Dynamics of spiral waves in perturbed ͑e.g., slightly inhomogeneous͒ two-dimensional autowave media can be described asymptotically in terms of Aristotelean dynamics, so that the velocities of the spiral wave drift in space and time are proportional to the forces caused by the perturbation. These forces are defined as convolutions of the perturbation with the so-called response functions. In this paper, we find the response functions numerically for the spiral waves in the complex Ginzburg-Landau equation, and show that they exponentially decrease with distance. ͓S1063-651X͑98͒06603-3͔ PACS number͑s͒: 82.40. Bj, 02.60.Cb, 64.60.Ht, 87.10.ϩe Problem formulation. Spiral waves are observed in twodimensional nonlinear active systems of various natures, e.g., Belousov-Zhabotinsky reaction ͓1͔ cardiac tissue ͓2͔, social microorganisms ͓3͔, neural tissue ͓4͔, and catalytic oxidation of CO ͓5͔. They attract attention as model self-organizing structures, and demonstrate remarkable stability. In this paper, we show that spiral waves have a very selective sensitivity to perturbations. Spiral waves are often studied in terms of ''reactiondiffusion'' PDE systems,where R ជ R 2 , u(R ជ ,t)ϭ(u 1 ,u 2 , . . . ) T R l is a columnvector of reagent concentrations, fR l are nonlinear reaction rates, DR l ϫl is matrix of diffusion coefficients, l у2and hR l is a perturbation. As shown in ͓6͔, if the last term in ͑1͒ is of a more general form of parametric perturbation h(u,R ជ ,t), this still reduces to ͑1͒ in the first order in , so without loss of generality here we consider the simpler form h(R ជ ,t). Physical origin of the perturbation may be various; the most frequent in applications is inhomogeneity of medium parameters, but the analysis can be also extended to external influence, anisotropy, etc. The simplest case of spiral wave is that of the steadily rotating spiral,where is its angular velocity and PϭP(R ជ ), ⌰ϭ⌰(R ជ ) are polar coordinates. This may be observed in perfectly homogeneous unbounded stationary media, i.e., at hϭ0. In the presence of perturbations, the spiral will drift in space and accelerate or decelerate its rotation, i.e., ''drift in time.'' This can be represented bywhere R ជ c ϭ(X c ,Y c ) is the vortex rotation center and ⌽ is its initial rotation phase.The asymptotic theory of such drifts has been developed in ͓6͔. It leads to Aristotelian motion equations, where the drift velocities are proportional to the forces caused by perturbation h,In the first approximation, the forces are linear convolutiontype functionals of the perturbation,where r ជ R 2 is the radius vector in the frame of references attached to the spiral wave, where the polar coordinates are ϭP͑R ជ ϪR ជ c ͒, ϭ⌰͑R ជ ϪR ជ c ͒ϩtϪ⌽. ͑6͒We call kernels W (0,1) response functions ͑RF's͒. They determine the influence of particular perturbations at a particular site and instant onto the phase ͑temporal RF, W (0) ) and location ͑spatial RF, W (1) ) of the spiral wave. As seen in Eq. ͑5͒, graphs of these functions rotate together with their spiral ...
We describe an asymptotic approach to gated ionic models of single-cell cardiac excitability. It has a form essentially different from the Tikhonov fast-slow form assumed in standard asymptotic reductions of excitable systems. This is of interest since the standard approaches have been previously found inadequate to describe phenomena such as the dissipation of cardiac wave fronts and the shape of action potential at repolarization. The proposed asymptotic description overcomes these deficiencies by allowing, among other non-Tikhonov features, that a dynamical variable may change its character from fast to slow within a single solution. The general asymptotic approach is best demonstrated on an example which should be both simple and generic. The classical model of Purkinje fibers (Noble in J. Physiol. 160:317-352, 1962) has the simplest functional form of all cardiac models but according to the current understanding it assigns a physiologically incorrect role to the Na current. This leads us to suggest an "Archetypal Model" with the simplicity of the Noble model but with a structure more typical to contemporary cardiac models. We demonstrate that the Archetypal Model admits a complete asymptotic solution in quadratures. To validate our asymptotic approach, we proceed to consider an exactly solvable "caricature" of the Archetypal Model and demonstrate that the asymptotic of its exact solution coincides with the solutions obtained by substituting the "caricature" right-hand sides into the asymptotic solution of the generic Archetypal Model. This is necessary, because, unlike in standard asymptotic descriptions, no general results exist which can guarantee the proximity of the non-Tikhonov asymptotic solutions to the solutions of the corresponding detailed ionic model.
Weak periodic external perturbations of an autowave medium can cause large-distance directed motion of the spiral wave. This happens when the period of the perturbation coincides with, or is close to the rotation period of a spiral wave, or its multiple. Such motion is called resonant or parametric drift. It may be used for low-voltage defibrillation of heart tissue. Theory of the resonant drift exists, but so far was used only qualitatively. In this paper, we show good quantitative agreement of the theory with direct numerical simulations. This is done for Complex Ginzburg-Landau Equation, one of the simplest autowave models.
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