The Dynamics of Scroll Waves in Excitable Media

Keener, James P.; Tyson, John J.

doi:10.1137/1034001

Cited by 145 publications

(92 citation statements)

References 55 publications

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“…This phenomenon has been described previously by Kida [7] for the velocity law v = κb which describes the motion of a thin inviscid fluid vortex filament (the full set of similarity reductions to v = κb are identical to those for v = κn, which are discussed in [13]). More general isotropic laws of motion (which may also involve an acceleration term) such as the two models for the evolution of scroll waves [9] and elastic rods [10], noted in the introduction, will be invariant under the same Lie point symmetry groups as (114) and thus also display rotating travelling wave and helical similarity reductions. We have also demonstrated that the subclass of laws of form v = κ c+1 ϒ κ τ n + κ τ b are, in addition, invariant under a rescaling (note that v = κn and v = κb both fall within this subclass).…”

Section: Discussionmentioning

confidence: 99%

“…The self-induced motion of scroll waves in excitable media also falls into this category for certain special classes of these materials, though the general law for the selfinduced motion of scroll waves (derived by Keener [9]) also contains terms with derivatives of the torsion. We note that (3) also applies to small aspect ratio fluid vortices [4,11].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The evolution of space curves by curvature and torsion

Richardson

King

2002

J. Phys. A: Math. Gen.

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We apply Lie group based similarity methods to the study of a new, and widely relevant, class of objects, namely motions of a space curve. In particular, we consider the motion of a curve evolving with a curvature κ and torsion τ dependent velocity law. We systematically derive the Lie point symmetries of all such laws of motion and use these to catalogue all their possible similarity reductions. This calculation reveals special classes of law with high degrees of symmetry (and a correspondingly large number of similarity reductions). Of particular note is one class which is invariant under general linear transformations in space. This has potential applications in pattern and signal recognition.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The evolution of space curves by curvature and torsion

Richardson

King

2002

J. Phys. A: Math. Gen.

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“…For example, if f (θ) has only one local maximum in [0, 2π], the new fixed point appears for α = α S given by equation (6). If f (θ) has only one minimum, the new fixed point appears for α = α S given by equation (7). The appearance of these fixed points on the limit cycle, when we vary the parameter α across the value α S , is the SNH bifurcation.…”

Section: The Saddle-node Homoclinic (Snh) Bifurcationmentioning

confidence: 99%

Excitability in a model with a saddle-node homoclinic bifurcation

Dilão¹,

Volford²

2004

Discrete &Amp; Continuous Dynamical Systems - B

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In order to describe excitable reaction-diffusion systems, we derive a two-dimensional model with a Hopf and a semilocal saddle-node homoclinic bifurcation. This model gives the theoretical framework for the analysis of the saddle-node homoclinic bifurcation as observed in chemical experiments, and for the concepts of excitability and excitability threshold. We show that if diffusion drives an extended system across the excitability threshold then, depending on the initial conditions, wave trains, propagating solitary pulses and propagating pulse packets can exist in the same extended system. The extended model shows chemical turbulence for equal diffusion coefficients and presents all the known types of topologically distinct activity waves observed in chemical experiments. In particular, the approach presented here enables to design experiments in order to decide between excitable systems with sharp and finite width thresholds.

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“…Several studies have modeled specific cases using this equation, e.g., Refs. [15,16]. For a filament tension of α > 0, the filament contracts and vanishes in finite time.…”

Section: Introductionmentioning

confidence: 99%

Analysis of anchor-size effects on pinned scroll waves and measurement of filament rigidity

et al. 2014

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Inert, spherical heterogeneities can pin three-dimensional scroll waves in the excitable Belousov-Zhabotinsky reaction. Three pinning sites cause initially circular rotation backbones to approach equilateral triangles. The resulting stationary shapes show convex deviations that increase with decreasing anchor radii. This dependence is interpreted as a transition between filament termination at large surfaces and true, local pinning of a continuous curve. The shapes of the filament segments are described by a hyperbolic cosine function which is predicted by kinematic theory that considers filament tension and rigidity. The latter value is measured as (1.0 ± 0.7) × 10 −6 cm 4 /s.

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The Dynamics of Scroll Waves in Excitable Media

Cited by 145 publications

References 55 publications

The evolution of space curves by curvature and torsion

The evolution of space curves by curvature and torsion

Excitability in a model with a saddle-node homoclinic bifurcation

Analysis of anchor-size effects on pinned scroll waves and measurement of filament rigidity

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