Stability of Impulsively Forced Excitable Fibers to Perturbations of the Forcing Period
Fibers of electrically coupled nerve or cardiac cells are among the best-known examples of excitable media. Such fibers are often forced periodically at one end by an impulsive electrical stimulus current, eliciting sequences of traveling pulses. If the excitable medium happens to be cardiac tissue, it is natural to ask whether a sudden change in the period of the forcing (e.g., the heart rate) might induce an abnormal pattern of electrical wave propagation. In this manuscript, we analyze the transient response of an excitable medium following a change in the period of impulsive forcing. There are two specific questions that we shall address: First, under what conditions is a periodic train of identical traveling pulses stable to small perturbations in the period of forcing? Second, in the stable regime, what can be said analytically regarding the transient behavior in response to a perturbation in the period of forcing? Instead of using the traditional reaction-diffusion model for wave propagation in excitable fibers, we analyze a kinematic model which describes the progress of each propagating wave front and wave back. The linearization of the kinematic model, presented as a recursive sequence of ordinary differential equations, can be solved exactly in terms of generalized Laguerre polynomials integrated against an exponential kernel. The solution gives the desired approximation to the transient behavior following a change in the forcing period and, with the aid of some basic functional analytic formalism, a criterion for linear stability of a periodic pulse train. In the appendix, we illustrate how this framework applies to a specific model of an excitable cardiac fiber.