{"type":"text","value":"Daunizeau ","key":"xgGpsknuq9"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"x3fTStNYPj"}],"key":"hS5iPNbENX"},{"type":"text","value":", 2012","key":"MYLq9m0Kxu"}],"key":"ncsuSD7uwi"}],"key":"tSHiiKALVO"},{"type":"text","value":", induces complex system dynamics (","key":"fuY4RcFqlQ"},{"type":"crossReference","kind":"figure","identifier":"attractors","label":"attractors","children":[{"type":"text","value":"Figure ","key":"M82vbOy9FV"},{"type":"text","value":"2","key":"G2wBtjutiZ"}],"template":"Figure %s","enumerator":"2","resolved":true,"html_id":"attractors","key":"mhDy10sGMB"},{"type":"text","value":"B).","key":"viCFseNZis"}],"key":"WnzmdQaCBx"},{"type":"paragraph","position":{"start":{"line":230,"column":0},"end":{"line":236,"column":0}},"children":[{"type":"text","value":"In order to enhance interpretability, we obtained the first two principal components (","key":"JJ2pOq3P5H"},{"type":"abbreviation","title":"Principal Component","children":[{"type":"text","value":"PC","key":"FW16WlXzlx"}],"key":"fI4i1Ocdqh"},{"type":"text","value":"s) of the states sampled from the stochastic relaxation procedure.\nThe resulting two-dimensional embedding (","key":"xDDmCkUF3G"},{"type":"crossReference","kind":"figure","identifier":"attractors","label":"attractors","children":[{"type":"text","value":"Figure ","key":"qv5aIpUH8I"},{"type":"text","value":"2","key":"xv4NyeLqY7"}],"template":"Figure %s","enumerator":"2","resolved":true,"html_id":"attractors","key":"HqPRpJOg0n"},{"type":"text","value":"B, bottom plot) exhibited high consistency across different values of ","key":"SkYWJpwjXC"},{"type":"inlineMath","value":"\\beta","html":"β\\betaβ","key":"XETjJpPWTK"},{"type":"text","value":" and ","key":"BgHHtpYLXm"},{"type":"inlineMath","value":"\\sigma","html":"σ\\sigmaσ","key":"hhTxppgbeg"},{"type":"text","value":" (","key":"bVwwEcoTzn"},{"type":"crossReference","kind":"figure","identifier":"attractors","label":"attractors","children":[{"type":"text","value":"Figure ","key":"qumbq3mQE6"},{"type":"text","value":"2","key":"cyBrT47lt6"}],"template":"Figure %s","enumerator":"2","resolved":true,"html_id":"attractors","key":"pGcvjVHyRW"},{"type":"text","value":"E).\nFor all subsequent analyses, we set ","key":"z6WpX0F1Xu"},{"type":"inlineMath","value":"\\sigma=0.37","html":"σ=0.37\\sigma=0.37σ=0.37","key":"NlIrUfP5Ir"},{"type":"text","value":" (based a coarse optimization procedure aimed at reconstructing the bimodal distribution of empirical data, ","key":"O7uDVhqBmK"},{"type":"crossReference","kind":"figure","identifier":"attractors","label":"attractors","children":[{"type":"text","value":"Figure ","key":"b0v0BvznUG"},{"type":"text","value":"2","key":"rgEkTfzh06"}],"template":"Figure %s","enumerator":"2","resolved":true,"html_id":"attractors","key":"weum11UVbv"},{"type":"text","value":"E right). On the low-dimensional embedding, which we refer to as the ","key":"Er2kRmI19J"},{"type":"emphasis","children":[{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"L6ryb7ZaQ7"}],"key":"peJaY2bzvp"},{"type":"text","value":" projection","key":"FkOax2a0VR"}],"key":"GYlxZLT1ST"},{"type":"text","value":", we observed a clear separation of the attractor states (","key":"rP1N5x0rXF"},{"type":"crossReference","kind":"figure","identifier":"attractors","label":"attractors","children":[{"type":"text","value":"Figure ","key":"GmwygSSZDl"},{"type":"text","value":"2","key":"sJwJF87TeD"}],"template":"Figure %s","enumerator":"2","resolved":true,"html_id":"attractors","key":"qeFHZVMFb0"},{"type":"text","value":"C), with the two symmetric pairs of attractor states located at the extremes of the first and second ","key":"HHncj0LOVN"},{"type":"abbreviation","title":"Principal Component","children":[{"type":"text","value":"PC","key":"fm9ShPnxFT"}],"key":"Ql3DiZJZt7"},{"type":"text","value":".\nTo map the attractor basins on the space spanned by the first two ","key":"nYBcJN621T"},{"type":"abbreviation","title":"Principal Component","children":[{"type":"text","value":"PC","key":"dGf9grIBOa"}],"key":"aLI5089b4e"},{"type":"text","value":"s (","key":"bSIZfrncFN"},{"type":"crossReference","kind":"figure","identifier":"attractors","label":"attractors","children":[{"type":"text","value":"Figure ","key":"paqI0G72zk"},{"type":"text","value":"2","key":"anKE7Uteip"}],"template":"Figure %s","enumerator":"2","resolved":true,"html_id":"attractors","key":"QCnm04M2b6"},{"type":"text","value":"C), we obtained the attractor state of each point visited during the stochastic relaxation and fit a multinomial logistic regression model to predict the attractor state from the first two ","key":"NbPF7f9oS1"},{"type":"abbreviation","title":"Principal Component","children":[{"type":"text","value":"PC","key":"hyEYXbyUAI"}],"key":"WTOlK1xjbh"},{"type":"text","value":"s.\nThe resulting model accurately predicted attractor states of arbitrary brain activity patterns, achieving a cross-validated accuracy of 96.5%.\nThe attractor basins were visualized by using the decision boundaries obtained from this model. (","key":"GpHnP7VMTr"},{"type":"crossReference","kind":"figure","identifier":"attractors","label":"attractors","children":[{"type":"text","value":"Figure ","key":"WOnAK8FZaA"},{"type":"text","value":"2","key":"HtaZHizSYi"}],"template":"Figure %s","enumerator":"2","resolved":true,"html_id":"attractors","key":"BOhvH7hrKK"},{"type":"text","value":"C). We propose the 2-dimensional ","key":"M6c4lTSz1T"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"C1QiQc716Q"}],"key":"XLDZU619H7"},{"type":"text","value":" projection depicted on (","key":"b5QUsRDbZU"},{"type":"crossReference","kind":"figure","identifier":"attractors","label":"attractors","children":[{"type":"text","value":"Figure ","key":"CwJGgkNDxX"},{"type":"text","value":"2","key":"KuF5l0nWvr"}],"template":"Figure %s","enumerator":"2","resolved":true,"html_id":"attractors","key":"RtKoo0nJhL"},{"type":"text","value":"C) as a simplified representation of brain dynamics, and use it as a basis for all subsequent analyses in this work.","key":"PbFyOPg6Tw"}],"key":"JPCCKCDop4"},{"type":"heading","depth":3,"position":{"start":{"line":237,"column":0},"end":{"line":238,"column":0}},"children":[{"type":"text","value":"Reconstruction of resting state brain dynamics","key":"YHrV3QR6V1"}],"identifier":"reconstruction-of-resting-state-brain-dynamics","label":"Reconstruction of resting state brain dynamics","html_id":"reconstruction-of-resting-state-brain-dynamics","implicit":true,"key":"G4pEEHApfd"},{"type":"paragraph","position":{"start":{"line":239,"column":0},"end":{"line":241,"column":0}},"children":[{"type":"text","value":"The spatial patterns of the obtained attractor states exhibit high neuroscientific relevance and closely resemble previously described large-scale brain systems. 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The first pair of attractors (mapped on ","key":"vp9FJZX7lH"},{"type":"abbreviation","title":"Principal Component","children":[{"type":"text","value":"PC","key":"NSB7OUimMh"}],"key":"dHAgzDBfmg"},{"type":"text","value":"1, horizontal axis) resemble the two complementary “macro” systems described, among others, by ","key":"deiLN0y03G"},{"type":"citeGroup","kind":"narrative","children":[{"type":"cite","label":"golland2008data","identifier":"golland2008data","children":[{"type":"text","value":"Golland ","key":"ouVUOYuOVr"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"ghPUvPKNHL"}],"key":"VfMX3hsawC"},{"type":"text","value":" (2008)","key":"I4FFFjpQVU"}],"key":"VrM3Ru22rh"}],"key":"XyzmUOXDnW"},{"type":"text","value":" and ","key":"mM6gE09Siz"},{"type":"citeGroup","kind":"narrative","children":[{"type":"cite","label":"cioli2014differences","identifier":"cioli2014differences","children":[{"type":"text","value":"Cioli ","key":"kXEmdvaVnP"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"DzIdXTdPCy"}],"key":"muBQvKdTx7"},{"type":"text","value":" (2014)","key":"LDasb6QUPn"}],"key":"FgPI0jBisH"}],"key":"a1o5YCMesb"},{"type":"text","value":" as well as the two \"primary\" brain states observed by ","key":"pxiAb9F6vj"},{"type":"citeGroup","kind":"narrative","children":[{"type":"cite","label":"chen2018human","identifier":"chen2018human","children":[{"type":"text","value":"Chen ","key":"xzWB2XvpaU"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"Uvf6NSy7ft"}],"key":"EiViMM5Sjd"},{"type":"text","value":" (2018)","key":"pFSHt5UJQr"}],"key":"KuDzohE1DG"}],"key":"FR8aVuWnLR"},{"type":"text","value":" and the 'unimodal-to-transmodal' principal gradient of ","key":"GfHm4JjkyP"},{"type":"citeGroup","kind":"narrative","children":[{"type":"cite","label":"margulies2016situating","identifier":"margulies2016situating","children":[{"type":"text","value":"Margulies ","key":"DCwTeHSULq"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"HPTLtUBpwa"}],"key":"z7KpCmEENW"},{"type":"text","value":" (2016)","key":"ZoiOo58mM8"}],"key":"Cm2GACkMaI"}],"key":"aH8wTEgjSy"},{"type":"text","value":" and ","key":"ro4FmRr0TI"},{"type":"citeGroup","kind":"narrative","children":[{"type":"cite","label":"huntenburg2018large","identifier":"huntenburg2018large","children":[{"type":"text","value":"Huntenburg ","key":"mqfYH5Dbl3"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"noYECegHlT"}],"key":"Cle22TH9Tr"},{"type":"text","value":" (2018)","key":"HLjxxrmdvM"}],"key":"vXPgBSCXry"}],"key":"cV0w86bS80"},{"type":"text","value":". A common interpretation of these two patterns is that they represent (i) an “extrinsic” system linked to the immediate sensory environment and (ii) an \"intrinsic\" system for higher-level internal context.\nThe other pair of attractors spans an orthogonal axis, and resemble to patterns commonly associated with perception–action cycles ","key":"QPtyjOTo5T"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"fuster2004upper","identifier":"fuster2004upper","children":[{"type":"text","value":"Fuster, 2004","key":"gZjtQfefQw"}],"key":"wdC7skEFP1"}],"key":"DCz8QOYTHv"},{"type":"text","value":", and described as a gradient across sensory-motor modalities ","key":"ZI8fR6XQPL"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"huntenburg2018large","identifier":"huntenburg2018large","children":[{"type":"text","value":"Huntenburg ","key":"KRK9RlmpNB"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"K1Fi8ysoto"}],"key":"vY17vMMzvy"},{"type":"text","value":", 2018","key":"s715vivJuV"}],"key":"vnKrw45lXt"}],"key":"XRVVwFrEGE"},{"type":"text","value":", recruiting regions associated with active inference (e.g. motor cortices) and perceptual inference (e.g visual areas).","key":"KPuhQmlAVq"}],"key":"ZhEft9NdZ9"},{"type":"container","kind":"figure","identifier":"rest-validity","label":"rest-validity","children":[{"type":"image","url":"/face_validity-85a51599b8224a0d26a8283f9370e3a0.png","alt":"Connectome-based Hopfield networks reconstruct characteristics of real resting state brain activity.

