Traveling pulse solutions in a three-component FitzHugh-Nagumo Model

TL;DR

This paper analyzes traveling pulse solutions in a three-component FitzHugh-Nagumo model using geometric singular perturbation and action functional methods, deriving conditions for existence, bifurcations, and pulse profiles.

Contribution

It introduces a novel combination of geometric singular perturbation and action functional techniques to study pulse solutions and bifurcations in a three-component FitzHugh-Nagumo model.

Findings

Derived explicit conditions for 1-pulse and 2-pulse existence.

Identified saddle-node bifurcation points for pulse solutions.

Discussed potential Hopf bifurcations near bifurcation points.

Abstract

We use geometric singular perturbation techniques combined with an action functional approach to study traveling pulse solutions in a three-component FitzHugh--Nagumo model. First, we derive the profile of traveling $1$-pulse solutions with undetermined width and propagating speed. Next, we compute the associated action functional for this profile from which we derive the conditions for existence and a saddle-node bifurcation as the zeros of the action functional and its derivatives. We obtain the same conditions by using a different analytical approach that exploits the singular limit of the problem. We also apply this methodology of the action functional to the problem for traveling $2$-pulse solutions and derive the explicit conditions for existence and a saddle-node bifurcation. From these we deduce a necessary condition for the existence of traveling $2$-pulse solutions. We end this article with a discussion related to Hopf bifurcations near the saddle-node bifurcation.