The Dynamics of Forced Excitable Systems

Abstract

An excitable system is one whose dynamics have the following properties: (i) there is a rest point or a steady state that is globally attracting relative to some large set in phase space, and (ii) there is a region in state space that can be idealized as a surface of codimension one that locally partitions the phase space into two sets D and A. the rest point lies in D (the decaying set) and all orbits through initial points in D return to the rest point without any substantial growth in any of the state variables. Thus an impulsive perturbation of the rest point that leaves the state in V decays without significant growth, and the responses are called subthreshold. By contrast, perturbations that carry the dynamics into A (the amplifying set) can lead to a large change in one or more of the state variables, even though the system eventually returns to the rest state. The surface that locally separates the amplifying and decaying sets is called the threshold surface, and perturbations that carry the state into A are called superthreshold. Excitable dynamics occur in many biological processes, including activation of contraction in cardiac tissue, nerve conduction, and cell signalling in development. In the Fitzhugh-Nagumo equations [10], the Hodgkin- Huxley equations [15], models of the cellular slime mold Dictyostelium discoideum [22, 20, 21], the Field-Noyes model of the Zhabotinskii-Belousov reaction [9] and other models, the parameters can be chosen so that the dynamics are excitable. Frequently these systems also have periodic solutions for parameters close to those that produce excitable behaviour.