MATH 723: Mathematical Neuroscience

Instructor: Georgi Medvedev
Lectures: M 6-9, Matheson 304
Office hours: 4:30-5:50
Office: 292 Korman Center
E-mail: medvedev[[at]]

The course web page from the last year is here.


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

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

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 2: Elements of the theory of differential equations: Hyperbolic Equilibria.

Topics: Hartman-Grobman theorem, hyperbolic equilibria, invariant (stable and unstable) manifolds, stability of equilibria,
Lyapunov function, Example: Neuronal competition model (phase plane analysis, linearization, interpretation)
(cf. Shpiro et al, J. Neurophysiol 97:462-473, 2007. )

Lecture 3: Elements of the theory of differential equations: Periodic orbis.   Homework 2.

Topics: Limit cycles, Poincare-Bendixson theorem
Example 1: Pendulum with friction and constant torque
Behavior near a 2D limit cycle: local coordinates, Poincare map, Floquet multipliers, stability of the limit cycle.
Example 2: Averaging (cf Section 3.4; notes for Lecture 1)

Lecture 4: Elements of the bifurcation theory.   Lecture notes.

Topics: Role of the bifurcation theory in analyzing neural models: Examples. Saddle-node and Andronov-Hopf bifurcations of equilibria in
one-parameter families of flows. Review of saddle-node and period-doubling bifurcations for one-dimensional maps. Logistic map. Period-doubling route to chaos.

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

Topics: The HH model, underlying assumptions and derivation. A 2D system approximation of the HH model: two-time scales.
The FitzHugh-Nagumo model. Slow-fast decomposition.

Reading: A.L. Hodgkin and A.F. Huxley, Quantitative description of membrane current and its application to conduction and excitation in nerve,
J. Physiol. (1952), 117, 500-544. [pdf]

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).

Animations:     Na-K exchange    Voltage-gated channels.

Lecture 6: Neuronal excitability.

Topics: Slow-fast analysis of the FitzHugh-Nagumo model. Representative phase portraits of 2D models featuring type I and type II excitability. Review of the saddle-node on the invariant circle and Andronov-Hopf bifurcations in the context excitability. Spiking thereshold. Canards.

Read: Chapter 7 (Izhikevich, Dynamical Neuroscience).
J. Rinzel and B. Ermentraut, Analysis of Neural Excitability and Oscillations, Chapter from Methods in Neuronal Modeling, C. Koch and I. Segev, eds. [pdf]

Lecture 7: Excitability / Bursting.

Topics: Excitability (continued). Homoclinic bifurcation in the context of the analysis of the square-wave bursting.

Student Presentations:
Slides Presenter: Chantal McMahon
Integration of Ca decay after repetative firing. Presenter: Michael Willy
Rhythms in reciprocally inhibitory model neurons. Presenter: Venkat Palgat
Slides Presenter: Anup Umranikar
Increased leak conductance alters ISI variability. Presenter: Sugania Karunakaran

Matlab tutorials

An Introduction to Matlab by David Griffiths

MATLAB Primer by Kermit Sigmon