Vadim N. Biktashev

Tension of organizing filaments of scroll waves

We consider the asymptotic theory for the dynamics of organizing filaments of three-dimensional scroll waves. For a generic autowave medium where two dimensional vortices do not meander, we show that some of the coefficients of the evolution equation are always zero. This simpler evolution equation predicts a monotonic change of the total filament length with time, independently of initial conditions. Whether the filament will shrink or expand is determined by a single coefficient, the filament tension, that depends on the medium parameters. We illustrate the behaviour of scroll wave filaments with positive and negative tension by numerical experiments. In particular, we show that in the case of negative filament tension, the straight filament is unstable, and its evolution may lead to a multiplication of vortices.

Vulnerability in an excitable medium: analytical and numerical studies of initiating unidirectional propagation

Cardiac tissue can display unusual responses to certain stimulation protocols. In the wake of a conditioning wave of excitation, spiral waves can be initiated by applying stimuli timed to occur during a period of vulnerability (VP). Although vulnerability is well known in cardiac and chemical media, the determinants of the VP and its boundaries have received little theoretical and analytical study. From numerical and analytical studies of reaction-diffusion equations, we have found that 1) vulnerability is an inherent property of Beeler-Reuter and FitzHugh-Nagumo models of excitable media; 2) the duration of the vulnerable window (VW) the one-dimensional analog of the VP, is sensitive to the medium properties and the size of the stimulus field; and 3) the amplitudes of the excitatory and recovery processes modulate the duration of the VW. The analytical results reveal macroscopic behavior (vulnerability) derived from the diffusion of excitation that is not observable at the level of isolated cells or single reaction units.

Reentrant waves and their elimination in a model of mammalian ventricular tissue

The vulnerability to reentrant wave propagation, its characteristics (period, meander, and stability), the effects of rotational transmural anisotropy, and the control of reentrant waves by small amplitude perturbations and large amplitude defibrillating shocks are investigated theoretically and numerically for models based on high order, stiff biophysically derived excitation equations.

Autowave principles for parallel image processing

Abstract A highly parallel autowave method for pattern analysis and topological feature detection is presented. It is invariant against translations, rotations and scaling of the input pattern. The method yields an increase in computational speed of 3 to 6 orders of magnitude in comparison with a sequential (von Neumann) computer. The method can be realized in principle using only one chip with simple uniform connections of elements.

RE-ENTRANT ACTIVITY AND ITS CONTROL IN A MODEL OF MAMMALIAN VENTRICULAR TISSUE

We characterize the meander of re-entrant excitation in a model of a sheet of mammalian ventricular tissue, and its control by resonant drift under feedback driven stimulation. The Oxsoft equations for excitability in a guinea pig single ventricular cell were incorporated in a two dimensional reaction-diffusion system to model homogeneous, isotropic tissue with a plane wave conduction velocity of 0.35 m s-1. Re-entrant spiral wave solutions have a spatially extended transient motion (linear core) that settles down into rotation with an irregular period of 100-110 ms around an irregular, multi-lobed spiky core. In anisotropic tissue this would appear as a linear conduction block. The typical velocity of drift of the spiral wave induced by low amplitude resonant forcing is 0.4 cm s-1.

A cross-diffusion model of forest boundary dynamics

A simple mathematical model of mono-species forest with two age classes which takes into account seed production and dispersal is presented in the paper. This reaction — diffusion type model is then reduced by means of an asymptotic procedure to a lower dimensional reaction — cross-diffusion model. The existence of standing and travelling wave front solutions corresponding to the forest boundary is shown for the later model. On the basis of the analysis, possible changes in forest boundary dynamics caused by antropogenic impacts are discussed.

Pursuit-evasion predator-prey waves in two spatial dimensions

We consider a spatially distributed population dynamics model with excitable predator-prey kinetics, where species propagate in space due to their taxis with respect to each others gradient in addition to, or instead of, their diffusive spread. Earlier, we have described new phenomena in this model in one spatial dimension, not found in analogous systems without taxis: reflecting and self-splitting waves. Here we identify new phenomena in two spatial dimensions: unusual patterns of meander of spirals, partial reflection of waves, swelling wave tips, attachment of free wave ends to wave backs, and as a result, a novel mechanism of self-supporting complicated spatiotemporal activity, unknown in reaction-diffusion population models.

Control of Re-Entrant Activity in a Model of Mammalian Atrial Tissue

We evaluate the feasibility of using resonant drift under feedback driven stimulation to control re-entrant excitation in atrial muscle. We simulate a two-dimensional sheet of atrial tissue, where the local kinetics are described by the Earm-Hilgemann-Noble equations for a rabbit atrial cell, and the effects of small amplitude spatially uniform forcing of the whole sheet are computed. Repetitive forcing can induce a drift of a spiral wave in the two-dimensional model, with a drift velocity of up to 10 cm s-1. For a 4 cm x 4 cm atrial surface this resonant drift can move the re-entrant spiral to the inexcitable boundaries, eliminating re-entry in less than 10 s when the amplitude of the repetitive stimulation is 10% that of the single shock defibrillation threshold.

Deterministic Brownian motion in the hypermeander of spiral waves

Abstract We explain the phenomenon of hypermeander of spiral waves, observed in numerical with various models of excitable media, as a chaotic attractor in the quotient system with respect to the Euclidean group. Such an attractor should lead to a motion of the spiral wave tip analogous to that of a Brownian particle, with mean square of displacement of the tip growing linearly at large times. This prediction is confirmed by numerical experiments with hypermeandering spiral waves.

SPIRAL WAVE MEANDER AND SYMMETRY OF THE PLANE

We present a general group-theoretic approach that explains the main qualitative features of the meander of spiral wave solutions on the plane. The approach is based on the well-known space reduction method and is used to separate the motions in the system into superposition of those ‘along’ orbits of the Euclidean symmetry group, and ‘across’ the group orbits. It can be interpreted as passing to a reference frame attached to the spiral wave’s tip. The system of ODEs governing the tip movement is obtained. It is the system that describes the movements along the group orbits. The motions across the group orbits are described by a PDE which lacks the Euclidean symmetry. Consequences of the Euclidean symmetry on the spiral wave dynamics are discussed. In particular, we explicitly derive the model system for bifurcation from rigid to biperiodic rotation, suggested earlier by Barkley [1994] from a priori symmetry considerations.