Computation of the Drift Velocity of Spiral Waves using Response Functions

TL;DR

This paper demonstrates that Response Functions can accurately predict the drift velocities of spiral waves under various perturbations in reaction-diffusion systems, validated through FitzHugh-Nagumo and Barkley models.

Contribution

It introduces a quantitative method using Response Functions to predict spiral wave drift velocities for different perturbations, validated against numerical simulations.

Findings

Response functions accurately predict drift velocities.

Good agreement between predictions and simulations.

Applicable to various perturbation types in reaction-diffusion systems.

Abstract

Rotating spiral waves are a form of self-organization observed in spatially extended systems of physical, chemical, and biological nature. In the presence of a small perturbation, the spiral wave's centre of rotation and fiducial phase may change over time, i.e. the spiral wave drifts. In linear approximation, the velocity of the drift is proportional to the convolution of the perturbation with the spiral's Response Functions (RFs), which are the eigenfunctions of the adjoint linearized operator corresponding to the critical eigenvalues $\lambda = 0, \pm i\omega$. Here we demonstrate that the response functions give quantitatively accurate prediction of the drift velocities due to a variety of perturbations: a time dependent, periodic perturbation (inducing resonant drift); a rotational symmetry breaking perturbation (inducing electrophoretic drift); and a translational symmetry breaking perturbation (inhomogeneity induced drift) including drift due to a gradient, step-wise and localised inhomogeneity. We predict the drift velocities using the response functions in FitzHugh-Nagumo (FHN) and Barkley models, and compare them with the velocities obtained in direct numerical simulations. In all cases good quantitative agreement is demonstrated.