Spatial localization in the FitzHugh-Nagumo model

TL;DR

This paper provides a detailed bifurcation analysis of spatially localized states in the FitzHugh-Nagumo model, revealing their transition behaviors and oscillatory dynamics through analytical and numerical methods.

Contribution

It introduces a comprehensive bifurcation analysis of localized states in the FitzHugh-Nagumo model, highlighting their transition between snaking regimes and oscillatory behaviors.

Findings

Localized states undergo a transition from standard to collapsed homoclinic snaking.

Oscillatory dynamics depend on time-scale separation and diffusion.

Analytical and numerical methods reveal stability regimes of localized structures.

Abstract

The FitzHugh-Nagumo model, originally introduced to study neural dynamics, has since found applications across diverse fields, including cardiology and biology. However, the formation and bifurcation structure of spatially localized states in this model remain underexplored. In this work, we present a detailed bifurcation analysis of such localized structures in one spatial dimension in the FitzHugh-Nagumo model. We demonstrate that these localized states undergo a smooth transition between standard and collapsed homoclinic snaking as the system shifts from pattern-uniform to uniform-uniform bistability. Additionally, we explore the oscillatory dynamics exhibited by these states when varying the time-scale separation and diffusion coefficient. Our study leverages a combination of analytical and numerical techniques to uncover the stability and dynamic regimes of spatially localized structures, offering new insights into the mechanisms governing spatial localization in this widely used model system.