BME 213
NONLINEAR DYNAMICS IN ELECTROPHYSIOLOGY

This course examines the electrophysiological behavior of nerves and muscle using methods of nonlinear dynamics. The topics include:

1. Introduction to BME 213. Review of the Hodgkin-Huxley model of the giant axon of the squid. Asymptotic reduction of Hodgkin-Huxley model to FitzHugh-Nagumo model.

2. Introduction to nonlinear dynamics. Concepts of phase space, trajectories, nullclines, and fixed points. Constructing phase-plane portraits. Linear stability analysis and classification of fixed points.

3. Excitable membrane as a dynamic system. Phase-plane analysis of the FitzHugh-Nagumo model. Interpretation of physiological phenomena in phase-space.

4. Limit cycles and the oscillatory response of the membrane. Periodic oscillations in linear vs nonlinear systems. Limit cycles in the FitzHugh-Nagumo model of excitable membrane. Influence of external stimuli on nonlinear oscillators.

5. Phase resetting in biological oscillators. Winfree's critical point theory and its applications to electrophysiology. Induction of rotors in the heart by cross-field stimulation.

6. Introduction to chaos. Chaos in continuous and discrete systems. Bifurcation diagrams, chaotic attractors. Chaos in discrete systems. Poincaire maps.

7. Chaos control. Ott, Grebogy, and Yorke method of controlling the dynamics of chaotic systems using small perturbations. Controlling irregular cardiac rhythm: theory and practical realization in Garfinkel's experiment.

Prerequisites: MTH 108 or equivalent; BME 101 or consent of the instructor.

Grading: Participation in class, homeworks, computer labs, and tests. Each student will prepare and present a project based on individual readings from the literature. (4 units)