Interactions of spiral waves in inhomogeneous excitable media

Vinson, Michael

doi:10.1016/s0167-2789(97)00256-x

Cited by 25 publications

(11 citation statements)

References 27 publications

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“…The numerical values of the model parameters were identical with those used in [15]. The calculations were performed in a threedimensional domain using a cylindrical coordinate system z; ; .…”

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confidence: 99%

“…However, due to the analytic intractability of the problem, similar progress in theoretical filament interaction dynamics is only recently forthcoming. Much of the literature deals with topological concerns [16], selfinteraction or stability [12,15,17], or motion of a single filament [18]; such experiments have been performed mostly with the complex Ginzburg-Landau equations.…”

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Interaction Dynamics of a Pair of Vortex Filament Rings

Bray

Wikswo

2003

Phys. Rev. Lett.

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Vortex filament-filament interactions are believed to underlie lethal arrhythmias in cardiac tissue but their dynamics remain poorly understood. We numerically replicate an experimentally postulated reentrant filament configuration as a pair of adjacent circular filaments (scroll rings) with common symmetry axes and varying initial radii and separation distances. The interaction properties are quantified in terms of the scroll-ring lifetime T(L) and direction of initial velocity V0. Two cases were examined, differing only in the direction of the wave around the filament, and observed drastic differences in T(L) between the cases as the separation distance between the rings was decreased. We conclude that ring interactions present unexpected behaviors associated with competing interaction and decay mechanisms.

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Interaction Dynamics of a Pair of Vortex Filament Rings

Bray

Wikswo

2003

Phys. Rev. Lett.

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“…We see that a domain wall has been pushed by D toward L, and in Fig. 1(b) this wall is now close to L. At this point, the lesser spiral L interacts strongly with the domain wall [12] and we observe that it is swept away at the speed of the domain wall,…”

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confidence: 69%

Effect of Inhomogeneity on Spiral Wave Dynamics

1999

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The effect of weak inhomogeneity on spiral wave dynamics is studied within the framework of the two-dimensional complex Ginzburg-Landau equation description. The inhomogeneity gives spatial dependence to the frequency of spiral waves. This provides a mechanism for the formation of a dominant spiral domain that suppresses other spiral domains. The spiral vortices also slowly drift in the inhomogeneity, and results for the velocity are given. [S0031-9007(98)08283-0] PACS numbers: 82.40.Ck, 47.32.Cc, 47.54. + r Spiral waves occur in such diverse situations as cardiac arrythmias [1], reaction-diffusion systems (such as that describing the Belousov-Zhabotinsky reaction) [2,3], and slime mold colonies [4,5]. In this paper we consider the effect of an inhomogeneity of the supporting medium on spiral wave dynamics. For example, in the case of arrythmias, cardiac tissue is inherently inhomogeneous. For slime mold, an excitation inhomogeneity forms due to the sorting of the prestalk and prespore cells and the inhomogeneity results in spiral vortex motion [5]. As a potential example involving chemical reaction-diffusion systems, in Ref.[3] the chemical reaction rate was varied using its sensitivity to red laser light intensity. This could conveniently provide the means to create an inhomogeneity for a test of our theory by having the intensity vary over the entire system (other sources of reaction rate inhomogeneity are temperature inhomogeneity and inhomogeneity of the gel or porous glass medium in which chemical reaction-diffusion experiments are often done, etc.).We find that, for a weak inhomogeneity, the evolution proceeds on distinct time scales. In particular, we find that, after spirals first form, the inhomogeneity causes certain spirals that are favorably located in the inhomogeneous medium to widen their domains, crowding out and sweeping away less favorably located spirals. In addition, the spiral vortices slowly drift with a velocity linearly related to appropriate gradients of the background medium properties.Our studies of these effects are based on the twodimensional complex Ginzburg-Landau equation,where A͑x, y, t͒ is complex. This equation provides a universal description for extended media in which the homogeneous state is oscillatory and near a Hopf bifurcation [6]. The equation is obtained as a balance between weak growth ͓Re͑m͔͒, weak nonlinearity (jAj 2 A term), and weak spatial coupling (= 2 A term). In the case of a homogeneous medium a and b are real constants, and m can be set to unity by an appropriate rescaling of (1). A spiral wave solution to Eq. (1) then has the general form [7] A͑r, u, t͒ F͑r͒ exp͕i͓su 2 v o t 1 c͑r͔͖͒ . (2) The topological charge s 61 results in a 2ps phase increment of A for a counterclockwise circuit around the vortex center (r 0) where A 0. For large r, the solution (2) is locally a plane wave of wave number k o . Using the rescaling of m to unity, the frequency v o satisfies the dispersion relation v o a 1 ͑b 2 a͒k 2 o . The real functions F͑r͒ ϵ jAj and c͑r͒ have the f...

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“…Here the inhibitor consists of (relatively) immobile ion channels. Constant advecting velocities for excitable media in the context of cardiac dynamics is well known for spiral wave drift when subjected to periodic wave trains [25,11,48,14]. An imposed chaotic advecting velocity may be introduced in an analogous way to model the effect of spatially and temporarily random excitations.…”

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On bifurcations in a chaotically stirred excitable medium

Menon

Gottwald

2009

Physica D: Nonlinear Phenomena

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We study a one-dimensional filamental model of a chaotically stirred excitable medium. In a numerical simulation we systematically explore its rich bifurcation scenarios involving saddle-nodes, Hopf bifurcations and hysteresis loops. The bifurcations are described in terms of two parameters signifying the excitability of the reacting medium and the strength of the chaotic stirring, respectively. The solution behaviour, in particular at the bifurcation points, is analytically described by means of a nonperturbative variational method. Using this method we reduce the partial differential equations to either algebraic equations for stationary solutions and bifurcations, or to ordinary differential equations in the case of non-stationary solutions and bifurcations. We present numerical simulations corroborating our analytical results.

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Interactions of spiral waves in inhomogeneous excitable media

Cited by 25 publications

References 27 publications

Interaction Dynamics of a Pair of Vortex Filament Rings

Interaction Dynamics of a Pair of Vortex Filament Rings

Effect of Inhomogeneity on Spiral Wave Dynamics

On bifurcations in a chaotically stirred excitable medium

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