\nA The four attractor states of the fcHNN model from study 1 reflect brain activation\npatterns with high neuroscientific relevance, representing sub-systems previously associated with \"internal context\"\n(blue), \"external context\" (yellow), \"action\" (red) and \"perception\" (green)\n.\nB The attractor states show excellent replicability in two external datasets (study 2 and 3, mean correlation 0.93).\nC The fcHNN projection (first two PCs of the fcHNN state space) explains significantly more variance (p<0.0001) in the real\nresting state fMRI data than principal components derived from the real resting state data itself and generalizes\nbetter (p<0.0001) to out-of-sample data (study 2). Error bars denote 99% bootstrapped confidence intervals.\nD The fcHNN of study 1 seeded with real activation maps (gray dots) of an example participant. All activation maps converge to one of the four attractor states during the relaxation procedure (without noise) and the system reaches equilibrium. Trajectories are colored by attractor state.\nE Illustration of the stochastic relaxation procedure in the same fcHNN model, seeded from a single starting point (activation pattern). The system does not converge to an attractor state but instead transverses the state space in a way restricted by the topology of the connectome and the \"gravitational pull\" of the attractor states. The shade of the trajectory changes with increasing number of iterations. The trajectory is smoothed with a moving average over 10 iterations for visualization purposes.\nF Real resting state fMRI data of an example participant from study 1, plotted on the fcHNN projection. The shade of the trajectory changes with increasing number of iterations.\nG Flow map of the mean trajectories (i.e. the timeframe-to-timeframe transition direction) in fcHNN-generated data, as compared to a shuffled null model representing zero temporal autocorrelation. The flow map reveals that the \"gravitational pull\" of the attractor states gives rise to a characteristic temporal autocorrelation structure.\nH A similar pattern can be found in real data (flow analysis of all participants from study 1 pooled, as compared to a shuffled null model representing zero temporal autocorrelation).\nI The fcHNN analysis accurately predicts (p<0.0001) the fraction of time spent on the basis of the four attractor\nstates in real restring state fMRI data (study 1) and,\nJ, reconstructs the characteristic bimodal distribution of the real resting state data.\nK Stochastic fcHNNs are capable of self-reconstruction: the timeseries resulting from the stochastic relaxation procedure\nmirror the co-variance structure of the functional connectome the fcHNN model was initialized 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Error bars denote 99% bootstrapped confidence intervals.\n","key":"VDwL5K68Fk"},{"type":"strong","children":[{"type":"text","value":"D","key":"xm50w07LC4"}],"key":"VrNXPDkVq0"},{"type":"text","value":" The ","key":"cd50dUtjod"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"OtFbqNSOPb"}],"key":"t1zAFam3G2"},{"type":"text","value":" of study 1 seeded with real activation maps (gray dots) of an example participant. All activation maps converge to one of the four attractor states during the relaxation procedure (without noise) and the system reaches equilibrium. Trajectories are colored by attractor state.\n","key":"pR0zCI8qH2"},{"type":"strong","children":[{"type":"text","value":"E","key":"ELf0odsbWu"}],"key":"DSAskhWFB9"},{"type":"text","value":" Illustration of the stochastic relaxation procedure in the same ","key":"JT5Xh4PZCl"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"FK4uPw1cAo"}],"key":"uKNwZANP39"},{"type":"text","value":" model, seeded from a single starting point (activation pattern). The system does not converge to an attractor state but instead transverses the state space in a way restricted by the topology of the connectome and the \"gravitational pull\" of the attractor states. The shade of the trajectory changes with increasing number of iterations. The trajectory is smoothed with a moving average over 10 iterations for visualization purposes.\n","key":"jy72nxJnnN"},{"type":"strong","children":[{"type":"text","value":"F","key":"lhqRHxC5Kp"}],"key":"N2jQFxJFkx"},{"type":"text","value":" Real resting state ","key":"dbMVrDKmfx"},{"type":"abbreviation","title":"functional Magnetic Resonance Imaging","children":[{"type":"text","value":"fMRI","key":"vllVIqIDfR"}],"key":"N1RAVzKMhZ"},{"type":"text","value":" data of an example participant from study 1, plotted on the ","key":"xXug8pRe2d"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"EUws06DPst"}],"key":"G2RR00J1vO"},{"type":"text","value":" projection. The shade of the trajectory changes with increasing number of iterations.\n","key":"jTopqaFWUg"},{"type":"strong","children":[{"type":"text","value":"G","key":"KblGmEKnnY"}],"key":"axertk3OpR"},{"type":"text","value":" Flow map of the mean trajectories (i.e. the timeframe-to-timeframe transition direction) in ","key":"UWtS94mCO9"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"tErrAVquRf"}],"key":"zjNP4fVdFd"},{"type":"text","value":"-generated data, as compared to a shuffled null model representing zero temporal autocorrelation. The flow map reveals that the \"gravitational pull\" of the attractor states gives rise to a characteristic temporal autocorrelation structure.\n","key":"VJSrZ7MQkt"},{"type":"strong","children":[{"type":"text","value":"H","key":"jbPW0JplFM"}],"key":"HnERVa9vvM"},{"type":"text","value":" A similar pattern can be found in real data (flow analysis of all participants from study 1 pooled, as compared to a shuffled null model representing zero temporal autocorrelation).\n","key":"VBYqHY2xhs"},{"type":"strong","children":[{"type":"text","value":"I","key":"YKeXgGBgjx"}],"key":"Ko8iyWTDxQ"},{"type":"text","value":" The ","key":"fOUWne7zxn"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"vBiCEbGwco"}],"key":"dzQM0aII0y"},{"type":"text","value":" analysis accurately predicts (p<0.0001) the fraction of time spent on the basis of the four attractor\nstates in real restring state ","key":"q46I8SPf9O"},{"type":"abbreviation","title":"functional Magnetic Resonance Imaging","children":[{"type":"text","value":"fMRI","key":"BTsN119PUK"}],"key":"CYOfR46wCM"},{"type":"text","value":" data (study 1) and,\n","key":"CV0Ui2jbfh"},{"type":"strong","children":[{"type":"text","value":"J","key":"dRo6FcLblU"}],"key":"xAdBOXhUOy"},{"type":"text","value":", reconstructs the characteristic bimodal distribution of the real resting state data.\n","key":"spLm7EblOW"},{"type":"strong","children":[{"type":"text","value":"K","key":"vobrpSqudP"}],"key":"qF2mI9eT7x"},{"type":"text","value":" Stochastic ","key":"hG48vixj9T"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"maCDpZ7r4t"}],"key":"kYFKqGkPia"},{"type":"text","value":"s are capable of self-reconstruction: the timeseries resulting from the stochastic relaxation procedure\nmirror the co-variance structure of the functional connectome the ","key":"ypXI2zMhEr"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"O4k3k6jweA"}],"key":"qbBJRal04n"},{"type":"text","value":" model was initialized with.","key":"FHpVPh9sBj"}],"key":"bMBFwnjVFG"}],"key":"hCtYohMzvD"}],"enumerator":"3","html_id":"rest-validity","key":"zHmNNbArK7"},{"type":"paragraph","position":{"start":{"line":265,"column":0},"end":{"line":268,"column":0}},"children":[{"type":"text","value":"The discovered attractor states demonstrate remarkable replicability (mean Pearson's\ncorrelation 0.93) across the discovery dataset (study 1) and two independent replication datasets\n(","key":"n0TCrHLifT"},{"type":"crossReference","children":[{"type":"text","value":"study 2 and 3","key":"rAYAAsHQUE"}],"urlSource":"#tab-samples","identifier":"tab-samples","label":"tab-samples","kind":"table","template":"Table %s","enumerator":"1","resolved":true,"html_id":"tab-samples","remote":true,"url":"/methods","dataUrl":"/methods.json","key":"e6JSn3ZELi"},{"type":"text","value":", ","key":"yYnxzrHKFW"},{"type":"crossReference","kind":"figure","identifier":"rest-validity","label":"rest-validity","children":[{"type":"text","value":"Figure ","key":"ezgnZEA8Aq"},{"type":"text","value":"3","key":"PkrJjFU7Qm"}],"template":"Figure %s","enumerator":"3","resolved":true,"html_id":"rest-validity","key":"v0d4UoeTPz"},{"type":"text","value":"C). Moreover, they were found to be robust to noise added to the connectome (","key":"BHEz5WBAge"},{"type":"crossReference","kind":"figure","identifier":"si_noise_robustness_weights","label":"si_noise_robustness_weights","children":[{"type":"text","value":"Supplementary Figure ","key":"jnRW72APvt"},{"type":"text","value":"7","key":"lK4u1kz6CN"}],"template":"Figure %s","enumerator":"7","resolved":true,"html_id":"si-noise-robustness-weights","remote":true,"url":"/supplement","dataUrl":"/supplement.json","key":"BSPdMFx1lc"},{"type":"text","value":").","key":"DHxAQzLZ3z"}],"key":"aKZ67NOyFA"},{"type":"paragraph","position":{"start":{"line":269,"column":0},"end":{"line":272,"column":0}},"children":[{"type":"text","value":"Further analysis in study 1 showed that connectome-based Hopfield models accurately reconstructed multiple\ncharacteristics of true resting-state data.\nFirst, the first two components of the ","key":"APJLv7hLd9"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"YTIdVdVYRI"}],"key":"TpvFaw44R5"},{"type":"text","value":" projection accounted for a substantial amount of variance in the real resting-state ","key":"O1chNhRUuN"},{"type":"abbreviation","title":"functional Magnetic Resonance Imaging","children":[{"type":"text","value":"fMRI","key":"ePf6QuVMAB"}],"key":"ZN58ZfdUNA"},{"type":"text","value":" data in study 1 (mean ","key":"lEk7q07GRV"},{"type":"inlineMath","value":"R^2=0.399","html":"R2=0.399R^2=0.399R2=0.399","key":"bxAeBFhIEi"},{"type":"text","value":") and generalized well to out-of-sample data (study 2, mean ","key":"JyGj7kTL8w"},{"type":"inlineMath","value":"R^2=0.396","html":"R2=0.396R^2=0.396R2=0.396","key":"Ssb2BtYkzI"},{"type":"text","value":") (","key":"KTC5Ab95cQ"},{"type":"crossReference","kind":"figure","identifier":"rest-validity","label":"rest-validity","children":[{"type":"text","value":"Figure ","key":"IhdYv55q9C"},{"type":"text","value":"3","key":"xufZO2fED6"}],"template":"Figure %s","enumerator":"3","resolved":true,"html_id":"rest-validity","key":"wWAgzwf3GO"},{"type":"text","value":"E). Remarkably, the explained variance of the ","key":"CaY6WZHaa8"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"PUEklAZhpq"}],"key":"hT4Nj9T9EW"},{"type":"text","value":" projection significantly exceeded that of a ","key":"nJVk2p4yZd"},{"type":"abbreviation","title":"Principal Component","children":[{"type":"text","value":"PC","key":"SzKL8KNwLr"}],"key":"LVTvb1JPqo"},{"type":"text","value":"A performed directly on the real resting-state ","key":"C2ciqLkydd"},{"type":"abbreviation","title":"functional Magnetic Resonance Imaging","children":[{"type":"text","value":"fMRI","key":"xDmhgHbmT1"}],"key":"Xt5INXSF5u"},{"type":"text","value":" data itself (","key":"IwqWk8b25D"},{"type":"inlineMath","value":"R^2=0.37","html":"R2=0.37R^2=0.37R2=0.37","key":"IiBLy3xErh"},{"type":"text","value":" and ","key":"stinCEYcvF"},{"type":"inlineMath","value":"0.364","html":"0.3640.3640.364","key":"rVBniYrrdS"},{"type":"text","value":" for in- and out-of-sample analyses).","key":"hBx5K8eNpa"}],"key":"d369o6aDJI"},{"type":"paragraph","position":{"start":{"line":273,"column":0},"end":{"line":275,"column":0}},"children":[{"type":"text","value":"Second, ","key":"HthsUyq5Lm"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"seVQKCU0bZ"}],"key":"WNetmmm0ct"},{"type":"text","value":" analyses accurately reconstructed various aspects of true resting state brain dynamics.