Spiral Waves and the Dynamical System Approach

Abstract

Spirals are common in Nature: the snail's shell and the ordering of seeds in the sunflower are amongst the most widely-known occurrences. While these are static, dynamic spirals can also be observed in excitable systems such as heart tissue, retina, certain chemical reactions, slime mold aggregates, flame fronts, etc. The images associated with these spirals are often breathtaking, but spirals have also been linked to cardiac arrhythmias, a potentially fatal heart ailment. In the literature, very specific models depending on the excitable system of interest are used to explain the observed behaviour of spirals (such as anchoring or drifting). Barkley first noticed that the Euclidean symmetry of these models, and not the model itself, is responsible for the observed behaviour. But in experiments, the physical domain is never Euclidean. The heart, for instance, is finite, anisotropic and littered with inhomogeneities. To capture this loss of symmetry, LeBlanc and Wulff introduced forced Euclidean symmetry-breaking (FESB) in the analysis. To accurately model the physical situation, two basic types of symmetry-breaking perturbations are used: translational symmetry-breaking (TSB) and rotational symmetry-breaking (RSB) terms. In this paper, we provide an overview of currently know results about spiral wave dynamics under FESB.