Existence of periodic solutions of the FitzHugh-Nagumo equations for an explicit range of the small parameter

TL;DR

This paper rigorously proves the existence of periodic solutions in the FitzHugh-Nagumo equations for a specific small parameter range using innovative topological and computational methods, extending understanding of nerve impulse models.

Contribution

It introduces a novel combination of topological techniques and validated continuation methods to establish periodic solutions in a fast-slow reaction-diffusion system for explicit parameter ranges.

Findings

Existence of periodic solutions confirmed for .0015  0.0015

Development of a new method combining covering relations and isolating segments

Successful application of interval Newton-Moore method at .0015

Abstract

The FitzHugh-Nagumo model describing propagation of nerve impulses in axon is given by fast-slow reaction-diffusion equations, with dependence on a parameter $\epsilon$ representing the ratio of time scales. It is well known that for all sufficiently small $\epsilon>0$ the system possesses a periodic traveling wave. With aid of computer-assisted rigorous computations, we prove the existence of this periodic orbit in the traveling wave equation for an explicit range $\epsilon \in (0, 0.0015]$. Our approach is based on a novel method of combination of topological techniques of covering relations and isolating segments, for which we provide a self-contained theory. We show that the range of existence is wide enough, so the upper bound can be reached by standard validated continuation procedures. In particular, for the range $\epsilon \in [1.5 \times 10^{-4}, 0.0015]$ we perform a rigorous continuation based on covering relations and not specifically tailored to the fast-slow setting. Moreover, we confirm that for $\epsilon=0.0015$ the classical interval Newton-Moore method applied to a sequence of Poincar\'e maps already succeeds. Techniques described in this paper can be adapted to other fast-slow systems of similar structure.