An Invariant Winding Number for the FitzHugh--Nagumo System with Applications to Cardiac Dynamics

Abstract

The FitzHugh--Nagumo (FHN) system of PDEs is a generic model for excitable media, often used to build a qualitative understanding of electrophysiological phenomena. A well-characterized traveling pulse solution to FHN serves as a model for action potentials in cardiac tissue and other contexts. The stability of the traveling pulse has been well studied, but the more global problem of predicting when an arbitrary initial condition will converge to the uniform rest solution and when it will converge to the traveling pulse remains unsolved. Here we prove the existence of an invariant winding number in an asymptotic limit of the FHN system called the singular FHN system (SFHN) that provides a crucial step toward a global convergence result. This result is valid for a relatively general class of nonlinearities. In addition, we provide evidence that our SFHN results extend with limitations to FHN and outline conditions under which the SFHN approximation fails. The invariant winding number provides explanations for several observations of physiological relevance. For example, it explains the requirements on stimulus protocols that allow the formation and elimination of reentrant rhythms in cardiac tissue.