Effective dynamics of twisted and curved scroll waves using virtual filaments

Abstract

Scroll waves are three-dimensional excitation patterns that rotate around a central filament curve; they occur in many physical, biological, and chemical systems. We explicitly derive the equations of motion for scroll wave filaments in reaction-diffusion systems with isotropic diffusion up to third order in the filament's twist and curvature. The net drift components define at every instance of time a virtual filament which lies close to the instantaneous filament. Importantly, virtual filaments obey simpler, time-independent laws of motion which we analytically derive here and illustrate with numerical examples. Stability analysis of scroll waves is performed using virtual filaments, showing that filament curvature and twist add as quadratic terms to the nominal filament tension. Applications to oscillating chemical reactions and cardiac tissue are discussed.

Nonlinear physics of electrical wave propagation in the heart: a review

“…Recently, Dierckx et al [326] have calculated higher orders in the perturbation theory in generic models of excitable media.…”

“…Defining p(σ, t), the unit vector pointing orthogonally from the center filament to the line of instantaneous scroll wave tips, the scroll wave twist can be defined as τ = (p × ∂ s p) · T. Then, a phenomenological extension of Eqs. (78)- (80) for a twisted scroll wave was developed in [160] (a more rigorous derivation can be found in [326]), giving…”

Measurement and structure of spiral wave response functions

“…Overlap integrals of spiral wave RFs thus appear in the equations of motion for three-dimensional scroll wave filaments, [13][14][15][16] or in the theory of 2D spiral waves drifting due to a constant external field, 17 surface curvature, 18 or mechano-electrical feedback. 19 In one spatial dimension, the translational RF of the wave front determines the velocitycurvature relation 20,21 and the shape of the RF itself can be used to shape reaction-diffusion patterns.…”

Dynamics of Filaments of Scroll Waves

“…A higher-order asymptotic equation for filament motion was obtained by Dierckx and Verschelde [2013]. Unlike the leading order (10), it is coupled to t = 75 t = 150 t = 300 Wavefronts are cut out by clipping planes halfway through the volume, to reveal the filaments [Dierckx et al, 2012].…”

“…The vector-function R(s, t) in (14) is the "virtual filament" which has to be defined more precisely than (9), and the coefficients are defined as integrals involving spiral wave response functions, similar to but more complicated than (11); see [Dierckx and Verschelde, 2013] for detail.…”