Pulses in FitzHugh--Nagumo systems with rapidly oscillating coefficients

TL;DR

This paper investigates pulse solutions in FitzHugh--Nagumo systems with rapidly oscillating coefficients, establishing existence, stability, and approximation of pulses through a two-scale limit system.

Contribution

It introduces a two-scale FitzHugh--Nagumo model for rapidly oscillating coefficients and proves the existence and stability of pulses within this framework.

Findings

Existence of pulse solutions in the limit two-scale system.

Stability of these pulse solutions.

Approximation of original system pulses by the two-scale model.

Abstract

This paper is devoted to pulse solutions in FitzHugh--Nagumo systems that are coupled parabolic equations with rapidly periodically oscillating coefficients. In the limit of vanishing periods, there arises a two-scale FitzHugh--Nagumo system, which qualitatively and quantitatively captures the dynamics of the original system. We prove existence and stability of pulses in the limit system and show their proximity on any finite time interval to pulse-like solutions of the original system.