\nPanel D on ","key":"sYlZ866Rkm"},{"type":"crossReference","kind":"figure","identifier":"rest-validity","label":"rest-validity","children":[{"type":"text","value":"Figure ","key":"q1c5otR4SF"},{"type":"text","value":"3","key":"ZC1jIFhsR5"}],"template":"Figure %s","enumerator":"3","resolved":true,"html_id":"rest-validity","key":"BOW8aQijGl"},{"type":"text","value":" shows that, with the conventional Hopfield relaxation procedure, individual activation maps converge to one of the four attractor states. When weak noise is introduced to the system (stochastic relaxation, panel E), the system does not converge to an attractor state but the resulting path is still influenced by the attractor states' gravity. The empirical timeseries data exhibits a similar pattern not only visually (panel F), but also when quantifying the average trajectories of flow, as compared to null-models of zero temporal autocorrelation (randomized timeframe order), reflecting the \"gravitational pull\" of attractor states (","key":"IC1664cgUz"},{"type":"crossReference","kind":"figure","identifier":"rest-validity","label":"rest-validity","children":[{"type":"text","value":"Figure ","key":"Lhr9tDktbN"},{"type":"text","value":"3","key":"hbIlmQ89cI"}],"template":"Figure %s","enumerator":"3","resolved":true,"html_id":"rest-validity","key":"v5HiVrUmwW"},{"type":"text","value":" G and H, see ","key":"HGoA21k42H"},{"type":"crossReference","children":[{"type":"text","value":"Methods","key":"vlzzVfEowM"}],"urlSource":"#evaluation-resting-state-dynamics","identifier":"evaluation-resting-state-dynamics","label":"evaluation-resting-state-dynamics","kind":"heading","template":"{name}","resolved":true,"html_id":"evaluation-resting-state-dynamics","remote":true,"url":"/methods","dataUrl":"/methods.json","key":"jHLfNeGucP"},{"type":"text","value":" fro analysis details).","key":"mBJxNJ4wh1"}],"key":"WgtwgYPpxH"},{"type":"paragraph","position":{"start":{"line":276,"column":0},"end":{"line":277,"column":0}},"children":[{"type":"text","value":"During stochastic relaxation, the ","key":"SGVu6oZyjw"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"aDHlwbu4f9"}],"key":"jwlLJWpp8k"},{"type":"text","value":" model was found to spend approximately three-quarters of the time on the basis of the first two attractor states and one-quarter on the basis of the second pair of attractor states (approximately equally distributed between pairs). We observed strikingly similar temporal occupancies in the real data ","key":"EUf4fo7hIS"},{"type":"crossReference","kind":"figure","identifier":"rest-validity","label":"rest-validity","children":[{"type":"text","value":"Figure ","key":"nuu0tmejG4"},{"type":"text","value":"3","key":"fH9kGsHQPE"}],"template":"Figure %s","enumerator":"3","resolved":true,"html_id":"rest-validity","key":"BSlUW8QLds"},{"type":"text","value":"D), statistically significant with various null models (","key":"LCAflXDHM9"},{"type":"crossReference","kind":"figure","identifier":"si_state_occupancy_null_models","label":"si_state_occupancy_null_models","children":[{"type":"text","value":"Supplementary Figure ","key":"Iu2WJ09Ybq"},{"type":"text","value":"4","key":"m1pacEcZ9C"}],"template":"Figure %s","enumerator":"4","resolved":true,"html_id":"si-state-occupancy-null-models","remote":true,"url":"/supplement","dataUrl":"/supplement.json","key":"sgoCFOScbj"},{"type":"text","value":"). Fine-grained details of the bimodal distribution observed in the real resting-state ","key":"OXHlwCBKkx"},{"type":"abbreviation","title":"functional Magnetic Resonance Imaging","children":[{"type":"text","value":"fMRI","key":"Frb2dVNlZy"}],"key":"lFdZjzQ9f7"},{"type":"text","value":" data were also convincingly reproduced by the ","key":"eUCJ6So9dn"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"lUeOqQVEh1"}],"key":"CG1RWiMX7T"},{"type":"text","value":" model (","key":"M7ZfpVkxR4"},{"type":"crossReference","kind":"figure","identifier":"rest-validity","label":"rest-validity","children":[{"type":"text","value":"Figure ","key":"zcxkwMx0sa"},{"type":"text","value":"3","key":"QnADZas9H8"}],"template":"Figure %s","enumerator":"3","resolved":true,"html_id":"rest-validity","key":"AF006O6VMk"},{"type":"text","value":"F and ","key":"nKwDM02IKE"},{"type":"crossReference","kind":"figure","identifier":"attractors","label":"attractors","children":[{"type":"text","value":"Figure ","key":"dtTaJcfEfh"},{"type":"text","value":"2","key":"b6QGnsN9Xo"}],"template":"Figure %s","enumerator":"2","resolved":true,"html_id":"attractors","key":"QY2W1omGi9"},{"type":"text","value":"E).","key":"FDgYHbJcAo"}],"key":"fCCaesdsnm"},{"type":"paragraph","position":{"start":{"line":278,"column":0},"end":{"line":279,"column":0}},"children":[{"type":"text","value":"Finally, ","key":"dHXH5565ng"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"e0NtT1MiUG"}],"key":"O1F84euZZs"},{"type":"text","value":"s were found to generate signal that preserves the covariance structure of the real functional connectome, indicating that dynamic systems of this type (including the brain) inevitably \"leak\" their underlying structure into the activity time series, strengthening the construct validity of our approach (","key":"CkIuJTxDeq"},{"type":"crossReference","kind":"figure","identifier":"rest-validity","label":"rest-validity","children":[{"type":"text","value":"Figure ","key":"FPHrQBEuKH"},{"type":"text","value":"3","key":"zw3J8uX1WK"}],"template":"Figure %s","enumerator":"3","resolved":true,"html_id":"rest-validity","key":"zbNFrBq24t"},{"type":"text","value":"D).","key":"WXmYbww9m0"}],"key":"nlY0ZAB4sz"},{"type":"heading","depth":3,"position":{"start":{"line":280,"column":0},"end":{"line":281,"column":0}},"children":[{"type":"text","value":"An explanatory framework for task-based brain activity","key":"thOlL5Wzbk"}],"identifier":"an-explanatory-framework-for-task-based-brain-activity","label":"An explanatory framework for task-based brain activity","html_id":"an-explanatory-framework-for-task-based-brain-activity","implicit":true,"key":"mb8okdz8bF"},{"type":"paragraph","position":{"start":{"line":282,"column":0},"end":{"line":283,"column":0}},"children":[{"type":"text","value":"Next to reproducing various characteristics of spontaneous brain dynamics, ","key":"Atd3BcYjFR"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"RKCsEorX2r"}],"key":"PJGUrwaOPs"},{"type":"text","value":"s can also be used to model responses to various perturbations. We obtained task-based ","key":"NKrlsv7QNh"},{"type":"abbreviation","title":"functional Magnetic Resonance Imaging","children":[{"type":"text","value":"fMRI","key":"zA0BmKJ5D6"}],"key":"Wdve9T2Nk7"},{"type":"text","value":" data from a study by ","key":"qyIUKf1m6Z"},{"type":"citeGroup","kind":"narrative","children":[{"type":"cite","label":"woo2015distinct","identifier":"woo2015distinct","children":[{"type":"text","value":"Woo ","key":"ZWUW8jpcEj"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"xSFcjPHlmB"}],"key":"trHqp933mc"},{"type":"text","value":" (2015)","key":"fQdfIWps91"}],"key":"GNPJ8DoxxK"}],"key":"rAhYaU20HS"},{"type":"text","value":" (","key":"ocEH5Q16w3"},{"type":"crossReference","children":[{"type":"text","value":"study 4","key":"Wq5QAfCAXa"}],"urlSource":"#tab-samples","identifier":"tab-samples","label":"tab-samples","kind":"table","template":"Table %s","enumerator":"1","resolved":true,"html_id":"tab-samples","remote":true,"url":"/methods","dataUrl":"/methods.json","key":"Z13r0BjqYw"},{"type":"text","value":", n=33, see ","key":"t46PUVtC5E"},{"type":"crossReference","kind":"figure","identifier":"rest-validity","label":"rest-validity","children":[{"type":"text","value":"Figure ","key":"YzaqgKKUnM"},{"type":"text","value":"3","key":"CEolvjU3Ip"}],"template":"Figure %s","enumerator":"3","resolved":true,"html_id":"rest-validity","key":"Ysg6KL0aQh"},{"type":"text","value":"), investigating the neural correlates of pain and its self-regulation.","key":"jgoCLsXYnQ"}],"key":"pnNlGM0NRf"},{"type":"paragraph","position":{"start":{"line":284,"column":0},"end":{"line":285,"column":0}},"children":[{"type":"text","value":"We found that activity changes due to pain (taking into account hemodynamics, see ","key":"wp0jMw5vMF"},{"type":"crossReference","children":[{"type":"text","value":"Methods","key":"xT6pJldVcw"}],"urlSource":"#evaluation-task-based-dynamics","identifier":"evaluation-task-based-dynamics","label":"evaluation-task-based-dynamics","kind":"heading","template":"{name}","resolved":true,"html_id":"evaluation-task-based-dynamics","remote":true,"url":"/methods","dataUrl":"/methods.json","key":"jReBzIAbCS"},{"type":"text","value":") were characterized on the ","key":"nPwi8IbjE0"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"hQeQUK1xf1"}],"key":"oB4pLwLJVD"},{"type":"text","value":" projection by a shift towards the attractor state of action/execution (permutation test for mean projection difference by randomly swapping conditions, p<0.001, ","key":"Ow1h7lTuPm"},{"type":"crossReference","kind":"figure","identifier":"task-validity","label":"task-validity","children":[{"type":"text","value":"Figure ","key":"pV5tCGK7nq"},{"type":"text","value":"4","key":"uzBHTnnBpR"}],"template":"Figure %s","enumerator":"4","resolved":true,"html_id":"task-validity","key":"w0RjvY918g"},{"type":"text","value":"A, left). Energies, as defined by the ","key":"AyOxZs5IL4"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"sUVltO3IiE"}],"key":"LKVTVvozGC"},{"type":"text","value":", were also significantly different between the two conditions (p<0.001), with higher energies during pain stimulation.","key":"cqYb6JXEBJ"}],"key":"dFaxmihlLf"},{"type":"paragraph","position":{"start":{"line":286,"column":0},"end":{"line":287,"column":0}},"children":[{"type":"text","value":"When participants were instructed to up- or downregulate their pain sensation (resulting in increased and decreased pain reports and differential brain activity in the nucleus accumbens, NAc (see ","key":"DNUHdJW9Vg"},{"type":"cite","label":"woo2015distinct","identifier":"woo2015distinct","children":[{"type":"text","value":"Woo ","key":"OG7tnT8N3X"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"zai1vlxK0L"}],"key":"ZlftUoLXZq"},{"type":"text","value":", 2015","key":"YeRYF1sy0f"}],"key":"hZn8USTGzT"},{"type":"text","value":" for details), we observed further changes of the location of momentary brain activity patterns on the ","key":"KtDv6Bt2a2"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"Zu2BX0KDwi"}],"key":"SW4Ifb8ulC"},{"type":"text","value":" projection (p<0.001, ","key":"B2Shmq9jHt"},{"type":"crossReference","kind":"figure","identifier":"task-validity","label":"task-validity","children":[{"type":"text","value":"Figure ","key":"af9NcFIIAS"},{"type":"text","value":"4","key":"BJ8dw8zIrj"}],"template":"Figure %s","enumerator":"4","resolved":true,"html_id":"task-validity","key":"XeW6Uj0l1C"},{"type":"text","value":"A, right), with down-regulation pulling brain dynamics towards the attractor state of internal context and perception. Interestingly, self-regulation did not trigger significant energy changes (p=0.36).","key":"KuIlUS1GDx"}],"key":"tztxd7VwLp"},{"type":"container","kind":"figure","identifier":"task-validity","label":"task-validity","children":[{"type":"image","url":"/task_validity-e7bb5a8057fc47eba1be0aee743c3783.png","alt":"Empirical Hopfield-networks reconstruct real task-based brain activity.

\nA Functional MRI time-frames during pain stimulation from study 4 (second fcHNN projection plot)\nand self-regulation (third and fourth) are distributed differently on the fcHNN projection than brain states\nduring rest (first projection, permutation test, p<0.001 for all). Energies, as defined by the Hopfield model, are also\nsignificantly different between rest and the pain conditions (permutation test, p<0.001), with higher energies during\npain stimulation. Triangles denote participant-level mean activations in the various blocks (corrected for\nhemodynamics). Small circle plots show the directions of the change for each individual (points) as well as the mean direction\nacross participants (arrow), as compared to the reference state (downregulation for the last circle plot, rest for all\nother circle plots).\nB Flow-analysis (difference in the average timeframe-to-timeframe transition direction) reveals a non-linear difference in brain dynamics during pain and rest (left). When introducing weak pain-related signal in the fcHNN model during stochastic relaxation, it accurately reproduces these non-linear flow differences (right).\nC Simulating activity in the Nucleus Accumbens (NAc) (the region showing significant activity differences in ) reconstructs the observed non-linear flow difference between up- and downregulation (left).\nD Schematic representation of brain dynamics during pain and its up- and downregulation, visualized on the fcHNN projection. In the proposed framework, pain does not simply elicit a direct response in certain regions, but instead, shifts spontaneous brain dynamics towards the \"action\" attractor, converging to a characteristic \"ghost attractor\" of pain. Down-regulation by NAc activation exerts force towards the attractor of internal context, leading to the brain less frequent \"visiting\" pain-associated states.