# MATH 723:
Mathematical Neuroscience

#### Instructor: Georgi Medvedev

Lectures: M 6-9, Matheson 311

Office hours: 4:30-5:50

Office: 292 Korman Center

E-mail: medvedev@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.

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

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Lecture 2:
Elements of the theory of differential equations (continued). Homework.

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Topics: More on phase plane analysis. Limit cycles. Poincare map.
Introduction to Matlab.

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.

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Lecture 3:
Elements of the theory of differential equations (continued).
Elements of the bifurcation theory.
Lecture notes.
Homework.

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Topics: Stable and unstable manifolds. More phase plane analysis.
Bifurcations: motivation, goals, and examples.

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Lecture 4:
Elements of the bifurcation theory (continued).
The Hodgkin-Huxley model.
Lecture notes.
pdf
Homework.

#### Topics: Saddle-node and Andronov-Hopf bifurcations of equilibria.
Bifurcations of the fixed points in the families of maps.
The HH model: assumptions and derivation.

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).

####
The HH model: assumptions and derivation. Nondimensional 2D system
approximating the HH model: two-time scales. The FitzHugh-Nagumo model.
Elements of the theory for slow-fast systems.

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Lecture 6:
The HH model: excitability
Lecture notes.

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The bifurcation mechansims of excitability: type I and type II models.
Phase plane analysis of the reduced systems: equilibria, basin of attraction,
separatices (threshold). Dynamical interpretation of selected phenomena
of experimetal electrophysiology: slow modulation, spike frequency adaptation,
threshold of spiking, response to slowly changing current, FI curves.

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Lecture 7:
Excitability (continued). Bursting.
Lecture notes.

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The Chay model. Slow-fast decomposition. Homoclinic bifuurcation.
Reduction to 1D map. A few other models of bursting.

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Lecture 8:
Bursting (continued).
Lecture notes.

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Bifurcation scenarios. Transition from spiking to bursting. Chaos.

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Lecture 9:
A pair of neurons coupled via reciprocal inhibition.

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Central pattern generators. Half-center oscillators. Postinhibitory
rebound. Release and escape mechanisms.

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MATLAB Primer
by Kermit Sigmon