Analytical approximations for spiral waves

TL;DR

This paper develops a non-perturbative analytical approach to approximate the behavior of spiral waves in excitable media, providing relations between rotation frequency and core radius that align well with numerical solutions.

Contribution

It introduces an implicit analytical relation between spiral wave rotation frequency and core radius, improving existing models for free and pinned spirals in excitable media.

Findings

Analytical relation between frequency and core radius matches numerical solutions.

Improved model for pinned spiral waves with defect radius R+.

Analytical approximations for spiral shapes are provided.

Abstract

We propose a non-perturbative attempt to solve the kinematic equations for spiral waves in excitable media. From the eikonal equation for the wave front we derive an implicit analytical relation between rotation frequency $\Omega$ and core radius $R_{0}$. For free, rigidly rotating spiral waves our analytical prediction is in good agreement with numerical solutions of the linear eikonal equation not only for very large but also for intermediate and small values of the core radius. An equivalent $\Omega\left(R_{+}\right)$ dependence improves the result by Keener and Tyson for spiral waves pinned to a circular defect with radius $R_{+}$ with Neumann boundaries at the periphery. Simultaneously, analytical approximations for the shape of free and pinned spirals are given. We discuss the reasons why the ansatz fails to correctly describe the result for the dependence of the rotation frequency on the excitability of the medium.