Basic Introduction to Maths and Python for Neuroscience

Developed and taught by John S Butler

Description of Module

This is a very short introduction into simple mathematical functions that are used in behavourial and neurophysiolgical papers. The python code below is motivated by data and plots from papers to illusrate the use and power of the line, sigmoid function, and sinewaves to analyse and interpret data.

Module Content

Behavioural Examples

Applying simple examples of the line and a psychometric function used Behavioural and Clinical Neuroscience to illustrate Python functions, the tutorials and solutions open in colab.

Tutorial 1 Plotting the line [1].

Tutorial	Solution

Tutorial 2 Two lines [1].

Tutorial	Solution

Tutorial 3 The Psychometric Function [2].

Tutorial	Solution

Tutorial 4 The Psychometric function for multisensory integration [2].

Tutorial	Solution

Neurophysiolgical Examples

Using python to implement simple examples of Spike train analysis, tuning functions, frequencies and fast fourier transform.

Tutorial 1 Single Spike Train [3].

Tutorial	Solution

Tutorial 2 Multiple Spike Trains [3].

Tutorial	Solution

Tutorial 3 The Tuning Function [4].

Tutorial	Solution

Tutorial 4 Frequencies [5].

Tutorial	Solution

Tutorial 5 Fourier Transform [5].

Tutorial	Solution

The sound of different spike patterns [6]

References

[1] Butler, John S., et al. "Non-parametric bootstrapping method for measuring the temporal discrimination threshold for movement disorders." Journal of neural engineering 12.4 (2015): 046026.

[2] Ernst, Marc O., and Martin S. Banks. "Humans integrate visual and haptic information in a statistically optimal fashion." Nature 415.6870 (2002): 429-433.

[3] Meredith, M. A., & Stein, B. E. (1986). Visual, auditory, and somatosensory convergence on cells in superior colliculus results in multisensory integration. Journal of neurophysiology, 56(3), 640-662.

[4] Britten, Kenneth H., et al. "The analysis of visual motion: a comparison of neuronal and psychophysical performance." Journal of Neuroscience 12.12 (1992): 4745-4765.

[5] Fiebelkorn, I. C., Foxe, J. J., Butler, J. S., Mercier, M. R., Snyder, A. C., & Molholm, S. (2011). Ready, set, reset: stimulus-locked periodicity in behavioral performance demonstrates the consequences of cross-sensory phase reset. Journal of Neuroscience, 31(27), 9971-9981.

[6] Izhikevich, E. M. (2003). Simple model of spiking neurons. IEEE Transactions on neural networks, 14(6), 1569-1572.

Supplemental References

Dayan, P., & Abbott, L. F. (2001). Theoretical neuroscience: computational and mathematical modeling of neural systems. Computational Neuroscience Series.

More Advanced Modules

Mathematical Tools for Neuroscience by Ella Batty

Butler, J. (2023, December 14). Numerical Methods and Machine Learning for Differential Equations with Applications in Python. Zenodo. https://doi.org/10.5281/zenodo.10376815

Neuromatch Academy Materials

't Hart, B. M., Achakulvisut, T., Blohm, G., Kording, K., Peters, M. A. K., Akrami, A., Alicea, B., et al. (2021, February 15). Neuromatch Academy: a 3-week, online summer school in computational neuroscience. OSF Preprints. Retrieved from [https://osf.io/9fp4v/]

Neuromatch Academy GitHub Repository

Neuromatch Computational Neuroscience Summer School

Neuromatch Deep Learning Summer School

Supplemental Popular Reading List

Lindsay, G. (2021). Models of the Mind: How Physics, Engineering and Mathematics Have Shaped Our Understanding of the Brain. Bloomsbury Publishing.

Strogatz, S. (2004). Sync: The emerging science of spontaneous order. Penguin UK.

Humphries, M. (2021). The Spike. In The Spike. Princeton University Press.