Laws of mutual spiral wave interaction in excitable media

TL;DR

This paper develops a law analogous to Newton's gravitational law to describe the interactions and drift of spiral waves in excitable media, with implications for understanding complex systems like cardiac fibrillation.

Contribution

It introduces a novel framework for quantifying spiral wave interactions using a force-based approach, extending the understanding of spiral dynamics in reaction-diffusion systems.

Findings

Spiral wave interactions are governed by forces determined through boundary integrals.

The spiral 'mass' varies over time and depends on the region of influence.

The interaction forces do not obey Newton's action-reaction law.

Abstract

Interacting rotating spiral waves have been observed in complex systems, such as cardiac fibrillation, cognitive processing in the brain cortex and oscillating chemical reactions, during dynamical regimes that are still poorly understood. We present the equivalent of Newton's gravitational attraction law for spiral waves on planar reaction-diffusion systems. The spiral waves' phases and positions determine their regions of influence, separated by collision interfaces. At the collision interfaces, wave front deflections cause spiral drift that pushes the interfaces forward. As a result, the spiral wave drift velocity is proportional to the total force exerted on on it, which can be determined by a boundary integral over its region of influence. The proportionality factor between force and response is akin to the `mass' of the spiral. However, this spiral mass depends on the region of influence of the spiral and thus also varies over time. The forces between spiral wave pairs are not directed along the line connecting their centers, violating Newton's law of action and reaction. Our solution to the N-body interaction problem for spirals in extended excitable media encompasses both pairwise interactions and spiral wave drift in bounded domains, with application to cardiac fibrillation.