Complex dynamics of spiral waves and motion of curves

Mikhailov, Alexander S.; Davydov, V. A.; Zykov, V. S.

doi:10.1016/0167-2789(94)90054-x

Cited by 190 publications

(147 citation statements)

References 43 publications

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“…The motion of the free edge of the surface depends on the mean curvature near the edge (H 0 ) and on the geodetic curvature κ, and it grows (G > 0) or contracts (G < 0) with a velocity G = G 0 − 2γ H 0 − γ κ. The five kinematic parameters describing the evolution of a scroll wave (V 0 , G 0 , D, γ and γ ) can be determined from the particular reaction-diffusion model (Mikhailov et al 1994).…”

Section: Kinematic Theorymentioning

confidence: 99%

“…The motion of the waves follows the dynamics of the free edge corresponding to the filament and, therefore, a quasisteady approximation can be applied (Mikhailov et al 1994). For the particular case of scroll rings, the geodetic and mean curvatures can be expressed in terms of the angle of rotation around the filament α and the radius R of the scroll ring: κ = −R −1 cos(α) and H 0 = (k − R −1 sin(α))/2.…”

Section: Kinematic Theorymentioning

confidence: 99%

“…Application of a weak periodic, spatially uniform external forcing through a temporal modulation of the medium excitability provides an efficient control mechanism for spiral waves (Mikhailov et al 1994;Mantel and Barkley 1996). Under resonance conditions, i.e., periodic forcing with the same frequency of the spiral rotation, the spiral wave starts to drift along a straight line with the direction determined by the modulation phase.…”

Section: Temporal Forcingmentioning

confidence: 99%

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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: Kinematic Theorymentioning

confidence: 99%

Section: Kinematic Theorymentioning

confidence: 99%

Section: Temporal Forcingmentioning

confidence: 99%

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

2012

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“…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.…”

mentioning

confidence: 99%

“…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.…”

mentioning

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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