The large core limit of spiral waves in excitable media: A numerical approach

TL;DR

This paper introduces a modified numerical method to analyze large core spiral waves in excitable media, enabling investigation near criticality where spiral cores become infinitely large and rotation frequency approaches zero.

Contribution

The paper presents a stable numerical approach with new boundary conditions to study spiral waves near criticality, overcoming limitations of previous methods.

Findings

Confirmed linear scaling of rotation frequency and core radius near criticality.

Enabled analysis of spiral waves with infinite core radius and zero rotation frequency.

Provided a practical method to study spiral wave bifurcations close to criticality.

Abstract

We modify the freezing method introduced by Beyn & Thuemmler, 2004, for analyzing rigidly rotating spiral waves in excitable media. The proposed method is designed to stably determine the rotation frequency and the core radius of rotating spirals, as well as the approximate shape of spiral waves in unbounded domains. In particular, we introduce spiral wave boundary conditions based on geometric approximations of spiral wave solutions by Archimedean spirals and by involutes of circles. We further propose a simple implementation of boundary conditions for the case when the inhibitor is non-diffusive, a case which had previously caused spurious oscillations. We then utilize the method to numerically analyze the large core limit. The proposed method allows us to investigate the case close to criticality where spiral waves acquire infinite core radius and zero rotation frequency, before they begin to develop into retracting fingers. We confirm the linear scaling regime of a drift bifurcation for the rotation frequency and the core radius of spiral wave solutions close to criticality. This regime is unattainable with conventional numerical methods.