MATH 723: Mathematical Neuroscience

Instructor: Georgi Medvedev
Lectures: M 6-9, Matheson 311
Office hours: 4:30-5:50
Office: 292 Korman Center

The course web page from the last year is here.


Lecture 1: Elements of the theory of differential equations.   Lecture notes.   Homework.

Topics: Fixed points and periodic obits, linearization, Hartman-Grobman theorem, linear systems, phase plane.

Lecture 2: Elements of the theory of differential equations (continued).  Homework.

Topics: More on phase plane analysis. Limit cycles. Poincare map. Introduction to Matlab.

Problem 2) Follow the steps described in this file .
The present excercise introduces some basic matlab tools for ploting 2D graphs,
as well as shows how to use MATLAB function ode23 for numerical solution of initial value problems.
Note that you can get help on any command or function in MATLAB by typing: 'help '.
For example, 'help plot'. Use HELP to learn more about the functions used in this excercise. Try also LOOKFOR.

Lecture 3: Elements of the theory of differential equations (continued). Elements of the bifurcation theory.   Lecture notes.   Homework.

Topics: Stable and unstable manifolds. More phase plane analysis. Bifurcations: motivation, goals, and examples.

Lecture 4: Elements of the bifurcation theory (continued). The Hodgkin-Huxley model.   Lecture notes.   pdf   Homework.

Topics: Saddle-node and Andronov-Hopf bifurcations of equilibria. Bifurcations of the fixed points in the families of maps. The HH model: assumptions and derivation.
Matlab codes:
HH_function.m. Use this function to numerically integrate the HH model in Matlab (e.g., using ode15s).
HH_stimulate.m was used for the numerical experiment shown in Figure 4 (see lecture notes).

Lecture 5: The Hodgkin-Huxley model.   Lecture notes.   pdf   Homework.

The HH model: assumptions and derivation. Nondimensional 2D system approximating the HH model: two-time scales. The FitzHugh-Nagumo model. Elements of the theory for slow-fast systems.

Lecture 6: The HH model: excitability   Lecture notes.

The bifurcation mechansims of excitability: type I and type II models. Phase plane analysis of the reduced systems: equilibria, basin of attraction, separatices (threshold). Dynamical interpretation of selected phenomena of experimetal electrophysiology: slow modulation, spike frequency adaptation, threshold of spiking, response to slowly changing current, FI curves.

Lecture 7: Excitability (continued). Bursting.   Lecture notes.

The Chay model. Slow-fast decomposition. Homoclinic bifuurcation. Reduction to 1D map. A few other models of bursting.

Lecture 8: Bursting (continued).   Lecture notes.

Bifurcation scenarios. Transition from spiking to bursting. Chaos.

Lecture 9: A pair of neurons coupled via reciprocal inhibition.

Central pattern generators. Half-center oscillators. Postinhibitory rebound. Release and escape mechanisms.

Student Presentations:
Subthreshold oscillations and the onset of spikes in a model of medial enthornial cortex. Presenter: Timothy Jones
Dynamics of spiking neurons connected by both inhibitory and electrical synapses. Presenter: Mathew O'Connell
Bursting mappings. Presenter: Patrick Jasinscki
Synchronization of pulse coupled oscillators. Presenter: Ramil Berner

Matlab resources online:

MATLAB Primer by Kermit Sigmon