Vladimir S. Zykov

External forcing of spiral waves.

The effect of an external rhythm on rotating spiral waves in excitable media is investigated. Parameters of the unperturbed medium were chosen, such that the organizing spiral tip describes meandering (hypocyclic) trajectories, which are the most general shape for the experimentally observed systems. Periodical modulation of excitability in a model of the Belousov-Zhabotinsky (BZ) reaction forces meandering spiral tips to describe trajectories that are not found at corresponding stationary conditions. For different modulation periods, two types of resonance drift, phase-locked tip motion, a spectrum of hypocyclic trajectories, and complex multifrequency patterns were computed. The computational results are complemented by experimental data obtained for periodically changing illumination of the photosensitive BZ reaction. The observed drastic deformation of the tip trajectory is considered as an efficient means to study and to control wave processes in excitable media.

Comparative study on photosynthetic activity of chloroplasts in acid and alkaline zones of Chara corallina

A novel experimental approach has been applied to investigate the relationship between pH banding in Chara cells and photosynthetic activity of chloroplasts located in cell regions adjacent to acid and alkaline bands. The combination of pH microelectrode technique with pulse amplitude modulation (PAM) microfluorimetry enabled parallel measurements of longitudinal pH profiles and chlorophyll fluorescence yield in acid and alkaline zones of individual Chara cells. The scanning with a pH-microelectrode along the cell length revealed the light-dependent pH pattern, i.e., alternating acid and alkaline bands with pH differences as large as 2 - 3 pH units. In parallel, measurements of chlorophyll fluorescence yield under actinic light were performed using PAM microfluorometry. It was found that the effective photochemical yield of photosystem II is substantially higher in acid than in alkaline zones. The results clearly show that the banding pattern is not confined solely to the plasmalemma but is also exhibited in alternating photosynthetic performance of the underlying chloroplast layer. Apparently, the acid regions enriched with CO2 ensure sufficient flow of this substrate to the Calvin cycle reactions, thus promoting the photosynthetic rate, whereas the alkaline zones devoid of CO2 favor radiative losses of absorbed solar energy in chloroplasts.

Spiral waves on circular and spherical domains of excitable medium

Abstract The evolution of spiral waves on a circular domain and on a spherical surface is studied by numerical integration of a reaction-diffusion system. Two different asymptotic regimes are observed for both domains. The first regime is a rigid rotation of an excitation wave around the symmetry axis of the domain. The second one is a compound rotation including a drift of the rotation center of the spiral wave either along the boundary of the disk or along the equator of the sphere. In this case the shape of the wave and its rotation velocity are periodically changing in time. Simplified analytical estimates are presented to describe the rigid rotation. The computational results are complemented by experimental observation of spiral wave evolution in a small circular domain of a thin layer of Belousov-Zhabotinsky reagent.

Measurements of kinematical parameters of spiral waves in media of low excitability

The dynamics of spiral waves rotating in a thin layer of the light sensitive Belousov–Zhabotinsky reaction mixture are studied under a homogeneous and steady illumination. At a given composition of the excitable medium, the spiral waves meand, when no or low intensity light is applied, or rigidly rotate, when the light intensity is increased sufficiently. There exists, however, a critical value of light intensity above which no wave activity is supported by the medium, since its excitability is too strongly reduced by the illumination. In the vicinity of this critical value the basic kinematical parameters of rigidly rotating spirals (such as the rotation period, wavelength, propagation velocity, and the diameter of the spiral core) are measured as a function of the illumination intensity. The experimental observations are in good agreement with the predictions based on an earlier proposed kinematical theory of spiral waves in media of low excitability.

Stability of travelling fronts in a piecewise-linear reaction-diffusion system

A stability analysis of fronts is performed analytically for a Rinzel-Keller-type model with equal diffusion constants. Exact solutions are obtained for the propagating front, the bifurcation diagram of the front velocity and the growth rate of disturbances. The effect of the most unstable eigenmode on the front shape is displayed. Moreover, we give the eigenvalues of the stability operator and show explicitly how a non-moving front becomes unstable at the bifurcation point while a moving front becomes stable.

