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
The course web page from 2008
The Hodgkin-Huxley model.
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.
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).
2D approximations of the HH model.
A 2D system approximation of the HH model: two-time scales.
The FitzHugh-Nagumo model. Slow-fast decomposition.
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.
X-J. Wang and J. Rinzel, Alternating and synchronous rhythms in reciprocally
inhibitory model neurons, Neural Computation 4, 84-97 (1992)
Elements of the theory of differential equations.
Topics: Fixed points and periodic obits, linearization, Hartman-Grobman
theorem, linear systems, phase plane.
Lectures 5 and 6:
Elements of the bifurcation theory.
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.
Introduction to XPPAUT.
Examples of the xpp codes from this class: chay.ode
Lectures 8 and 9:
Ref.: Chapter 7 in Dynamical Systems in Neuroscience by Eugene Izhikevich.
Lecture 10, 11:
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,
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:
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:
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]
An Introduction to Matlab by David Griffiths
by Kermit Sigmon