\nE Visualizing meta-analytic activation maps (see Supplementary Table %s for details) on the fcHNN projection captures intimate relations between the corresponding tasks and F serves as a basis for a fcHNN-based theoretical interpretative framework for spontaneous and task-based brain dynamics. In the proposed framework, task-based activity is not a mere response to external stimuli in certain brain locations but a perturbation of the brain's characteristic dynamic trajectories, constrained by the underlying functional connectivity. From this perspective, \"activity maps\" from conventional task-based fMRI analyses capture time-averaged differences in these whole brain dynamics.","urlSource":"figures/task_validity.png","key":"KRFdQiWpIQ"},{"type":"caption","children":[{"type":"paragraph","position":{"start":{"line":290,"column":0},"end":{"line":303,"column":0}},"children":[{"type":"captionNumber","kind":"figure","label":"task-validity","identifier":"task-validity","html_id":"task-validity","enumerator":"4","children":[{"type":"text","value":"Figure ","key":"gWs2UxF4iI"},{"type":"text","value":"4","key":"DeUzfbxKDB"},{"type":"text","value":":","key":"lIWcvJ0GKD"}],"template":"Figure %s:","key":"TKkNTELS53"},{"type":"strong","children":[{"type":"text","value":"Empirical Hopfield-networks reconstruct real task-based brain activity.","key":"vtDqGalXue"}],"key":"WTr8dfsaBZ"},{"type":"text","value":" ","key":"XuzR2ZYtKy"},{"type":"break","key":"l5P1ekHlhA"},{"type":"text","value":"\n","key":"oKyL5bmzko"},{"type":"strong","children":[{"type":"text","value":"A","key":"d9lzozs0DO"}],"key":"mRVF3jfNuT"},{"type":"text","value":" Functional MRI time-frames during pain stimulation from ","key":"SKC7e4x4RX"},{"type":"crossReference","children":[{"type":"text","value":"study 4","key":"o9zCM9ts3o"}],"urlSource":"#tab-samples","identifier":"tab-samples","label":"tab-samples","kind":"table","template":"Table %s","enumerator":"1","resolved":true,"html_id":"tab-samples","remote":true,"url":"/methods","dataUrl":"/methods.json","key":"d2EEVIPxJd"},{"type":"text","value":" (second ","key":"nHo3DsENQn"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"t8k4Xk5ENA"}],"key":"cawkABDs8M"},{"type":"text","value":" projection plot)\nand self-regulation (third and fourth) are distributed differently on the ","key":"Y81ZhqpS52"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"w9Gxi08Ahx"}],"key":"OIpnjb16Ar"},{"type":"text","value":" projection than brain states\nduring rest (first projection, permutation test, p<0.001 for all). Energies, as defined by the Hopfield model, are also\nsignificantly different between rest and the pain conditions (permutation test, p<0.001), with higher energies during\npain stimulation. Triangles denote participant-level mean activations in the various blocks (corrected for\nhemodynamics). Small circle plots show the directions of the change for each individual (points) as well as the mean direction\nacross participants (arrow), as compared to the reference state (downregulation for the last circle plot, rest for all\nother circle plots).\n","key":"Dvu2Hkcp4y"},{"type":"strong","children":[{"type":"text","value":"B","key":"PpklIn4Mjh"}],"key":"TuW8oFtxJf"},{"type":"text","value":" Flow-analysis (difference in the average timeframe-to-timeframe transition direction) reveals a non-linear difference in brain dynamics during pain and rest (left). When introducing weak pain-related signal in the ","key":"h9S1Ba5R4K"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"QUK7FeZ1oi"}],"key":"u0dMLmwqDu"},{"type":"text","value":" model during stochastic relaxation, it accurately reproduces these non-linear flow differences (right).\n","key":"ICYJN30VfE"},{"type":"strong","children":[{"type":"text","value":"C","key":"HkMPM7i4yd"}],"key":"r3KpCO3uNn"},{"type":"text","value":" Simulating activity in the Nucleus Accumbens (NAc) (the region showing significant activity differences in ","key":"D7myR9ObTX"},{"type":"cite","label":"woo2015distinct","identifier":"woo2015distinct","children":[{"type":"text","value":"Woo ","key":"OT5pcr8wwe"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"Oar0UhUXIh"}],"key":"KzA9MWmEwC"},{"type":"text","value":", 2015","key":"KZ6ELnsTyQ"}],"key":"dlAaT9QIn8"},{"type":"text","value":") reconstructs the observed non-linear flow difference between up- and downregulation (left).\n","key":"S3TY5QBLBd"},{"type":"strong","children":[{"type":"text","value":"D","key":"ENIaHXZUYa"}],"key":"UAONXKiRBZ"},{"type":"text","value":" Schematic representation of brain dynamics during pain and its up- and downregulation, visualized on the ","key":"BCb31uIIgn"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"k1eNFdXmdH"}],"key":"PHbaxSe35u"},{"type":"text","value":" projection. In the proposed framework, pain does not simply elicit a direct response in certain regions, but instead, shifts spontaneous brain dynamics towards the \"action\" attractor, converging to a characteristic \"ghost attractor\" of pain. Down-regulation by NAc activation exerts force towards the attractor of internal context, leading to the brain less frequent \"visiting\" pain-associated states.\n","key":"yklEske0tc"},{"type":"strong","children":[{"type":"text","value":"E","key":"MyZ7MiI1ao"}],"key":"MGl6aB19dB"},{"type":"text","value":" Visualizing meta-analytic activation maps (see ","key":"iZ5XOKR8pF"},{"type":"crossReference","kind":"table","identifier":"si-tab-neurosynth","label":"si-tab-neurosynth","children":[{"type":"text","value":"Supplementary Table ","key":"YJZIxQQgHn"},{"type":"text","value":"1","key":"Z3Jct0ZOIU"}],"template":"Table %s","enumerator":"1","resolved":true,"html_id":"si-tab-neurosynth","remote":true,"url":"/supplement","dataUrl":"/supplement.json","key":"XcWBGurDnv"},{"type":"text","value":" for details) on the ","key":"cEge7kqvkj"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"zjqpdZiRJm"}],"key":"ZxohXulJa1"},{"type":"text","value":" projection captures intimate relations between the corresponding tasks and ","key":"lyUT0L4Htu"},{"type":"strong","children":[{"type":"text","value":"F","key":"E6EZMbzuYr"}],"key":"Fq7ryk9AJI"},{"type":"text","value":" serves as a basis for a ","key":"PI1YGHBCSY"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"olR3kTjpGd"}],"key":"zSH6tboRZh"},{"type":"text","value":"-based theoretical interpretative framework for spontaneous and task-based brain dynamics. In the proposed framework, task-based activity is not a mere response to external stimuli in certain brain locations but a perturbation of the brain's characteristic dynamic trajectories, constrained by the underlying functional connectivity. From this perspective, \"activity maps\" from conventional task-based ","key":"kSJ5gYOzzr"},{"type":"abbreviation","title":"functional Magnetic Resonance Imaging","children":[{"type":"text","value":"fMRI","key":"bnufGn9aJw"}],"key":"cKCvi9zHC3"},{"type":"text","value":" analyses capture time-averaged differences in these whole brain dynamics.","key":"B6BMXndNXv"}],"key":"RTOFvPXMub"}],"key":"LmhylEQYW9"}],"enumerator":"4","html_id":"task-validity","key":"aDNX4Q5Lf0"},{"type":"paragraph","position":{"start":{"line":305,"column":0},"end":{"line":307,"column":0}},"children":[{"type":"text","value":"Next, we conducted a \"flow analysis\" on the ","key":"SbcYuHdmNA"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"BnmCLdg13h"}],"key":"O3ACGYHei9"},{"type":"text","value":" projection, quantifying how the average timeframe-to-timeframe transition direction differs on the ","key":"q9SGMW7EBC"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"ruLWtFt1CX"}],"key":"itJqT9iylq"},{"type":"text","value":" projection between conditions (see ","key":"ZCUsVU6NgR"},{"type":"crossReference","children":[{"type":"text","value":"Methods","key":"GOeiOCamqW"}],"urlSource":"#evaluation-task-based-dynamics","identifier":"evaluation-task-based-dynamics","label":"evaluation-task-based-dynamics","kind":"heading","template":"{name}","resolved":true,"html_id":"evaluation-task-based-dynamics","remote":true,"url":"/methods","dataUrl":"/methods.json","key":"wJsEvG59vU"},{"type":"text","value":").\nThis analysis unveiled that during pain (","key":"n1l43wzKIf"},{"type":"crossReference","kind":"figure","identifier":"task-validity","label":"task-validity","children":[{"type":"text","value":"Figure ","key":"iIisnrdlUn"},{"type":"text","value":"4","key":"AX0FWKYE8l"}],"template":"Figure %s","enumerator":"4","resolved":true,"html_id":"task-validity","key":"nhiAJtAmbg"},{"type":"text","value":"B, left side), brain activity tends to gravitate towards a distinct point on the projection on the boundary the basins of the internal and action attractors, which we term the \"ghost attractor\" of pain (similar to ","key":"QYdD0bWAN1"},{"type":"cite","label":"vohryzek2020ghost","identifier":"vohryzek2020ghost","children":[{"type":"text","value":"Vohryzek ","key":"szlwrZhpkd"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"nw9qkLnsWb"}],"key":"N3c5WqxhOM"},{"type":"text","value":", 2020","key":"GXHvyqdv0h"}],"key":"VoRUwB1RDg"},{"type":"text","value":"). In case of downregulation (as compared to upregulation), brain activity is pulled away from the pain-related \"ghost attractor\" (","key":"itbFaFtjjJ"},{"type":"crossReference","kind":"figure","identifier":"task-validity","label":"task-validity","children":[{"type":"text","value":"Figure ","key":"BydwObrle4"},{"type":"text","value":"4","key":"ZavlVW2ooa"}],"template":"Figure %s","enumerator":"4","resolved":true,"html_id":"task-validity","key":"GLOztsPMNi"},{"type":"text","value":"C, left side), towards the attractor of internal context.","key":"LD5NvTyeuF"}],"key":"pMUlRntC7b"},{"type":"paragraph","position":{"start":{"line":308,"column":0},"end":{"line":309,"column":0}},"children":[{"type":"text","value":"Our ","key":"e7GXWylMHG"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"nosXtmBbjP"}],"key":"nZvwIhCAbI"},{"type":"text","value":" was able to accurately reconstruct these non-linear dynamics by adding a small amount of realistic \"control signal\" (similarly to network control theory ","key":"bMhgpLsGpr"},{"type":"citeGroup","kind":"narrative","children":[{"type":"cite","label":"liu2011controllability","identifier":"liu2011controllability","children":[{"type":"text","value":"Liu ","key":"TC4h1VIXoU"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"WABIehWs7G"}],"key":"YHrchkGfYN"},{"type":"text","value":" (2011)","key":"v7DJwyoTOQ"}],"key":"JavRJoBTCc"},{"type":"cite","label":"gu2015controllability","identifier":"gu2015controllability","children":[{"type":"text","value":"Gu ","key":"g7ejMKTMCi"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"YcxtIOFXVg"}],"key":"tHXv4N9dqP"},{"type":"text","value":" (2015)","key":"hLpxtANBuz"}],"key":"Y1qTnruTUt"}],"key":"UonbrbNEji"},{"type":"text","value":"). To simulate the alterations in brain dynamics during pain stimulation, we acquired a meta-analytic pain activation map ","key":"b8UxOFHPvj"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"zunhammer2021meta","identifier":"zunhammer2021meta","children":[{"type":"text","value":"Zunhammer ","key":"bQOEDpLScS"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"FH4JTiPXBI"}],"key":"vVOYRAJR9V"},{"type":"text","value":", 2021","key":"SBkr9yFUmX"}],"key":"QpaT219FtT"}],"key":"IZdPaJBe9r"},{"type":"text","value":" (n=603) and incorporated it as a control signal added to each iteration of the stochastic relaxation procedure. The ghost attractor found in the empirical data was present across a relatively wide range of signal-to-noise (SNR) values (","key":"MIcMjCeGsd"},{"type":"crossReference","kind":"figure","identifier":"si_pain_ghost_attractor_sim","label":"si_pain_ghost_attractor_sim","children":[{"type":"text","value":"Supplementary Figure ","key":"BWXaF3FoXv"},{"type":"text","value":"5","key":"U2U1w8G2A5"}],"template":"Figure %s","enumerator":"5","resolved":true,"html_id":"si-pain-ghost-attractor-sim","remote":true,"url":"/supplement","dataUrl":"/supplement.json","key":"E1wdX0D81k"},{"type":"text","value":"). Results with SNR=0.005 are presented on ","key":"lWi1On89Os"},{"type":"crossReference","kind":"figure","identifier":"task-validity","label":"task-validity","children":[{"type":"text","value":"Figure ","key":"Hg8fW4pdzd"},{"type":"text","value":"4","key":"EDiD8O7joX"}],"template":"Figure %s","enumerator":"4","resolved":true,"html_id":"task-validity","key":"UlGRErkFxl"},{"type":"text","value":"B, right side (Pearson's r = 0.46, p=0.005 based on randomizing conditions on a per-participant basis).","key":"U2uutexlCI"}],"key":"SV3J3yxTp7"},{"type":"paragraph","position":{"start":{"line":310,"column":0},"end":{"line":313,"column":0}},"children":[{"type":"text","value":"The same model was also able to reconstruct the observed non-linear differences in brain dynamics between the up- and downregulation conditions (Pearson's r = 0.62, p=0.023) without any further