A geometric theory for scroll wave filaments in anisotropic excitable media

Dierckx, Hans; Bernus, Olivier; Verschelde, Henri

doi:10.1016/j.physd.2008.09.006

Cited by 19 publications

(29 citation statements)

References 38 publications

(65 reference statements)

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“…Recent analytical studies (Verschelde et al 2007;Dierckx et al 2009) show that nonhomogeneous anisotropy can substantially affect scrolls and may generate the complex dynamics of scroll waves observed in numerical simulations (Fenton and Karma 1998b). Most of the studies on effects of anisotropy were performed for positive filament tension (Panfilov and Keener 1995b;Qu et al 2000a).…”

Section: Discussionmentioning

confidence: 99%

Negative Tension of Scroll Wave Filaments and Turbulence in Three-Dimensional Excitable Media and Application in Cardiac Dynamics

2012

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Scroll waves are vortices that occur in three-dimensional excitable media. Scroll waves have been observed in a variety of systems including cardiac tissue, where they are associated with cardiac arrhythmias. The disorganization of scroll waves into chaotic behavior is thought to be the mechanism of ventricular fibrillation, whose lethality is widely known. One possible mechanism for this process of scroll wave instability is negative filament tension. It was discovered in 1987 in a simple two variables model of an excitable medium. Since that time, negative filament tension of scroll waves and the resulting complex, often turbulent dynamics was studied in many generic models of excitable media as well as in physiologically realistic models of cardiac tissue. In this article, we review the work in this area from the first simulations in FitzHugh-Nagumo type models to recent studies involving detailed ionic models of cardiac tissue. We discuss the relation of negative filament tension and tissue excitability and the effects of discreteness in the tissue on the filament tension. Finally, we consider the application of the negative tension mechanism to computational cardiology, where it may be regarded as a fundamental mechanism that explains differences in the onset of arrhythmias in thin and thick tissue.

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Section: Discussionmentioning

confidence: 99%

Negative Tension of Scroll Wave Filaments and Turbulence in Three-Dimensional Excitable Media and Application in Cardiac Dynamics

2012

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“…In the works [22,23,25], the quantityω 0 was used, whereas in Ref. [61] a clockwise rotating spiral was assumed (K = 1).…”

Section: Response Functionsmentioning

confidence: 99%

“…In terms of practical applications, scroll wave filaments are frequently being used to describe time evolution of vortex patterns in the chemical BZ reaction [18], and the degree of complexity during ventricular fibrillation [16,[19][20][21]. Unfortunately, analytical derivations have so far only covered the leading order dynamics of scroll wave filaments [7,8,22,23], not capturing the twist-induced destabilization of filaments. Previous theoretical extensions of filament dynamics in this direction have been limited to phenomenological models [24,25].…”

Section: Introductionmentioning

confidence: 99%

Effective dynamics of twisted and curved scroll waves using virtual filaments

Dierckx

Verschelde

2013

Phys. Rev. E

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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.

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“…This observation suggests to enlarge Kuramoto's notion of surface tension [8] to the effective dynamical surface tension eff ðKÞ defined by Eq. (17). Consequently, the stability of systems with % 0 depends on their local curvature.…”

Section: P H Y S I C a L R E V I E W L E T T E R Smentioning

confidence: 99%

“…For homogenous media, the determinant g of the metric in Cartesian coordinates is constant; long-wavelength deviations from homogeneity can be treated in perturbation theory [17]. Under such a condition, the diffusion term is proven to be equal to a covariant Laplacian [15]:…”

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confidence: 99%

Accurate Eikonal-Curvature Relation for Wave Fronts in Locally Anisotropic Reaction-Diffusion Systems

2011

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1The dependency of wave velocity in reaction-diffusion (RD) systems on the local front curvature determines not only the stability of wave propagation, but also the fundamental properties of other spatial configurations such as vortices. This Letter gives the first derivation of a covariant eikonal-curvature relation applicable to general RD systems with spatially varying anisotropic diffusion properties, such as cardiac tissue. The theoretical prediction that waves which seem planar can nevertheless possess a nonvanishing geometrical curvature induced by local anisotropy is confirmed by numerical simulations, which reveal deviations up to 20% from the nominal plane wave speed. [5]. The waves of excitation in cardiac tissue that initiate cardiac contraction are also being intensively studied, as abnormal wave propagation in the heart may lead to the formation of vortices and results in dangerous cardiac arrhythmias [6,7]. The key parameter that characterizes a propagating wave is its velocity; it turns out that the velocity of wave propagation in RD systems is affected by several factors, the curvature of the wave front being one of the most important. The dependency of the wave velocity on front curvature, i.e., the eikonal-curvature relation, was found in many systems [4,[8][9][10], and it was also shown to be essential to the stability of the wave front and properties of vortices in the particular RD system. The simplest eikonal-curvature relation is given by the following linear relationship [8,9]:with c 0 the speed of a traveling plane wave in the given medium and a medium-dependent constant close to the scalar diffusivity of the isotropic medium [8,11]. Although stability analysis of Eq. (1) ensures stable propagating waves for positive , stability in critical media where lies close to zero is not guaranteed [12]. Remarkably, even when > 0, the linearized equation (1) Fortunately, the mathematical treatment of local anisotropy is greatly simplified when the curved-space formalism introduced in [15] is adopted. This formalism will not only allow us to generalize Eq. (1) to homogeneous excitable media of an arbitrary anisotropy type, but it also simplifies calculations such that the quadratic curvature corrections to the wave speed can be calculated for the first time. Our mathematical derivation provides an alternative and a generalization to Keener's seminal proof for the isotropic case, which relies on a boundary layer approximation and is thus restricted to the limit of steep wave fronts [9]. Here, Kuramoto's approach [8] is elaborated in a Riemannian context and pursued to higher order in curvature.To start the derivation, the set of coupled reactiondiffusion equations (RDE) is written in terms of a column matrix u of state variables u ðmÞ , with the summation convention for repeated spatial indices, @ t uðr; tÞ ¼ @ i ðD ij ðrÞ@ j Puðr; tÞÞ þ Fðuðr; tÞÞ:Local anisotropy of the medium is embodied by the spatially varying diffusion tensor D ij ðrÞ. The constant projection matrix P selects only those sta...

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A geometric theory for scroll wave filaments in anisotropic excitable media

Cited by 19 publications

References 38 publications

Negative Tension of Scroll Wave Filaments and Turbulence in Three-Dimensional Excitable Media and Application in Cardiac Dynamics

Negative Tension of Scroll Wave Filaments and Turbulence in Three-Dimensional Excitable Media and Application in Cardiac Dynamics

Effective dynamics of twisted and curved scroll waves using virtual filaments

Accurate Eikonal-Curvature Relation for Wave Fronts in Locally Anisotropic Reaction-Diffusion Systems

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