Simple and Complex Spiral Wave Dynamics

Spiral waves rotating in an excitable medium present a classical example of unusual nonlinear phenomena in distributed systems. In this paper we discuss the results of experimental studies of spiral wave dynamics in homogeneous excitable media which are modifications of the Belousov-Zhabotinsky system. A variety of dynamical regimes from very simple and well ordered to irregular complex ones are described that are created under different experimental conditions. Spiral wave dynamics is considered in stationary media with different excitability, under the influence of the boundary conditions, and under a periodic modulation of a parameter of the medium. The experimentally observed patterns are compared with the data of computer simulations on the basis of equations representing the properties of excitable media.

Transition from local to global feedback control of spiral wave dynamics

The dynamics of spiral waves rotating in a thin layer of the light-sensitive Belousov–Zhabotinsky reaction are controlled by a time-dependent uniform illumination. The intensity of the illumination is taken to be proportional to the average wave activity observed within a circular domain of the reaction layer. It is shown that an increase of the domain size, corresponding to a transition from local to global feedback control, drastically changes the conditions for stabilization and/or destabilization of the rigidly rotating spiral wave. A theoretical approach is proposed in order to explain the experimentally observed phenomena.

Fast propagation regions cause self-sustained reentry in excitable media

Significance Rotating spiral waves have been observed in a wide variety of nonlinear spatially distributed systems in physics, chemistry, and biology, called excitable media. In medicine, they are associated with cardiac arrhythmias. Nevertheless, how exactly these waves are initiated in excitable tissue remains largely unknown. Several candidate mechanisms have been proposed over the last century, largely based on the assumption of wave breaking due to unexcitable obstacles or the medium’s refractoriness. Here, we have identified a generic mechanism for self-sustained spiral wave generation in inhomogeneous excitable media by localized regions with the fast propagation velocity. We believe our theory creates constructive options to fight these deadly phenomena. Self-sustained waves of electrophysiological activity can cause arrhythmia in the heart. These reentrant excitations have been associated with spiral waves circulating around either an anatomically defined weakly conducting region or a functionally determined core. Recently, an ablation procedure has been clinically introduced that stops atrial fibrillation of the heart by destroying the electrical activity at the spiral core. This is puzzling because the tissue at the anatomically defined spiral core would already be weakly conducting, and a further decrease should not improve the situation. In the case of a functionally determined core, an ablation procedure should even further stabilize the rotating wave. The efficacy of the procedure thus needs explanation. Here, we show theoretically that fundamentally in any excitable medium a region with a propagation velocity faster than its surrounding can act as a nucleation center for reentry and can anchor an induced spiral wave. Our findings demonstrate a mechanistic underpinning for the recently developed ablation procedure. Our theoretical results are based on a very general and widely used two-component model of an excitable medium. Moreover, the important control parameters used to realize conditions for the discovered phenomena are applicable to quite different multicomponent models.

Flow-driven instabilities during pattern formation of Dictyostelium discoideum

The slime mold Dictyostelium discoideum is a well known model system for the study of biological pattern formation. In the natural environment, aggregating populations of starving Dictyostelium discoideum cells may experience fluid flows that can profoundly change the underlying wave generation process. Here we study the effect of advection on the pattern formation in a colony of homogeneously distributed Dictyostelium discoideum cells described by the standard Martiel–Goldbeter model. The external flow advects the signaling molecule cyclic adenosine monophosphate (cAMP) downstream, while the chemotactic cells attached to the solid substrate are not transported with the flow. The evolution of small perturbations in cAMP concentrations is studied analytically in the linear regime and by corresponding numerical simulations. We show that flow can significantly influence the dynamics of the system and lead to a flow-driven instability that initiate downstream traveling cAMP waves. We also show that boundary conditions have a significant effect on the observed patterns and can lead to a new kind of instability.

Stabilized wave segments in an excitable medium with a phase wave at the wave back

The propagation velocity and the shape of a stationary propagating wave segment are determined analytically for excitable media supporting excitation waves with trigger fronts and phase backs. The general relationships between the mediumʼs excitability and the wave segment parameters are obtained in the framework of the free boundary approach under quite usual assumptions. Two universal limits restricting the region of existence of stabilized wave segments are found. The comparison of the analytical results with numerical simulations of the well-known Kessler–Levine model demonstrates their good quantitative agreement. The findings should be applicable to a wide class of systems, such as the propagation of electrical waves in the cardiac muscle or wave propagation in autocatalytic chemical reactions, due to the generality of the free-boundary approach used.