optimization (SNR=0.005,\n","key":"lLVEpr1mCC"},{"type":"crossReference","kind":"figure","identifier":"task-validity","label":"task-validity","children":[{"type":"text","value":"Figure ","key":"bvuukhPGdW"},{"type":"text","value":"4","key":"Q91hQs9bQ9"}],"template":"Figure %s","enumerator":"4","resolved":true,"html_id":"task-validity","key":"CMCJxKXBND"},{"type":"text","value":"C, right side). The only change we made to the model was the addition (downregulation) or\nsubtraction (upregulation) of control signal in the NAc (the region in which ","key":"XLNmzl8jlD"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"woo2015distinct","identifier":"woo2015distinct","children":[{"type":"text","value":"Woo ","key":"PFzK3OA7nZ"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"kEXoBQPm8y"}],"key":"Vfqc88PPcX"},{"type":"text","value":", 2015","key":"XZ6o6W6h0f"}],"key":"ZDjdlSfehW"}],"key":"mObA51SW4n"},{"type":"text","value":" observed significant changes between up- and downregulation), introducing a signal difference of ","key":"pEv1qqptUs"},{"type":"inlineMath","value":"\\Delta","html":"Δ\\DeltaΔ","key":"gx1kIQbsHX"},{"type":"text","value":"SNR=0.005 (the same value we found optimal in the pain-analysis). Results were reproducible with lower NAc SNRs, too (","key":"gCS35ohRrv"},{"type":"crossReference","kind":"figure","identifier":"si_downreg_trajectory_sim","label":"si_downreg_trajectory_sim","children":[{"type":"text","value":"Supplementary Figure ","key":"TUO5eA50SF"},{"type":"text","value":"6","key":"KqPxO9hw3w"}],"template":"Figure %s","enumerator":"6","resolved":true,"html_id":"si-downreg-trajectory-sim","remote":true,"url":"/supplement","dataUrl":"/supplement.json","key":"HaaeS1mI1z"},{"type":"text","value":").","key":"AvoaocAZma"}],"key":"UriB1dH4Cd"},{"type":"paragraph","position":{"start":{"line":314,"column":0},"end":{"line":316,"column":0}},"children":[{"type":"text","value":"To provide a comprehensive picture on how tasks and stimuli other then pain map onto the ","key":"VYZFLmSK4j"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"f101cwhasr"}],"key":"Xz7TY5pSCg"},{"type":"text","value":" projection, we obtained various task-based meta-analytic activation maps from Neurosynth (see ","key":"WncIXgwBKH"},{"type":"crossReference","children":[{"type":"text","value":"Methods","key":"n4ByXMN3be"}],"urlSource":"#evaluation-task-based-dynamics","identifier":"evaluation-task-based-dynamics","label":"evaluation-task-based-dynamics","kind":"heading","template":"{name}","resolved":true,"html_id":"evaluation-task-based-dynamics","remote":true,"url":"/methods","dataUrl":"/methods.json","key":"pIj6eq8Czg"},{"type":"text","value":") and plotted them on the ","key":"c4rr69a46L"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"mi1u1ogSzU"}],"key":"KqpH1gmOBT"},{"type":"text","value":" projection (","key":"mEeYiRkpSG"},{"type":"crossReference","kind":"figure","identifier":"task-validity","label":"task-validity","children":[{"type":"text","value":"Figure ","key":"GUorGxDAMW"},{"type":"text","value":"4","key":"IOlSW8hQ2n"}],"template":"Figure %s","enumerator":"4","resolved":true,"html_id":"task-validity","key":"wQApz01FQ4"},{"type":"text","value":"E). This analysis reinforced and extended our interpretation of the four investigated attractor states and shed more light on how various functions are mapped on the axes of internal vs. external context and perception vs. action.\nIn the coordinate system of the ","key":"Hso52dt4hK"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"uPssWdhEZQ"}],"key":"e2t8O7cY2v"},{"type":"text","value":" projection, visual processing is labeled \"external-perception\", sensory-motor processes \"external-active\", language, verbal cognition and working memory is labelled \"internal-active\" and long-term memory as well as social and autobiographic schemata fall into the \"internal-perception\" regime (","key":"SuxFHRw3b6"},{"type":"crossReference","kind":"figure","identifier":"task-validity","label":"task-validity","children":[{"type":"text","value":"Figure ","key":"gBI4ejS2yd"},{"type":"text","value":"4","key":"jrrtySdQ8a"}],"template":"Figure %s","enumerator":"4","resolved":true,"html_id":"task-validity","key":"xyfx3FBtQ5"},{"type":"text","value":"F).","key":"zF463xi1S7"}],"key":"ocZ9D6Bm1J"},{"type":"heading","depth":3,"position":{"start":{"line":317,"column":0},"end":{"line":318,"column":0}},"children":[{"type":"text","value":"Clinical relevance","key":"oN3H7zDKwH"}],"identifier":"clinical-relevance","label":"Clinical relevance","html_id":"clinical-relevance","implicit":true,"key":"l1i8Yegh9e"},{"type":"paragraph","position":{"start":{"line":319,"column":0},"end":{"line":323,"column":0}},"children":[{"type":"text","value":"We obtained data from n=172 autism spectrum disorder (","key":"K3MlrKD9lU"},{"type":"abbreviation","title":"Autism Spectrum Disorder","children":[{"type":"text","value":"ASD","key":"dWsLzxprEM"}],"key":"C32BaoFvIl"},{"type":"text","value":") and typically developing control (TDC) individuals, acquired at the New York University Langone Medical Center, New York, NY, USA (NYU) and generously shared in the Autism Brain Imaging Data Exchange dataset (","key":"oT2Crg4Sdo"},{"type":"crossReference","children":[{"type":"text","value":"study 7","key":"d745MJZgqe"}],"urlSource":"#tab-samples","identifier":"tab-samples","label":"tab-samples","kind":"table","template":"Table %s","enumerator":"1","resolved":true,"html_id":"tab-samples","remote":true,"url":"/methods","dataUrl":"/methods.json","key":"uPzhQRwz5H"},{"type":"text","value":": ","key":"KtZOGRpFsl"},{"type":"abbreviation","title":"Autism Brain Imaging Data Exchange","children":[{"type":"text","value":"ABIDE","key":"GQOaZJkrx0"}],"key":"oWPzwk4pmy"},{"type":"text","value":", 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Neural Network","children":[{"type":"text","value":"fcHNN","key":"q8GZDqXv8T"}],"key":"DReIKkuKdU"},{"type":"text","value":"-projection separately for the ","key":"DKSBdyOe3Q"},{"type":"abbreviation","title":"Autism Spectrum Disorder","children":[{"type":"text","value":"ASD","key":"VCKugYQEBm"}],"key":"RGIMauWMqp"},{"type":"text","value":" and TDC groups (","key":"uY3T413oVu"},{"type":"crossReference","kind":"figure","identifier":"clinical-validity","label":"clinical-validity","children":[{"type":"text","value":"Figure ","key":"qcHcsR15Gi"},{"type":"text","value":"5","key":"hzAS4RMV3j"}],"template":"Figure %s","enumerator":"5","resolved":true,"html_id":"clinical-validity","key":"io0vcAjD8J"},{"type":"text","value":"A).\nFirst, we assigned all timeframes to one of the 4 attractor states with the ","key":"e9Ynkm6djE"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"AJm1PzJQjo"}],"key":"zeLyeBIKbG"},{"type":"text","value":" from study 1 and found several significant differences in the mean activity on the attractor basins (see ","key":"qafJFkTnXM"},{"type":"crossReference","children":[{"type":"text","value":"Methods","key":"TpzkbOPRIU"}],"urlSource":"#clinical-data","identifier":"clinical-data","label":"clinical-data","kind":"heading","template":"{name}","resolved":true,"html_id":"clinical-data","remote":true,"url":"/methods","dataUrl":"/methods.json","key":"u9aI1mFM2t"},{"type":"text","value":") of the ","key":"VETcCZZvcP"},{"type":"abbreviation","title":"Autism Spectrum Disorder","children":[{"type":"text","value":"ASD","key":"zvejJKYbP1"}],"key":"SU15EiSvfB"},{"type":"text","value":" group as compared to the respective controls (","key":"JtwIwPNmjR"},{"type":"crossReference","kind":"figure","identifier":"clinical-validity","label":"clinical-validity","children":[{"type":"text","value":"Figure ","key":"bIujBN52vF"},{"type":"text","value":"5","key":"jz72zcOJGx"}],"template":"Figure %s","enumerator":"5","resolved":true,"html_id":"clinical-validity","key":"LnMytrAHn6"},{"type":"text","value":"B).\nStrongest differences were found on the \"action-perception\" axis (","key":"XfUVdILE12"},{"type":"crossReference","kind":"table","identifier":"tab-clinical-results","label":"tab-clinical-results","children":[{"type":"text","value":"Table ","key":"xeJvdYWMYx"},{"type":"text","value":"1","key":"q7OgtAcvA8"}],"template":"Table %s","enumerator":"1","resolved":true,"html_id":"tab-clinical-results","key":"sJb7JUnob9"},{"type":"text","value":"), with increased activity of the sensory-motor and mid","key":"Ev8TjgDidL"},{"type":"abbreviation","title":"dorsolateral","children":[{"type":"text","value":"dl","key":"REnJsW1XB9"}],"key":"UfBeKuxEPI"},{"type":"text","value":"e cingular cortices during \"action-execution\" related states and increased visual and decreased sensory and auditory activity during \"perception\" states, likely reflecting the widely acknowledged, yet poorly understood, perceptual atypicalities in ","key":"ifPsVQRykk"},{"type":"abbreviation","title":"Autism Spectrum Disorder","children":[{"type":"text","value":"ASD","key":"fi1rBWNw8Q"}],"key":"IGcBNyqoa2"},{"type":"text","value":" ","key":"fhvAM7fu5L"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"hadad2019perception","identifier":"hadad2019perception","children":[{"type":"text","value":"Hadad & Schwartz, 2019","key":"E3cssGiPH1"}],"key":"afKvNB1NTP"}],"key":"x5IJRXHpEk"},{"type":"text","value":". ","key":"P4pvQbcRRS"},{"type":"abbreviation","title":"Autism Spectrum Disorder","children":[{"type":"text","value":"ASD","key":"jI24FayP6l"}],"key":"GDLd4xVpiZ"},{"type":"text","value":" related changes in the internal-external axis were characterized by more involvement of the posterior cingulate, the precuneus, the nucleus accumbens, the dorsolateral prefrontal cortex (","key":"bzyxk6QGvZ"},{"type":"abbreviation","title":"dorsolateral","children":[{"type":"text","value":"dl","key":"ZQ6obmfSMc"}],"key":"eB6moi04XQ"},{"type":"abbreviation","title":"Prefrontal Cortex","children":[{"type":"text","value":"PFC","key":"Oq1YtsKNco"}],"key":"ibeW0B7OVa"},{"type":"text","value":"), the cerebellum (Crus II, lobule VII) and inferior temporal regions during activity of the internalizing subsystem (","key":"ASquPfdtiS"},{"type":"crossReference","kind":"table","identifier":"tab-clinical-results","label":"tab-clinical-results","children":[{"type":"text","value":"Table ","key":"KzTclo4EJA"},{"type":"text","value":"1","key":"yebYDxpSkn"}],"template":"Table %s","enumerator":"1","resolved":true,"html_id":"tab-clinical-results","key":"dcFlGffiOQ"},{"type":"text","value":"). While similar, default mode network (DMN)-related changes have often been attributed to an atypical integration of information about the “self” and the “other” ","key":"jVfjEGedYd"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"padmanabhan2017default","identifier":"padmanabhan2017default","children":[{"type":"text","value":"Padmanabhan ","key":"NBx9Qw541F"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"OvilTEDF8c"}],"key":"lfSR1u8VcC"},{"type":"text","value":", 2017","key":"aJtJrGzWZF"}],"key":"pgTSDE9wGj"}],"key":"gZWVsi7Y5c"},{"type":"text","value":", a more detailed ","key":"d7TxJNok5g"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"pD2jxDiKZn"}],"key":"A6x4YiXU9m"},{"type":"text","value":"-analysis may help to further disentangle the specific nature of these changes.","key":"NQipLuWGb4"}],"key":"jehGGxlF1u"},{"type":"container","kind":"figure","identifier":"clinical-validity","label":"clinical-validity","children":[{"type":"image","url":"/state_analysis-ee5a9d498a950579340cc02d477e17dc.svg","alt":"Connectome-based Hopfield analysis of autism spectrum disorder.

\nA The distribution of time-frames on the fcHNN-projection separately for ASD patients and typically developing control (TDC) participants.

\nB We quantified attractor state activations in the Autism Brain Imaging Data Exchange datasets (study 7) as the\nindividual-level mean activation of all time-frames belonging to the same attractor state. This analysis captured alterations similar to those previously associated to ASD-related perceptual atypicalities (visual, auditory and somatosensory cortices) as well as atypical integration of information about the “self” and the “other” (default mode network regions). All results are corrected for multiple comparisons across brain regions and attractor states (122*4 comparisons) with Bonferroni-correction. See and Supplementary Figure %s for detailed results.

\nC The comparison of data generated by fcHNNs initialized with ASD and TDC connectomes, respectively, revealed a characteristic pattern of differences in the system's dynamics, with increased pull towards (and potentially a higher separation between) the action and perception attractors and a lower tendency of trajectories going towards the internal and external attractors.

