On the singularly perturbed Hodgkin-Huxley equations
by
Cecilia Cavaterra
Università di Milano (Italy)
Coauthors: Maurizio Grasselli

In 1952, A.L. Hodgkin and A.F. Huxley proposed a well-known nonlinear one-dimensional reaction-diffusion system to model the nerve conduction in the giant axon of the squid Loligo. In this system, the parabolic equation which governs the electrical potential in the nerve can be replaced by a more appropriate hyperbolic type equation characterized by a (small) parameter (i.e., the inductance coefficient). The resulting system is a singular perturbation of the original Hodgkin-Huxley equations. The global dynamics of such equations, perturbed or not, is rather complex. This fact justifies a qualitative approach whose first goal is to prove the existence of small geometric sets which describe the nontransient dynamics (typically, global and exponential attractors). In addition, one may wonder how these invariant objects depend on the inductance of the system when it vanishes. These issues were carefully analyzed by W.E. Fitzgibbon, M. Parrott and Y. You in 1996 and by C. Galusinki in 1998. In this talk we intend to present some new results on the large time dynamics of the singularly perturbed H-H equations and its dependence on the inductance coefficient.

Date received: February 25, 2008

Copyright © 2008 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cawu-13.