In some recent email exchanges I've realized that when people by some coincidence make it to this blog, they rarely end up visiting my main website, and even if they do, they rarely browse through the teaching materials. This is not really a complaint, I hardly ever visit my website myself, but there are some materials there that I go back to every once in a while (though I have copies on my laptop). These are the lecture notes I made for a lecture on mathematical foundations of neuroscience.
As a bit of a background, in 2009 after I defended my PhD and before I joined Brain Corporation I was briefly an Adjunct Professor at the Faculty of Mathematics and Computer Science Nicolaus Copernicus University in Torun. During that time I decided to refresh everything I gathered about mathematics of neuroscience and prepare a lecture series complete with exercises, lots of pictures, graphs, and all the necessary theory. And even though 9 years have passed since then, the lectures hold up pretty well, hence why not bring that content to a broader audience?
The lecture consists of 15 main pdf presentations, a number of sample exercises as well as a large collection of animations and matlab/octave scripts. Unfortunately the lecture itself was not recorded. Anyway, if you are interested in this stuff, enjoy [please report any broken links/typos]:
Lecture presentations:
- Lecture 1 - Basic facts about the brain and its analysis, Sample questions
- Lecture 2 - Electrophysiology of neurons, Sample questions
- Lecture 3 - Electrophysiology of neurons (continued), Sample questions
- Lecture 4 - 1d systems, Sample questions
- Lecture 5 - 2d systems, Sample questions
- Lecture 6 - Bifurcations in 2d systems, Sample questions
- Lecture 7 - Bifurcations II, Sample questions
- Lecture 8 - Classes of excitability, Sample questions
- Lecture 9 - Simple models of neurons and synapses, Sample questions
- Lecture 10 - Bursting, Sample questions
- Lecture 11 - Bursting continued Sample questions
- Lecture 12 - Synchronization Sample questions
- Lecture 13 - Coupled Oscillators Sample questions
- Lecture 14 - Synchronization of neurons and synaptic plasticity Sample questions
- Lecture 15 - Building large models step by step Sample questions
Programs/scripts/animations:
- DTI tracking with Slicer 3d - a QuickTime animation
- Slicer3d program webpage - an opensource medical visualization software. Also check the tutorials which include the nescessary data files.
- Numerical stability demo - Matlab/Octave script
- Saddle node bifurcation of a 1d system - Matlab/Octave script
- Quadratic integrate and fire neuron - Matlab/Octave script
- Saddle-node bifurcation in 1d system - a QuickTime animation
- Saddle-node on invariant cicrle bifurcation in the Na-p-K model - a QuickTime animation
- Phase portrait plot - Matlab/Octave script and sample fields to plot: sample field (pdf image), sample field 2 (pdf image), sample field 3 (pdf image), sample field 4 (pdf image)
- Fold Cycle bifurcation in the canonical model - a QuickTime animation
- Saddle Homoclinic Orbit bifurcation in the Na-p-K model - a QuickTime animation
- Supercritical Andronov-Hopf bifurcation in the Na-p-K model - a QuickTime animation
- Subcritical Andronov-Hopf bifurcation in the Na-p-K model - a QuickTime animation
- Saddle node bifurcation in the Na-p-K model - a QuickTime animation
- Dynamic regimes near Bogdanov-Takens bifurcation - a QuickTime animation
- A script simulating an array of coupled neurons - good student project seed - Matlab/Octave script
- Lecture notes of Georgi Medvedev, useful supplementary material
- Publications of Eugene Izhikevich, useful supplementary material
- A script simulating a piece of an axon with the cable equation - good student project seed - Matlab/Octave script
- An animation of an action potential propagating spatialy along the fiber - a QuickTime animation
- Spiking Neuron Models Single Neurons, Populations, Plasticity a book by Wulfram Gerstner and Werner M. Kistler - useful supplementary material
- A script simulating a piece of an axon with the cable equation (FitzHugh-Nagumo) - good student project seed - Matlab/Octave script
- A spiking network with CUDA - A project by Jayram Moorkanikara Nageswaran et.al
- Hysteresis loop - a QuickTime animation
- Hysteresis loop - Matlab/Octave script
- INa-IK-IK(M) burster - Matlab/Octave script
- An animation of an action potential propagating spatialy along the fiber in a bursting model (fold/homoclinic bursting) - a QuickTime animation
- An animation of an action potential propagating spatialy along the fiber in a bursting model (circle/circle bursting) - a QuickTime animation
- INa-IK-IK(M) circle/circle burster spatial solution - Matlab/Octave script
- A mesh resembling a neuron in 3d - Blender file
- Another mesh resembling a neuron in 3d - Blender file
- INa-IK-IK(M) Fold/Homoclinic burster spatial solution - Matlab/Octave script
- Propagating Waves in Neocortical Slices Research results of JIAN-YOUNG WU, PH.D.
- Some more movies of Jian-Young Wu
- Okinawa computational neuroscience course - a very good summer school, I encourage my students to apply!
- Neuron - software for neuron simulations
- Web page of Bard Ermentrout an author of XPP and many publications on mathematical neuroscience
- Solution of a bursting neuron in 2d - note the artifacts related to the second order derrivative mask
- Solution of a bursting neuron in 2d (source) - Matlab/Octave script
- Rotating waves in a grid of coupled oscillators - a QuickTime animation
- Spiral waves in a grid of coupled oscillators - a QuickTime animation
- 2d grid of coupled oscillators (source) - Matlab/Octave script
- Computing PRC - Matlab/Octave script
- Poincare phase map - Matlab/Octave script
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