\nAbbreviations: MCC: middle cingulate cortex, ACC: anterior cingulate cortex, pg: perigenual, PFC: prefrontal cortex, dm: dorsomedial, dl: dorsolateral, STG: superior temporal gyrus, ITG: inferior temporal gyrus, Caud/Acc: caudate-accumbens, SM: sensorimotor, V1: primary visual, A1: primary auditory, SMA: supplementary motor cortex, ASD: autism spectrum disorder, TDC: typically developing control.","urlSource":"figures/state_analysis.svg","key":"XGk0Qer5wN"},{"type":"caption","children":[{"type":"paragraph","position":{"start":{"line":326,"column":0},"end":{"line":332,"column":0}},"children":[{"type":"captionNumber","kind":"figure","label":"clinical-validity","identifier":"clinical-validity","html_id":"clinical-validity","enumerator":"5","children":[{"type":"text","value":"Figure ","key":"eKRC1qp7dB"},{"type":"text","value":"5","key":"cnSF1oDfvb"},{"type":"text","value":":","key":"hXYr0knl7z"}],"template":"Figure %s:","key":"zq65uE1c7g"},{"type":"strong","children":[{"type":"text","value":"Connectome-based Hopfield analysis of autism spectrum disorder.","key":"HNiijf5xKU"}],"key":"T3bYP31GSb"},{"type":"text","value":" ","key":"NSTkEIEP7c"},{"type":"break","key":"Sjpn3D0OZ9"},{"type":"text","value":"\n","key":"JkyqLH3GLJ"},{"type":"strong","children":[{"type":"text","value":"A","key":"yxcrFsvZNm"}],"key":"K4jHL3j6iW"},{"type":"text","value":" The distribution of time-frames on the ","key":"kL1Gx1dP9e"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"Zp9JRptXMb"}],"key":"dC2lntCa4K"},{"type":"text","value":"-projection separately for ","key":"eY2eduivt6"},{"type":"abbreviation","title":"Autism Spectrum Disorder","children":[{"type":"text","value":"ASD","key":"SUkqSRPuI2"}],"key":"dsGidWqFvG"},{"type":"text","value":" patients and typically developing control (TDC) participants. ","key":"trybrtdrSd"},{"type":"break","key":"Jqel43CqQO"},{"type":"text","value":"\n","key":"VZ1x8qQcbY"},{"type":"strong","children":[{"type":"text","value":"B","key":"U6crKt7vVN"}],"key":"mBb3wTLyKS"},{"type":"text","value":" We quantified attractor state activations in the Autism Brain Imaging Data Exchange datasets (","key":"AE1CeHLHgF"},{"type":"crossReference","children":[{"type":"text","value":"study 7","key":"PPTvEQk2Ub"}],"urlSource":"#tab-samples","identifier":"tab-samples","label":"tab-samples","kind":"table","template":"Table %s","enumerator":"1","resolved":true,"html_id":"tab-samples","remote":true,"url":"/methods","dataUrl":"/methods.json","key":"eebHrPrYps"},{"type":"text","value":") as the\nindividual-level mean activation of all time-frames belonging to the same attractor state. This analysis captured alterations similar to those previously associated to ","key":"oIm5J3UXiO"},{"type":"abbreviation","title":"Autism Spectrum Disorder","children":[{"type":"text","value":"ASD","key":"g4apWml4iD"}],"key":"TY9sBpGxLV"},{"type":"text","value":"-related perceptual atypicalities (visual, auditory and somatosensory cortices) as well as atypical integration of information about the “self” and the “other” (default mode network regions). All results are corrected for multiple comparisons across brain regions and attractor states (122*4 comparisons) with Bonferroni-correction. See ","key":"LmUwEaYyRm"},{"type":"crossReference","kind":"table","identifier":"tab-clinical-results","label":"tab-clinical-results","children":[{"type":"text","value":"Table ","key":"B7WfDuqcDC"},{"type":"text","value":"1","key":"BwGrBIDwvP"}],"template":"Table %s","enumerator":"1","resolved":true,"html_id":"tab-clinical-results","key":"KqkIlVQYUq"},{"type":"text","value":" and ","key":"eKkIZ5HZC4"},{"type":"crossReference","kind":"figure","identifier":"si_clinical_results_table","label":"si_clinical_results_table","children":[{"type":"text","value":"Supplementary Figure ","key":"bdVNCwIDxk"},{"type":"text","value":"8","key":"g5SeNLgj2E"}],"template":"Figure %s","enumerator":"8","resolved":true,"html_id":"si-clinical-results-table","remote":true,"url":"/supplement","dataUrl":"/supplement.json","key":"mET8q3zEI0"},{"type":"text","value":" for detailed results. 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We observed a highly similar pattern in the real data (Pearson's correlation: 0.66), statistically significant after permutation testing (shuffling the group assignment, p=0.009).","key":"Vjtei6FCnI"}],"key":"LQFNYB02fM"},{"type":"heading","depth":2,"position":{"start":{"line":386,"column":0},"end":{"line":387,"column":0}},"children":[{"type":"text","value":"Discussion","key":"S6plNtNfoo"}],"identifier":"discussion","label":"Discussion","html_id":"discussion","implicit":true,"key":"MmZvtMgOOS"},{"type":"paragraph","position":{"start":{"line":388,"column":0},"end":{"line":389,"column":0}},"children":[{"type":"text","value":"In this study, we have introduced and validated a simple yet robust computational generative framework that elucidates how activity propagation within the functional connectome orchestrates large-scale brain dynamics, leading to the spontaneous emergence of brain states and characteristic dynamic responses to perturbations.","key":"iSVhgxrtYU"}],"key":"TvzODQe5Cn"},{"type":"paragraph","position":{"start":{"line":390,"column":0},"end":{"line":392,"column":0}},"children":[{"type":"text","value":"The construct validity of our model is rooted in the activity flow principle, first introduced by ","key":"gDWjMBc7f9"},{"type":"citeGroup","kind":"narrative","children":[{"type":"cite","label":"cole2016activity","identifier":"cole2016activity","children":[{"type":"text","value":"Cole ","key":"bcdb1Ya7xq"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"lpYgj0MjBA"}],"key":"RvUE2Kgror"},{"type":"text","value":" (2016)","key":"CB1nnwXxbb"}],"key":"Dh3CwMdAe0"}],"key":"ROjDRmLniS"},{"type":"text","value":". The activity flow principle states that activity in a brain region can be predicted by a weighted combination of the activity of all other regions, where the weights are set to the functional connectivity of those regions to the held-out region. This principle has been shown to hold across a wide range of experimental and clinical conditions ","key":"GD8ZOA2fU2"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"cole2016activity","identifier":"cole2016activity","children":[{"type":"text","value":"Cole ","key":"HVfgnnxdQi"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"hDj5BBDzzY"}],"key":"XfKZLY8saq"},{"type":"text","value":", 2016","key":"X7A06RhJES"}],"key":"h6yvLSbBS6"},{"type":"cite","label":"ito2017cognitive","identifier":"ito2017cognitive","children":[{"type":"text","value":"Ito ","key":"titA5AdTz0"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"Hc2pCDdQpG"}],"key":"arjwh9HjKu"},{"type":"text","value":", 2017","key":"DIh3LlnLg8"}],"key":"baiBLgx3fC"},{"type":"cite","label":"mill2022network","identifier":"mill2022network","children":[{"type":"text","value":"Mill ","key":"W8iXlMzlZ1"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"XMSPhk8zEb"}],"key":"BbAqk6yXhW"},{"type":"text","value":", 2022","key":"Q1ixfHTmWi"}],"key":"Dbw6QVVbYH"},{"type":"cite","label":"hearne2021activity","identifier":"hearne2021activity","children":[{"type":"text","value":"Hearne ","key":"D65BRe69Nw"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"VJsRimxJSE"}],"key":"gjpVWX4Jjv"},{"type":"text","value":", 2021","key":"bexYOWt2Er"}],"key":"cWF9SYDkGk"},{"type":"cite","label":"chen2018human","identifier":"chen2018human","children":[{"type":"text","value":"Chen ","key":"PsgW6pvy7c"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"OVKKoJrna8"}],"key":"eWrF5aA7T7"},{"type":"text","value":", 2018","key":"auiTn4mGwM"}],"key":"BxkCaHnpFU"}],"key":"ycMZVLcmun"},{"type":"text","value":".\nThe proposed approach is based on the intuition that the repeated, iterative application of the activity flow equation in a system exhibits close analogies with a type of recurrent artificial neural networks, known as Hopfield networks ","key":"fAdJ1xcm4n"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"hopfield1982neural","identifier":"hopfield1982neural","children":[{"type":"text","value":"Hopfield, 1982","key":"lu1XLruYo2"}],"key":"At2FTg6lSR"}],"key":"Ul7GUdd9yq"},{"type":"text","value":".","key":"xj9B9xipV5"}],"key":"mQD3015jwe"},{"type":"paragraph","position":{"start":{"line":393,"column":0},"end":{"line":395,"column":0}},"children":[{"type":"text","value":"Hopfield networks have been widely acknowledged for their relevance for brain function, including the ability to store and recall memories ","key":"wGNXaYiLq4"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"hopfield1982neural","identifier":"hopfield1982neural","children":[{"type":"text","value":"Hopfield, 1982","key":"dGlgLHQxSr"}],"key":"sTpMAx7D4V"}],"key":"NTk5ZE43mr"},{"type":"text","value":", self-repair ","key":"LDfyKijcHR"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"murre2003selfreparing","identifier":"murre2003selfreparing","children":[{"type":"text","value":"Murre ","key":"kdRPKLYIV3"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"uo1x5SDv1b"}],"key":"nqjm1E8UGY"},{"type":"text","value":", 2003","key":"I6HZwId9xt"}],"key":"ffZ5HluFG7"}],"key":"Ulwp78Tg9X"},{"type":"text","value":",\na staggering robustness to noisy or corrupted inputs ","key":"HrJATgGrME"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"hertz1991introduction","identifier":"hertz1991introduction","children":[{"type":"text","value":"Hertz ","key":"NBwpzho1vQ"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"zyt8Yg9Dho"}],"key":"fefa5l8kJa"},{"type":"text","value":", 1991","key":"vKLQX31Vco"}],"key":"AvflP0L97c"}],"key":"cnxddsTJMy"},{"type":"text","value":" (see also ","key":"uO9rFJfbFY"},{"type":"crossReference","kind":"figure","identifier":"si_noise_robustness_weights","label":"si_noise_robustness_weights","children":[{"type":"text","value":"Supplementary Figure ","key":"kQPCOZE7Kc"},{"type":"text","value":"7","key":"ghB9mYKjNS"}],"template":"Figure %s","enumerator":"7","resolved":true,"html_id":"si-noise-robustness-weights","remote":true,"url":"/supplement","dataUrl":"/supplement.json","key":"z6eOz6oDqF"},{"type":"text","value":") and the ability to produce multistable dynamics organized by the \"gravitational pull\" of a finite number of attractor states ","key":"i8OECrdu9N"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"khona2022attractor","identifier":"khona2022attractor","children":[{"type":"text","value":"Khona & Fiete, 2022","key":"lmSUYkzQY6"}],"key":"ZGApNxpJWX"}],"key":"Ld68EuGXXj"},{"type":"text","value":". 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","key":"HAAebSGD09"},{"type":"text","value":"1","key":"xQHe0EFhCY"}],"template":"Figure %s","enumerator":"1","resolved":true,"html_id":"si-expl-variance-energy","remote":true,"url":"/supplement","dataUrl":"/supplement.json","key":"DlYjvX0eSe"},{"type":"text","value":", ","key":"H1o0jbMcHg"},{"type":"crossReference","kind":"figure","identifier":"si_att_state_emergence_over_beta","label":"si_att_state_emergence_over_beta","children":[{"type":"text","value":"3","key":"rGn0GOcmDj"}],"template":"Figure %s","enumerator":"3","resolved":true,"html_id":"si-att-state-emergence-over-beta","remote":true,"url":"/supplement","dataUrl":"/supplement.json","key":"htwKGCR60C"},{"type":"text","value":", ","key":"S2qWR4iXT4"},{"type":"crossReference","kind":"figure","identifier":"si_state_occupancy_null_models","label":"si_state_occupancy_null_models","children":[{"type":"text","value":"4","key":"BMWPKPQSPh"}],"template":"Figure %s","enumerator":"4","resolved":true,"html_id":"si-state-occupancy-null-models","remote":true,"url":"/supplement","dataUrl":"/supplement.json","key":"OVr06MPBDy"},{"type":"text","value":", ","key":"BSBlOnn4ZQ"},{"type":"crossReference","kind":"figure","identifier":"si_pain_ghost_attractor_sim","label":"si_pain_ghost_attractor_sim","children":[{"type":"text","value":"5","key":"rdojDDTdXg"}],"template":"Figure %s","enumerator":"5","resolved":true,"html_id":"si-pain-ghost-attractor-sim","remote":true,"url":"/supplement","dataUrl":"/supplement.json","key":"bsErawpBOn"},{"type":"text","value":", 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To underscore the potency of this simplicity and stability, in the present work, we avoided any unnecessary parameter optimization. It is likely, however, that extensive parameter optimization could further improve the performance of the model.","key":"qlslmARFMm"}],"key":"KtwTUyuEFz"},{"type":"paragraph","position":{"start":{"line":401,"column":0},"end":{"line":402,"column":0}},"children":[{"type":"text","value":"Another advantage of ","key":"jZftaJ7iAh"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"PaW8I2Drbq"}],"key":"sOy4tMh2E0"},{"type":"text","value":"s over more detailed models is that ","key":"oILJKWgnib"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"Vp8aL3x1FD"}],"key":"AK6PU23KNW"},{"type":"text","value":"s establish a simple and easily interpretable link between two highly prevalent metrics of brain function: functional connectivity and brain activity. This connection is not solely phenomenological, but also mathematical, facilitating the exploration and prediction of alterations in the system's dynamics in response to perturbations affecting both activity and connectivity.","key":"N51Fjez4P0"}],"key":"OscdwxlICu"},{"type":"paragraph","position":{"start":{"line":403,"column":0},"end":{"line":404,"column":0}},"children":[{"type":"text","value":"The proposed model also exhibits several advantages over linear network control theory-based ","key":"Z8Ff4WrEv4"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"gu2015controllability","identifier":"gu2015controllability","children":[{"type":"text","value":"Gu ","key":"j0s6Uxsdle"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"DqfGSQi2kq"}],"key":"rNrtupF44I"},{"type":"text","value":", 2015","key":"jVQVUp6fes"}],"key":"ULAu81kVrT"}],"key":"Q0HtZcK8cX"},{"type":"text","value":" approaches. First, the ","key":"ja7Nrc7h7q"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"NCBtFlXd07"}],"key":"y2oTKmZisZ"},{"type":"text","value":" approach works with direct activity flow estimates and does not require knowledge about the structural-functional coupling in the brain. Second, the ","key":"KUqJprXrfq"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"DOQMl6jP2n"}],"key":"WVHv8oW10z"},{"type":"text","value":" approach is based