Traveling Wave Solutions to a Coupled System of Spatially Discrete Nagumo Equations

Abstract

Abstract. We consider a coupled system of discrete Nagumo equations and derive traveling wave solutions to this system using McKean's caricature of the cubic. A certain form of this system is used to model ephaptic coupling between pairs of nerve axons. We study the difference g(c) = a 1 − a 2 between the detuning parameters a i that is required to make both waves move at the same speed c. Of particular interest is the effect of a coupling parameter α and an "alignment" parameter A on the function g. Numerical investigation indicates that for fixed A, there exists a time delay value β that results in g = 0, and for large enough wave speeds, multiple such β values exist. Also, numerical results indicate that the perturbation of α away from zero will yield additional solutions with positive wave speed when A = 1 2. We employ both analytical and numerical results to demonstrate our claims.Key words. discrete Nagumo equations, ephaptic coupling, traveling waves AMS subject classifications. 35K57, 74N99DOI. 10.1137/050624352 1. Introduction. In myelinated nerve axons, transmembrane ion flow occurs only at the spatially periodic nodes of Ranvier, and the activity at these nodes may affect the activity at nodes of neighboring fibers. This so-called ephaptic coupling is an electrical effect that causes neighboring fibers to interact and possibly synchronize with each other. Accompanying the problem of ephaptic coupling is the issue of the relative positioning of the nodes of Ranvier on the different fibers. That is, given two parallel nerve fibers, the nodes on one fiber may or may not align perfectly with the nodes on the other fiber.Our contribution in this paper is to derive a solution to a system that models these phenomena and use this model to show that coupling decreases the size of the range of propagation failure when the nodes of Ranvier are staggered, but that coupling increases the size of this range when the nodes are perfectly aligned. To do this, we consider a system of two myelinated nerve axons coupled ephaptically. In particular, our goal is to study the effect of this coupling and the effect of nonalignment on the propagation of action potentials. Different types of coupling between fibers are possible. In [2], Binczak, Eilbeck, and Scott model "saltatory" conduction present in these myelinated neurons with equations used to govern the behavior of electrical circuits and introduce the effect of ephaptic coupling between two myelinated neurons. In [1] a different type of coupling, called "ohmetric" coupling, is considered and it is shown that the introduction of this kind of coupling causes waves on two adjacent myelinated axons to match speeds with each other. Earlier work of Keener [13] and Bose and Jones [4,5] addressed the issue of ohmetric coupling.The strength of the ephaptic coupling α depends on the electrical resistance R int inside the axons (assumed to be the same for both axons) and the resistance R o of the medium between the axons. We must also consider the positioning of the axons