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]]drexel.edu

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.

The FitzHugh-Nagumo model. Slow-fast decomposition.

Release and escape mechanisms of anti-phase oscillations.

Reading:

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]

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.

Reading:

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.

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.

Reading:

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]

Reading:

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]

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An Introduction to Matlab by David Griffiths

MATLAB Primer by Kermit Sigmon