on a non-linear ","key":"LSUJnUbzRE"},{"type":"abbreviation","title":"Artificial Neural Network","children":[{"type":"text","value":"ANN","key":"pyonswxvMr"}],"key":"HJpQ1CwYhI"},{"type":"text","value":" architecture, thus, similarly to neuroconnectionist approaches, allows leveraging on knowledge about the ","key":"NMR8leYTKd"},{"type":"abbreviation","title":"Artificial Neural Network","children":[{"type":"text","value":"ANN","key":"g80FkETOne"}],"key":"Nd4Xq8cnLq"},{"type":"text","value":" architecture itself. Specifically, the ","key":"p6jtMaSIBz"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"DMZatoedUJ"}],"key":"nMtP6CEY8v"},{"type":"text","value":"s provide a mechanistic account for the emergence of large-scale canonical brain networks ","key":"GCfHZcPC75"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"zalesky2014time","identifier":"zalesky2014time","children":[{"type":"text","value":"Zalesky ","key":"MxVz56mDq6"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"dOnOO2BVtP"}],"key":"lDw7cKpkwD"},{"type":"text","value":", 2014","key":"aIe1loyYqb"}],"key":"zyLwzvrp97"}],"key":"dLtvSg77AQ"},{"type":"text","value":" and brain states or the presence of \"ghost attractors\" ","key":"erbGcZcqc3"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"deco2012ongoing","identifier":"deco2012ongoing","children":[{"type":"text","value":"Deco & Jirsa, 2012","key":"Lt4DITBsSR"}],"key":"d0YDh3peXh"},{"type":"cite","label":"vohryzek2020ghost","identifier":"vohryzek2020ghost","children":[{"type":"text","value":"Vohryzek ","key":"zz7QjHloEQ"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"xI6v1oMgKD"}],"key":"ulZPcqOfJJ"},{"type":"text","value":", 2020","key":"Js6RpXJimZ"}],"key":"oB3BBIx6UF"}],"key":"EjaqalVoB8"},{"type":"text","value":", via the key concept in the Hopfield network framework, the attractor states.","key":"yXFFUB2mF8"}],"key":"SI1pQBPiPr"},{"type":"paragraph","position":{"start":{"line":405,"column":0},"end":{"line":406,"column":0}},"children":[{"type":"text","value":"In comparison to conventional neuroconnectionist approaches, ","key":"foLoxCL31G"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"Hf0MTjPA8p"}],"key":"dzVycEoBgM"},{"type":"text","value":"s do not need to be trained to solve tasks and thus allow for the exploration of spontaneous brain dynamics. However, it is worth mentioning that, like any other ","key":"KFJaNC67LV"},{"type":"abbreviation","title":"Artificial Neural Network","children":[{"type":"text","value":"ANN","key":"jJQn2I6NzD"}],"key":"Wx7O6C3GhZ"},{"type":"text","value":"s, ","key":"NAE0hft3YH"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"R5HTPYxbni"}],"key":"nFXmh1ItSG"},{"type":"text","value":"s can also be further trained via established ","key":"OQF7krpi78"},{"type":"abbreviation","title":"Artificial Neural Network","children":[{"type":"text","value":"ANN","key":"HN3Osu9sAB"}],"key":"r3zMfzOPxj"},{"type":"text","value":" training techniques (e.g. via the Hebbian learning rule) to \"solve\" various tasks or to match altered dynamics during development or in clinical populations. In this interesting future direction, the training procedure itself becomes part of the model, providing testable hypotheses about the formation, and various malformations, of brain dynamics.","key":"TAmAV9BpqU"}],"key":"m3WlFnIrmj"},{"type":"paragraph","position":{"start":{"line":407,"column":0},"end":{"line":410,"column":0}},"children":[{"type":"text","value":"Given its simplicity, it is remarkable, if not surprising, how accurately the ","key":"S0ETJdwwZ2"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"Qw9n5u74ov"}],"key":"luiBpXo7az"},{"type":"text","value":" model is able to reconstruct and predict brain dynamics under a wide range of conditions. Particularly interesting is the result that the two-dimensional ","key":"U6M1EvYLh5"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"hw5F1RApsJ"}],"key":"LndCVvNirb"},{"type":"text","value":" projection can explain more variance in real resting state ","key":"Dhrbjvg3PZ"},{"type":"abbreviation","title":"functional Magnetic Resonance Imaging","children":[{"type":"text","value":"fMRI","key":"ZgjNQeIiU1"}],"key":"JGpxLPaYLz"},{"type":"text","value":" data than the first two principal components derived from the data itself.\nA plausible explanation for the remarkable reconstruction performance is that, trough their known noise tolerance, ","key":"kScEHvySt5"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"VoXBskLK59"}],"key":"EPXga1Qjin"},{"type":"text","value":"s are able to capture essential principles of the underlying dynamic processes even if our empirical measurements are corrupted by noise and low sampling rate.\nIndeed, ","key":"GV9btwCAW4"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"gtdRXONPNO"}],"key":"HaH3M13CkK"},{"type":"text","value":" attractor states were found to be robust to noisy weights (","key":"E2N2R1WDjz"},{"type":"crossReference","kind":"figure","identifier":"si_noise_robustness_weights","label":"si_noise_robustness_weights","children":[{"type":"text","value":"Supplementary Figure ","key":"ME5wJVuDZQ"},{"type":"text","value":"7","key":"LKD7nRvTNs"}],"template":"Figure %s","enumerator":"7","resolved":true,"html_id":"si-noise-robustness-weights","remote":true,"url":"/supplement","dataUrl":"/supplement.json","key":"xiOJN48YM1"},{"type":"text","value":") and highly replicable across datasets acquired at different sites, with different scanners and imaging sequences (study 2 and 3). The observed level of replicability allowed us to re-use the ","key":"phqWtH8Mch"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"W5gd5cyIrL"}],"key":"Q1XYZdrBQ6"},{"type":"text","value":" model constructed with the connectome of study 1 for all subsequent analyses, without any further fine-tuning or study-specific parameter optimization.","key":"l9bPnLatFt"}],"key":"LmUlC7zqbZ"},{"type":"paragraph","position":{"start":{"line":411,"column":0},"end":{"line":413,"column":0}},"children":[{"type":"text","value":"Conceptually, the notion of a global attractor model of the brain network is not new ","key":"Y1PEsfezRE"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"deco2012ongoing","identifier":"deco2012ongoing","children":[{"type":"text","value":"Deco & Jirsa, 2012","key":"cY8msdzHxk"}],"key":"lVmkowCyAw"}],"key":"EAFOesJwy6"},{"type":"text","value":". The present work shows, however, that the brain as an attractor network necessarily 'leaks' its code in form of the partial correlation across the regional timeseries, allowing us to uncover its large-scale attractor states. Moreover, we demonstrate that the brain's attractor states are not solely local minima in the state-space but act as a driving force for the dynamic trajectories of brain activity. Attractor-dynamics may be the main driving factor for the spatial and temporal autocorrelation structure of the brain, recently described to be predictive of network topology in relation to age, subclinical symptoms of dementia, and pharmacological manipulations with serotonergic drugs ","key":"BnIugC4LJW"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"shinn2023functional","identifier":"shinn2023functional","children":[{"type":"text","value":"Shinn ","key":"llZZvtpNxu"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"KhwTH7LCQn"}],"key":"ghHHRuopHS"},{"type":"text","value":", 2023","key":"dK5YCRvroY"}],"key":"KnpM95I0bT"}],"key":"VjqcCoBan4"},{"type":"text","value":".\nNevertheless, attractor states should not be confused with the conventional notion of brain states ","key":"pnOQFJy0WD"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"chen2015introducing","identifier":"chen2015introducing","children":[{"type":"text","value":"Chen ","key":"VSQRtCddRP"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"X4x0OXZ5CE"}],"key":"FBvmAZ59bU"},{"type":"text","value":", 2015","key":"NfJTHZfyMB"}],"key":"OH47wlZGMc"}],"key":"hYHxm6tROh"},{"type":"text","value":" and large-scale functional gradients ","key":"JnD1hK87D4"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"margulies2016situating","identifier":"margulies2016situating","children":[{"type":"text","value":"Margulies ","key":"fAg2Dq59xv"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"Ya3cbnzCxv"}],"key":"cm1IU1TlP2"},{"type":"text","value":", 2016","key":"vRPzDNrP61"}],"key":"ABK0AGbaFT"}],"key":"NHZacyBwnf"},{"type":"text","value":". In the ","key":"XOCuZjQ1Vu"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"Wz00HIZyho"}],"key":"Nd14noYcBL"},{"type":"text","value":" framework, attractor states can rather be conceptualized as \"Platonic idealizations\" of brain activity, that are continuously approximated - but never reached - by the brain, resulting in re-occurring patterns (brain states) and smooth gradual transitions (large-scale gradients).","key":"K2juQmKQ4K"}],"key":"gxXBf88zln"},{"type":"paragraph","position":{"start":{"line":414,"column":0},"end":{"line":416,"column":0}},"children":[{"type":"text","value":"Relying on previous work, we can establish a relatively straightforward (although somewhat speculative) correspondence between attractor states and brain function, mapping brain activation on the axes of internal vs. external context ","key":"Z8sjNzLrbK"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"golland2008data","identifier":"golland2008data","children":[{"type":"text","value":"Golland ","key":"pJ8T4xuMIG"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"P5zkXC2BmY"}],"key":"NlEs5LoTI7"},{"type":"text","value":", 2008","key":"byvqaxGPbU"}],"key":"l4XrsUADyk"},{"type":"cite","label":"cioli2014differences","identifier":"cioli2014differences","children":[{"type":"text","value":"Cioli ","key":"GrqbafFwLM"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"II8GlC1bM4"}],"key":"mMHLocVhY6"},{"type":"text","value":", 2014","key":"PHxxloZvUH"}],"key":"cm6sWKrwzr"}],"key":"iAY5iNWYSA"},{"type":"text","value":", as well as perception vs. action ","key":"cbo0i94zVe"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"fuster2004upper","identifier":"fuster2004upper","children":[{"type":"text","value":"Fuster, 2004","key":"epzw4XbinJ"}],"key":"LJSMZ2RMgD"}],"key":"WyCq2mVwjj"},{"type":"text","value":".\nThis four-attractor architecture exhibits an appealing analogy with Friston's free energy principle ","key":"Ao9ZaUTC5R"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"friston2006free","identifier":"friston2006free","children":[{"type":"text","value":"Friston ","key":"z9ENl0g61D"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"e0PmPFCVI9"}],"key":"FlwaWmsAND"},{"type":"text","value":", 2006","key":"yjBNSZCjve"}],"key":"MyrrbLPA4u"}],"key":"TTzYhK4O4p"},{"type":"text","value":" that postulate the necessary existence of subsystems for active and perceptual inference as well as a hierarchically organized (i.e. external and internal) subsystems that give rise to consciousness ","key":"lOihcitGEd"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"ramstead2023inner","identifier":"ramstead2023inner","children":[{"type":"text","value":"Ramstead ","key":"ozjKxzUj4t"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"TEo2ULsezl"}],"key":"fLYcWG8ZKh"},{"type":"text","value":", 2023","key":"ATfPAtlsM6"}],"key":"sZEyHgdPPZ"},{"type":"cite","label":"lee2023life","identifier":"lee2023life","children":[{"type":"text","value":"Lee ","key":"kvRZ6CF6OB"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"XBkoby2T72"}],"key":"kQwqwO8Zr7"},{"type":"text","value":", 2023","key":"bvcPZm54fo"}],"key":"RZTugvxJPh"}],"key":"RkQdIMprUj"},{"type":"text","value":".","key":"YIblrERVUu"}],"key":"AgM1HpcFJL"},{"type":"paragraph","position":{"start":{"line":417,"column":0},"end":{"line":420,"column":0}},"children":[{"type":"text","value":"Both conceptually and in terms of analysis practices, resting and task states are often treated as separate phenomena. However, in the ","key":"FUkSjzwdsG"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"HNiLyqLplN"}],"key":"yRgjTk0ub7"},{"type":"text","value":" framework, the differentiation between task and resting states is considered an artificial dichotomy.\nTask-based brain activity in the ","key":"RFIED6qyc9"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"ORr8Hue7qC"}],"key":"KU9MiIzCSh"},{"type":"text","value":" framework is not a mere response to external stimuli in certain brain locations but a perturbation of the brain's characteristic dynamic trajectories, with increased preference for certain locations on the energy landscape (\"ghost attractors\").