Epicyclic drifting in anisotropic excitable media with multiple inhomogeneities

Abstract

Spirals have been studied from a dynamical system perspective starting with Barkley's seminal papers linking a wide class of spiral wave dynamics to the Euclidean symmetry of the excitable media in which they are observed. However, in order to explain certain non-Euclidean phenomena, such as anchoring and epicyclic drifting, LeBlanc and Wulff introduced a single translational symmetry-breaking perturbation to the center bundle equation and showed that rotating waves may be attracted to a non-trivial solution manifold and travel epicyclically around the perturbation center. In this paper, we continue the (model-independent) investigation of the effects of inhomogeneities on spiral wave dynamics by studying epicyclic drifting in the presence of: a) $n$ simultaneous translational symmetry-breaking terms, with $n > 1$, and b) a combination of a single rotational symmetry-breaking term and a single translational symmetry-breaking term. These types of forced Euclidean symmetry-breaking provide a much more realistic model of certain excitable media such as cardiac tissue. However, the main theoretical tool used by LeBlanc and Wulff can only be applied to their particular perturbation: we show how an averaging theorem of Hale can be modified to analyze our two more general scenarios and state the conditions under which epicyclic drifting takes place in the general case. In the process, we recover LeBlanc and Wulff's specific result. Finally, we illustrate our results with the help of a simple numerical simulation of a modified bidomain model.