Interaction of spiral waves with external fields in excitable media

A theory for spiral wave drift induced by ac and polarized electric fields in chemical excitable media

Spiral waves are shown to undergo directional drifts in the presence of ac and polarized electric fields when their frequencies are twice of the spiral frequencies. Here, we propose a quantitative description for the spiral wave drift induced by weak electric fields, and provide the explicit equations for the spiral wave drift speed and direction. Numerical simulations are performed to demonstrate the quantitative agreement with analytical results in both weakly and highly excitable media.

Spiral wave drift in an electric field and scroll wave instabilities

Here, I present the numerical computation of speed and direction of the drift of a spiral wave in an excitable medium in the presence of an electric field. The drift speed presents a strong variation close to the parameter value where the drift-speed component along the field direction from parallel becomes antiparallel. Using a simple phenomenological model and results from a numerical linear stability analysis of scroll waves, I show that this behavior can be attributed to a resonance of the meander modes with the translation modes of the spiral wave. Extending this phenomenological model to scroll waves also clarifies the link between the drift and long wavelength instabilities of scroll waves.

Large-excitability asymptotics for scroll waves in three-dimensional excitable media

Three-dimensional scroll waves are considered in a reaction-diffusion model of excitable media in the large excitability limit. Coordinates based on the scroll filament are defined and shown to provide a natural extension of the coordinates used for two-dimensional spiral waves. The leading-order free-boundary equations for interface motion in three dimensions are explicitly derived in these coordinates. Three specific examples are considered: straight twisted scroll waves, axisymmetric scroll waves, and helical scroll waves. The equations for the fields at leading and first order in the core region are given.

Scroll waves in isotropic excitable media: linear instabilities, bifurcations, and restabilized states

Scroll waves are three-dimensional analogs of spiral waves. The linear stability spectrum of untwisted and twisted scroll waves is computed for a two-variable reaction-diffusion model of an excitable medium. Different bands of modes are seen to be unstable in different regions of parameter space. The corresponding bifurcations and bifurcated states are characterized by performing direct numerical simulations. In addition, computations of the adjoint linear stability operator eigenmodes are also performed and serve to obtain a number of matrix elements characterizing the long-wavelength deformations of scroll waves.

Theory of spiral wave dynamics in weakly excitable media: asymptotic reduction to a kinematic model and applications

In a weakly excitable medium, characterized by a large threshold stimulus, the free end of an isolated broken plane wave (wave tip) can either rotate (steadily or unsteadily) around a large excitable core, thereby producing a spiral pattern, or retract, causing the wave to vanish at boundaries. An asymptotic analysis of spiral motion and retraction is carried out in this weakly excitable large core regime starting from the free-boundary limit of the reaction-diffusion models, valid when the excited region is delimited by a thin interface. The wave description is shown to naturally split between the tip region and a far region that are smoothly matched on an intermediate scale. This separation allows us to rigorously derive an equation of motion for the wave tip, with the large scale motion of the spiral wave front slaved to the tip. This kinematic description provides both a physical picture and exact predictions for a wide range of wave behavior, including (i) steady rotation (frequency and core radius), (ii) exact treatment of the meandering instability in the free-boundary limit with the prediction that the frequency of unstable motion is half the primary steady frequency, (iii) drift under external actions (external field with application to axisymmetric scroll ring motion in three dimensions, and spatial- or/and time-dependent variation of excitability), and (iv) the dynamics of multiarmed spiral waves with the prediction that steadily rotating waves with two or more arms are linearly unstable. Numerical simulations of FitzHugh-Nagumo kinetics are used to test several aspects of our results. In addition, we discuss the semiquantitative extension of this theory to finite cores and pinpoint mathematical subtleties related to the thin interface limit of singly diffusive reaction-diffusion models.

Chirality dependent component of vortex advection in excitable media

An advective field induces drift of a vortex in excitable media. The component of the drift velocity C( perpendicular ) perpendicular to the field is known to change its sign with the chirality of the vortex. In an experiment with vortices in an electric field in a chemical excitable medium, we have found unexpectedly that C( perpendicular ) changes its sign also independently of chirality with changing composition of the medium. We did not succeed to explain this phenomenon by using existing mathematical models of chemical excitable media. The experiment described calls for more realistic models.(c) 1999 American Institute of Physics.