Problem 2) Follow the steps described in this
file .
MATLAB Primer
by Kermit Sigmon
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 3:
Elements of the theory of differential equations (continued).
Elements of the bifurcation theory.
Lecture notes.
Homework.
Topics: Stable and unstable manifolds. More phase plane analysis.
Bifurcations: motivation, goals, and examples.
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).
Lecture 5:
The Hodgkin-Huxley model.
Lecture notes.
pdf
Homework.
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.
Lecture 6:
The HH model: excitability
Lecture notes.
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.
Lecture 7:
Excitability (continued). Bursting.
Lecture notes.
The Chay model. Slow-fast decomposition. Homoclinic bifuurcation.
Reduction to 1D map. A few other models of bursting.
Lecture 8:
Bursting (continued).
Lecture notes.
Bifurcation scenarios. Transition from spiking to bursting. Chaos.
Lecture 9:
A pair of neurons coupled via reciprocal inhibition.
Central pattern generators. Half-center oscillators. Postinhibitory
rebound. Release and escape mechanisms.
Student Presentations:
Subthreshold oscillations and the onset of spikes in a
model of medial enthornial cortex. Presenter: Timothy Jones
Dynamics of spiking neurons connected by both inhibitory and electrical
synapses. Presenter: Mathew O'Connell
Bursting mappings. Presenter: Patrick Jasinscki
Synchronization of pulse coupled oscillators.
Presenter: Ramil Berner
Matlab resources online: