MATH 723: Mathematical Neuroscience

Instructor: Georgi Medvedev
Lectures: TR 3.30-4.50, Curtis 455
Office hours: T 2.30-3.20 or by appointment
Office: 292 Korman Center
E-mail: medvedev[[at]]

The course web page from 2008 here.


Lecture 1: The Hodgkin-Huxley model.   Lecture notes.

Topics: The HH model: underlying assumptions and derivation. Disparate timescales. Action potential generation. The Morris-Lecar model.

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 2: 2D approximations of the HH model.   Lecture notes.

A 2D system approximation of the HH model: two-time scales.
The FitzHugh-Nagumo model. Slow-fast decomposition.

Lecture 3: Geometric analysis of small networks.

A pair of mutually coupled cells. Synchronous and antiphase oscillations. Fast threshold modulation. Post-inhibitory rebound.
Release and escape mechanisms of anti-phase oscillations.

D. Somers and N. Kopell, Waves and synchrony in networks of oscillators of relaxation and non-relaxation type, Physica D 89 (1995), 169-183. [pdf]
X-J. Wang and J. Rinzel, Alternating and synchronous rhythms in reciprocally inhibitory model neurons, Neural Computation 4, 84-97 (1992) [pdf]

Lecture 4: Elements of the theory of differential equations.   Lecture notes.  

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

Lectures 5 and 6: Elements of the bifurcation theory.   Lecture notes.   Homework.

Topics: The role of the bifurcation theory in analyzing neural models: Examples. Saddle-node and Andronov-Hopf bifurcations of equilibria in
one-parameter families of flows. Saddle-node and period-doubling bifurcations for one-dimensional maps. Logistic map. Period-doubling route to chaos. Homoclinic bifurcations for planar vector fields.

Lecture 7: Introduction to   XPPAUT.

Examples of the xpp codes from this class:   chay.ode   fast.ode

Lectures 8 and 9: Neuronal Excitability.

Ref.: Chapter 7 in Dynamical Systems in Neuroscience by Eugene Izhikevich.

Lecture 10, 11: Bursting.   Homework.

Topics: Slow-fast analysis of bursting. Averaging. Poincare map. Transition from spiking to bursting.

Janet Best, Alla Borisyuk, Jonathan Rubin, David Terman, and Martin Wechselberger, The dynamic range of bursting in a model respiratory pacemaker network, SIAM J. Appl. Dyn. Syst., 4 (2005), no. 4, 1107-1139. [pdf]
G.S. Medvedev, Reduction of a model of an excitable cell to a one-dimensional map, Physica D, 202(1-2), 37-59, 2005. [pdf]
Eugene M. Izhikevich, Neural Excitability, Spiking, and Bursting, International Journal of Bifurcation and Chaos (2000), 10:1171--1266.

Lecture 12, 13: Noise.

Topics: Sources of noise, noise-induced patterns (spontaneous spiking, mixed-mode oscillations, bursting), randomly perturbed conductance-based models, Brownian motion, Euler-Maruyama scheme,
Ito integral, first exit problem, Kolmogorov equation, large deviations, Kramers' law,
Brownian particle in a double-well potential, spontaneous spiking in a type I neuron, stochastic resonance.

D.J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, vol. 43(3), 525-546, 2001. [pdf]
N. Berglund, Kramers' law: Validity, derivations and generalisations, Markov Processes Relat. Fields 19: 459-490 (2013)
Deville RE, Vanden-Eijnden E, Muratov CB, Two distinct mechanisms of coherence in randomly perturbed dynamical systems, PRE,72, 031105 (2005). [pdf]

Lecture 14, 15: Coupled networks.

Topics: Phase reduction. Phase responce curve. The derivation of the Kuramoto model of coupled phase oscillators. Averaging. Stability of synchronized and anti-phase solutions. Continuum limit (local vs nonlocal coupling). Synchronization. Algebraic tools for describing connectivity of the network. Sponteneous dynamics in electrically coupled networks of type I neurons.

Yoshiki Kuramoto, Cooperative Dynamics of Oscillator Community, Progress of Theoretical Physics Supplement. 01/1984; 79:223-240. [abstract]
G.S. Medvedev and S. Zhuravytska, The geometry of spontaneous spiking in neuronal networks, J. Nonlinear Sci., 2012. [abstract]

Student Presentations:
Lei Wang Slides

Michael Meyers Slides

Matlab tutorials

An Introduction to Matlab by David Griffiths

MATLAB Primer by Kermit Sigmon