Computation of the response functions of spiral waves in active media

Abstract

Rotating spiral waves are a form of self-organization observed in spatially extended systems of physical, chemical, and biological natures. A small perturbation causes gradual change in spatial location of spiral's rotation center and frequency, i.e., drift. The response functions ͑RFs͒ of a spiral wave are the eigenfunctions of the adjoint linearized operator corresponding to the critical eigenvalues =0, Ϯ i. The RFs describe the spiral's sensitivity to small perturbations in the way that a spiral is insensitive to small perturbations where its RFs are close to zero. The velocity of a spiral's drift is proportional to the convolution of RFs with the perturbation. Here we develop a regular and generic method of computing the RFs of stationary rotating spirals in reaction-diffusion equations. We demonstrate the method on the FitzHugh-Nagumo system and also show convergence of the method with respect to the computational parameters, i.e., discretization steps and size of the medium. The obtained RFs are localized at the spiral's core.

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“…Importantly, it was found that response functions (RFs) are closely related to the symmetries of the RDE (2) in two spatial dimensions [7,27], which allows one to compute them numerically for given diffusion matrix P and reaction kinetics F(u) [30,61]. Before studying the RDE in a rotating frame of reference, it is necessary to reconcile sign conventions for the angular frequency, since we will be using an extension of the DXSPIRAL program of Biktasheva et al [61] to compute RFs.…”

“…In this section, we briefly review response function theory and introduce sign conventions for going from the Cartesian basis of [7,22,25] to the complex-valued representation of [8,61].…”

“…Before studying the RDE in a rotating frame of reference, it is necessary to reconcile sign conventions for the angular frequency, since we will be using an extension of the DXSPIRAL program of Biktasheva et al [61] to compute RFs. In DXSPIRAL, the angular frequency of spirals ω 0 is always taken positive; the program possesses a sign flag K to denote whether rotation is clockwise (K = 1) or counterclockwise (K = −1).…”

“…[61] a clockwise rotating spiral was assumed (K = 1). To preserve full compatibility with those earlier works, we will explicitly write the sign flag K ∈ {−1, +1} throughout this paper.…”

“…In the works [8,31,61], including DXSPIRAL, a complex-valued basis of eigenmodes toL was used, given by…”