\nIn our analyses, the ","key":"BEDqRkDlLA"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"MC0XEyRLAR"}],"key":"oNbvo4fZU4"},{"type":"text","value":" approach capture and predict participant-level activity changes induced by pain and its self-regulation and gave a mechanistic account for how relatively small activity changes in a single region (NAcc) may result in a significantly altered pain experience.","key":"oBMvviclVG"}],"key":"o2mSqNAqHp"},{"type":"paragraph","position":{"start":{"line":421,"column":0},"end":{"line":422,"column":0}},"children":[{"type":"text","value":"Brain dynamics can not only be perturbed by task or other types of experimental or naturalistic interventions, but also by pathological alterations. Here we have demonstrated (study 7) that ","key":"CCIboNHbPJ"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"j2Dhp8XCJt"}],"key":"ohKmODRDTg"},{"type":"text","value":"-based analyses can characterize and predict altered brain dynamics in autism spectrum disorder (","key":"nEW6B0AhOW"},{"type":"abbreviation","title":"Autism Spectrum Disorder","children":[{"type":"text","value":"ASD","key":"vqjhtjotlx"}],"key":"wc5hhVJMAK"},{"type":"text","value":"). The observed ","key":"jXdTJ5ezRN"},{"type":"abbreviation","title":"Autism Spectrum Disorder","children":[{"type":"text","value":"ASD","key":"TVLaeusq6d"}],"key":"qleqsjkiVp"},{"type":"text","value":"-associated changes in brain dynamics are indicative of a reduced ability to flexibly switch between internal and external modes of processing, corroborating previous findings that in ","key":"El1DkukieL"},{"type":"abbreviation","title":"Autism Spectrum Disorder","children":[{"type":"text","value":"ASD","key":"kEg9Fxqj8l"}],"key":"ZwChduU7rg"},{"type":"text","value":", sensory-driven connectivity transitions do not converge to transmodal areas ","key":"BQPguV2ana"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"hong2019atypical","identifier":"hong2019atypical","children":[{"type":"text","value":"Hong ","key":"Ma9NeDJYaH"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"W3VRAw1sq6"}],"key":"SzldsFWPOu"},{"type":"text","value":", 2019","key":"n1DQRuVga3"}],"key":"aa7mfVUqgV"}],"key":"z8fG1DLXkV"},{"type":"text","value":". Such findings are in line with previous reports of a reduced influence of context on the interpretation of incoming sensory information in ","key":"UOMFr3Ve00"},{"type":"abbreviation","title":"Autism Spectrum Disorder","children":[{"type":"text","value":"ASD","key":"HLvODFRcsA"}],"key":"vLfI4y2EMw"},{"type":"text","value":" (e.g. the violation of Weber's law) ","key":"vccMiIDQBP"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"hadad2019perception","identifier":"hadad2019perception","children":[{"type":"text","value":"Hadad & Schwartz, 2019","key":"qWZntqjzCq"}],"key":"VVroymhbGL"}],"key":"rxy771nvxO"},{"type":"text","value":".","key":"lBthxhQa07"}],"key":"yZyg4pvtsU"},{"type":"paragraph","position":{"start":{"line":423,"column":0},"end":{"line":424,"column":0}},"children":[{"type":"text","value":"Together, our findings open up a series of exciting opportunities for the better understanding of brain function in health and disease.","key":"rvqgckrqvs"}],"key":"YW3pYOzJkS"},{"type":"paragraph","position":{"start":{"line":425,"column":0},"end":{"line":426,"column":0}},"children":[{"type":"text","value":"First, the 2-dimensional ","key":"msSXDLfrbY"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"MsPSGE8NEV"}],"key":"FSfthPFXhm"},{"type":"text","value":" projection offers a streamlined framework not only for the visualization, but also for the ","key":"TC7RHeUTaE"},{"type":"emphasis","children":[{"type":"text","value":"interpretation","key":"VyOVSTAUDM"}],"key":"u0jaZNLc5X"},{"type":"text","value":", of brain activity patterns, as it conceptualizes changes related to various behavioral or clinical states or traits as a shift in brain dynamics in relation to brain attractor states.","key":"mYMEEjg0XG"}],"key":"vycTtotxsh"},{"type":"paragraph","position":{"start":{"line":427,"column":0},"end":{"line":428,"column":0}},"children":[{"type":"text","value":"Second, ","key":"y8IZ8Kkwmg"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"geI3BhCmny"}],"key":"x2F8dO0Lm1"},{"type":"text","value":" analyses may provide insights into the causes of changes in brain dynamics, by for instance, identifying the regions or connections that act as an \"Achilles heel\" in generating such changes. Such analyses could, for instance, aid the differentiation of primary causes and secondary effects of particular activity or connectivity changes in various clinical conditions.","key":"sMTUMTS1mo"}],"key":"kDYpTLIpmf"},{"type":"paragraph","position":{"start":{"line":429,"column":0},"end":{"line":430,"column":0}},"children":[{"type":"text","value":"Third, the ","key":"CLqn4Zb547"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"qBNPnAYlNh"}],"key":"Z4pyAhN5X2"},{"type":"text","value":" approach can provide testable predictions about the effects of pharmacological interventions as well as non-invasive brain stimulation (e.g. transcranial magnetic or direct current stimulation, focused ultrasound, etc) and neurofeedback. Obtaining the optimal stimulation or treatment target within the ","key":"SFF3DtUkCs"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"m563RwhHKm"}],"key":"deMScaPDgX"},{"type":"text","value":" framework (e.g. by means of network control theory ","key":"ZHcgVOPrpe"},{"type":"citeGroup","kind":"parenthetical","children":[{"type":"cite","label":"liu2011controllability","identifier":"liu2011controllability","children":[{"type":"text","value":"Liu ","key":"eeikR7u8BQ"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"dCZSJkSdOT"}],"key":"oj0P7cRSfg"},{"type":"text","value":", 2011","key":"ZTvQ4T3m3O"}],"key":"vZ4EpOuAEL"}],"key":"CPfUNuJaJN"},{"type":"text","value":") is one of the most promising future directions with the potential to significantly advance the development of novel, personalized treatment approaches.","key":"Oulxqx0Hkr"}],"key":"PbPP90iIah"},{"type":"paragraph","position":{"start":{"line":431,"column":0},"end":{"line":433,"column":0}},"children":[{"type":"text","value":"In this initial work, we presented the simplest possible implementation of the ","key":"nozr5v7txU"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"b1g6yEtmng"}],"key":"Lb3S98nKXP"},{"type":"text","value":" concept. It is clear that the presented analyses exploit only a small proportion of the richness of the full state-space dynamics reconstructed by the ","key":"iQLSjE3LKT"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"uP5Lo6M6Au"}],"key":"v6k1ZY9CeI"},{"type":"text","value":" model.\nThere are many potential ways to further improve the utility of the ","key":"v1ZhjUgybN"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"kegNwmVWmr"}],"key":"MM91VtPhpx"},{"type":"text","value":" approach. Increasing the number of reconstructed attractor states (by increasing the temperature parameter), investigating higher-dimensional dynamics, fine-tuning the hyperparameters, testing the effect of different initializations and perturbations are all important direction for future work, with the potential to further improve the model's accuracy and usefulness.","key":"h3LDfDlCoV"}],"key":"t6k0pfT3UR"},{"type":"comment","value":"**other potential topics**:\n - is the functional connectome stationary? Why don't we use dynamic connectivity? See arguments by the Cole-group. Also, the fcHNN model can actually probably also reproduce task-based connectivity, when adding a task-related control signal to the stochastic relaxation procedure (as on Fig. 3). Thus it could be a model of how task-based connectivity and dynamic connectivity changes arise from the underlying rs-fMRI connectome. Maybe it could be even better to use \"latent-FC\" a'la McCormick, 2022, [](https://doi.org/10.1162/netn_a_00234))\n - why no HRF modelling (could be a possible extension, but it is also not part of the activity flow approach and we don't reconstruct time series, per-se, but rather activations)\n - the fcHNN model is not a model of brain function, but a model of brain dynamics. It does not strive to explain various brain regions ability to perform certain computations, but the brain's characteristic dynamic \"trajectories\", and how these are perturbed by tasks and other types of interventions.","position":{"start":{"line":434,"column":0},"end":{"line":438,"column":0}},"key":"PFxyjfKjzL"},{"type":"heading","depth":2,"position":{"start":{"line":439,"column":0},"end":{"line":440,"column":0}},"children":[{"type":"text","value":"Conclusion","key":"JicDkYaaS8"}],"identifier":"conclusion","label":"Conclusion","html_id":"conclusion","implicit":true,"key":"ZkkdCSQWZ4"},{"type":"paragraph","position":{"start":{"line":441,"column":0},"end":{"line":442,"column":0}},"children":[{"type":"text","value":"To conclude, here we have proposed a lightweight, high-level computational framework that accurately captures and predicts brain dynamics under a wide range of conditions. The framework models large-scale activity flow in the brain with a recurrent artificial neural network architecture that, instead of being trained to solve specific tasks or mimic certain dynamics, is simply initialized with the empirical functional connectome. The framework identifies neurobiologically meaningful attractor states and provides a model for how these restrict brain dynamics. The proposed framework, referred to as the connectome-based Hopfield neural network (","key":"IbdEpfMLrD"},{"type":"abbreviation","title":"functional connectome-based Hopfield Neural Network","children":[{"type":"text","value":"fcHNN","key":"k5yzU5ny3r"}],"key":"pywEMYLetH"},{"type":"text","value":") model, can accurately reconstruct and predict brain dynamics under a wide range of conditions, including resting state, task-induced activity changes, as well as brain disorders. Fc","key":"Lx95kDQq1G"},{"type":"abbreviation","title":"Hopfield Neural Network","children":[{"type":"text","value":"HNN","key":"aFKTJqYINI"}],"key":"iKzRDipyGC"},{"type":"text","value":"s establish a conceptual link between connectivity and activity and offer a simple, robust, and highly interpretable computational alternative to conventional descriptive approaches to investigating brain function. The generative nature of the proposed model opens up a series of exciting opportunities for future research, including predicting the effect, and understanding the mechanistic bases, of various interventions; thereby paving the way for designing novel treatment approaches.","key":"rOKeNawWnO"}],"key":"xdXMSXnDxO"}],"key":"ViZDLAAQRp"},{"type":"block","position":{"start":{"line":444,"column":0},"end":{"line":445,"column":0}},"children":[{"type":"heading","depth":2,"position":{"start":{"line":445,"column":0},"end":{"line":446,"column":0}},"children":[{"type":"text","value":"Acknowledgements","key":"doq1QQL41l"}],"identifier":"acknowledgements","label":"Acknowledgements","html_id":"acknowledgements","implicit":true,"key":"AGyyfwRVHB"},{"type":"paragraph","position":{"start":{"line":447,"column":0},"end":{"line":448,"column":0}},"children":[{"type":"text","value":"The work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation; projects ‘TRR289 - Treatment Expectation’, ID 422744262 and ‘SFB1280 - Extinction Learning’, ID 316803389) and by IBS-R015-D1 (Institute for Basic Science; C.W.-W.).","key":"A47cxvuTWO"}],"key":"vFfhFwIcBG"}],"data":{"part":"acknowledgements"},"key":"rTSm3AIt2p"},{"type":"block","position":{"start":{"line":448,"column":0},"end":{"line":449,"column":0}},"key":"WEOg8HLq1y"},{"type":"block","position":{"start":{"line":450,"column":0},"end":{"line":451,"column":0}},"children":[{"type":"heading","depth":2,"position":{"start":{"line":452,"column":0},"end":{"line":453,"column":0}},"children":[{"type":"text","value":"Analysis source code","key":"oT3vqttkNW"}],"identifier":"analysis-source-code","label":"Analysis source code","html_id":"analysis-source-code","implicit":true,"key":"ciBZUfrLZl"},{"type":"paragraph","position":{"start":{"line":453,"column":0},"end":{"line":454,"column":0}},"children":[{"type":"link","url":"https://github.com/pni-lab/connattractor","children":[{"type":"text","value":"https://​​github​​.com​​/pni​​-lab​​/connattractor","key":"i3zWQWAS4h"}],"urlSource":"https://github.com/pni-lab/connattractor","error":true,"key":"hhooJHbUrS"}],"key":"LmNHambHOM"},{"type":"heading","depth":2,"position":{"start":{"line":455,"column":0},"end":{"line":456,"column":0}},"children":[{"type":"text","value":"Project website","key":"SOjlTviDXQ"}],"identifier":"project-website","label":"Project website","html_id":"project-website","implicit":true,"key":"JPnWLQQVGg"},{"type":"paragraph","position":{"start":{"line":456,"column":0},"end":{"line":457,"column":0}},"children":[{"type":"link","url":"https://pni-lab.github.io/connattractor/","children":[{"type":"text","value":"https://​​pni​​-lab​​.github​​.io​​/connattractor​​/","key":"kNMeznFFr8"}],"urlSource":"https://pni-lab.github.io/connattractor/","key":"ssIyhNd3qh"}],"key":"M01LsKrRvr"},{"type":"heading","depth":2,"position":{"start":{"line":458,"column":0},"end":{"line":459,"column":0}},"children":[{"type":"text","value":"Data availability","key":"rLF1LtdetK"}],"identifier":"data-availability","label":"Data availability","html_id":"data-availability","implicit":true,"key":"OoqR6rdbqU"},{"type":"paragraph","position":{"start":{"line":459,"column":0},"end":{"line":460,"column":0}},"children":[{"type":"text","value":"Study 1, 2 and 4 is available at ","key":"i70xkzQBms"},{"type":"link","url":"http://openneuro.org","children":[{"type":"text","value":"openneuro.org","key":"JPWbi7mcW6"}],"urlSource":"http://openneuro.org","key":"LJe19z35Np"},{"type":"text","value":" (ds002608, ds002608, ds000140). Data for study 3 is available upon request. Data for study 5-6 is available at the github page of the project: ","key":"OT49iq13qz"},{"type":"link","url":"https://github.com/pni-lab/connattractor","children":[{"type":"text","value":"https://​​github​​.com​​/pni​​-lab​​/connattractor","key":"QG3dUQRvsl"}],"urlSource":"https://github.com/pni-lab/connattractor","error":true,"key":"TPsipee464"},{"type":"text","value":". Study 7 is available at https://fcon_1000.projects.nitrc.org/indi/abide/, preprocessed data is available at ","key":"hbwoIXKpaR"},{"type":"link","url":"http://preprocessed-connectomes-project.org/","children":[{"type":"text","value":"http://​​preprocessed​​-connectomes​​-project​​.org​​/","key":"xbYZabhl4u"}],"urlSource":"http://preprocessed-connectomes-project.org/","key":"jOhD9PYtfK"},{"type":"text","value":".","key":"Vg0WBNaNXp"}],"key":"jjeb1lQ767"}],"data":{"part":"data-availability"},"key":"mLApwe7QtX"},{"type":"block","position":{"start":{"line":460,"column":0},"end":{"line":461,"column":0}},"key":"RF1tUFr4yL"}],"key":"Hdzw5pjKzb"},"references":{"cite":{"order":["buzsaki2006rhythms","bassett2017network","liu2013time","zalesky2014time","margulies2016situating","huntenburg2018large","greene2023everyone","vidaurre2017brain","smith2012temporally","chen2018human","hutchison2013dynamic","barttfeld2015signature","meer2020movie","breakspear2017dynamic","murray2018biophysical","kriegeskorte2018cognitive","heinz2019towards","schirner2022dynamic","schiff1994controlling","papadopoulos2017development","chiem2021structure","scheid2021time","gu2015controllability","doerig2023neuroconnectionist","richards2019deep","cole2016activity","ito2017cognitive","hopfield1982neural","krotov2023new","koiran1994dynamics","daunizeau2012stochastic","robinson2005multiscale","golland2008data","cioli2014differences","fuster2004upper","woo2015distinct","vohryzek2020ghost","liu2011controllability","zunhammer2021meta","di2014autism","hadad2019perception","padmanabhan2017default","mill2022network","hearne2021activity","murre2003selfreparing","hertz1991introduction","khona2022attractor","cabral2017functional","deco2012ongoing","shinn2023functional","chen2015introducing","friston2006free","ramstead2023inner","lee2023life","hong2019atypical"],"data":{"buzsaki2006rhythms":{"number":1,"html":"Buzsaki, G. 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