# 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]]drexel.edu

####
The course web page from the last year is
here.

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

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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)

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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.

####
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.

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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]

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Lecture 7:
Excitability / Bursting.

####

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

**
**

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Matlab tutorials

An Introduction to Matlab by David Griffiths

MATLAB Primer
by Kermit Sigmon