Excitable front geometry in reaction-diffusion systems with anomalous dispersion

Mechanism and Phenomenology of an Oscillating Chemical Reaction

Chemical reactions, which are far from equilibrium, are capable of displaying oscillations in species concentrations and hence in colour, electrode potential, pH and/or temperature. The oscillations arise from the interplay between positive and negative kinetic feedback. Mechanisms for such reactions are presented, along with the rich phenomenology that these systems exhibit, from complex oscillations and chemical waves, to stationary concentration patterns. This review will focus on the Belousov-Zhabotinksy reaction but reference to other reactions will be made where appropriate.

Period-2 spiral waves supported by nonmonotonic wave dispersion

Rotating spiral waves appear ubiquitously in a wide range of nonlinear systems, and they play important roles in many biological phenomena. Recently, unusual spiral waves, which support period-2 dynamics, have been found in several different systems including cardiac tissues as well as nonlinear chemical reaction-diffusion systems. They are potentially significant as an intermediate dynamic state linking regularly rotating period-1 spiral waves to complex dynamic states such as cardiac fibrillations; for example, it is intrinsic of period-2 spiral waves to have "line defects" and their instability can lead to a spatiotemporal chaos. Previous mathematical models regarding period-2 spiral waves are mostly based on a coupled system of period-2 oscillators, but these are inappropriate for the description of a large class of systems that are composed of (nonoscillatory) excitable elements--a good example being the heart. In this paper we hypothesize that excitable media, which support a nonmonotonic conduction velocity dispersion relation, can sustain period-2 oscillatory spiral waves. We explicitly demonstrate that the new mechanism can create period-2 spirals by computer simulations on a simple mathematical model describing spiral wave front dynamics.

Electric-field-induced wave groupings of spiral waves with oscillatory dispersion relation

The dynamic behavior of spiral-shaped excitation patterns with oscillatory dispersion is investigated under the influence of externally applied direct current or alternating current. For these two types of electric field, wave-grouping phenomena are generally observed. For the direct current field, the spiral wave drifts approximately along a straight line and wave groupings appear in certain ranges of spatial polar angles when the strength of the external field is larger than a threshold. In terms of the Doppler effect induced by the drift of the spiral tip and the oscillatory dispersion, we propose a theory model to predict the spatial distribution of wave grouping and the critical strength of the current. In contrast, for the alternating current field, the spiral wave may stay stationary and wave grouping may appear in the whole space with a different manner. This finding indicates that movement of the spiral tip is not necessary for the appearance of wave grouping.

Input-output relation of FitzHugh-Nagumo elements arranged in a trifurcated structure

In this study, the propagation of an action potential in a network of excitable elements is studied numerically. The network we consider consists of excitable elements arranged in the shape of a trifurcated structure having three cables. The system has a branch point, a Y junction, at which the three cables are joined. Two types of external stimulations are considered: a single impulsive stimulation at one of the cable terminals, and a pair of stimuli applied to two different terminals. We have found three basic phases depending on the excitability of the elements for a single external stimulus as follows: (1) signal distributor--as the excitability gets higher, the pulse generated by a stimulus splits into two at the branch point, and two pulses are transmitted to the opposite terminals, (2) propagation block--the pulse in the lower excitable chain is blocked at the branch point, and (3) transient propagation--as the excitability is decreased further, we see that the pulse vanishes before reaching the branch point. By the interaction between the pulses that originate from different sources, signal transmission is recovered if the pulses arrive at the branch point nearly synchronously or after a specific delay time. The effects of the repetition of these two types of stimulation are also investigated. Complex spatiotemporal patterns occur due to pulse-pulse interaction and collisions at the branch point. The input-output relationship, which depends crucially on the repetition period and the time lag between the pair of stimuli, is characterized by the stimulus-response ratio and the interspike interval. We also show the effects of noise on the distribution of the interspike interval.

Wave trains in an excitable FitzHugh-Nagumo model: bistable dispersion relation and formation of isolas

We investigate the dispersion relations of nonlinear periodic wave trains in excitable systems which describe the dependence of the propagation velocity on the wavelength. Pulse interaction by oscillating pulse tails within a wave train leads to bistable wavelength bands, in which two stable and one unstable wave train coexist for the same wavelength. The essential spectra of the unstable wave trains exhibit a circle of eigenvalues with positive real parts which is detached from the imaginary axis. We describe the destruction of the bistable dispersion curve and the formation of isolas of wave trains in a sequence of transcritical bifurcations unfolding into pairs of saddle-node bifurcations. It turns out that additional dispersion curves of unstable wave trains play an important role in the destruction of the bistable dispersion curve.

Wave grouping of a meandering spiral induced by Doppler effects and oscillatory dispersion

A new type of meandering spiral pattern, in which the dense waves form groups while the sparse waves keep evenly spaced, is observed in a spatial open reactor using a ferroin-catalyzed Belousov-Zhabotinsky reaction. Such a phenomenon is related to both the Doppler effect of a meandering spiral and the oscillatory dispersion relation of the system. Simulation in the two-dimensional Oregonator reaction-diffusion model with an oscillatory dispersion relation gives very similar results.

Front aggregation in multiarmed excitation vortices

Using the Belousov-Zhabotinsky reaction, we study the pinning of multiarmed spiral waves to nonexcitable obstacles. With increasing obstacle size, the individual arms switch from a repulsive to an attractive state. This transition yields densely aggregated spiral arms and is caused by anomalous dispersion. A kinematic model reproduces the measurements quantitatively and identifies the transition as a supercritical pitchfork bifurcation.

Tracking waves and vortex nucleation in excitable systems with anomalous dispersion

We report experimental results obtained from a chemical reaction-diffusion system in which wave propagation is limited to a finite band of wavelengths and in which no solitary pulses exist. Wave patterns increase their size through repeated annihilation events of the frontier pulse that allow the succeeding pulses to advance farther. A related type of wave dynamics involves a stable but slow frontier pulse that annihilates subsequent waves in front-to-back collisions. These so-called merging dynamics give rise to an unexpected form of spiral wave nucleation. All of these phenomena are reproduced by a simple, three-species reaction-diffusion model that reveals the importance of the underlying anomalous dispersion relation.

Oscillatory dispersion and coexisting stable pulse trains in an excitable medium

For planar wave trains in excitable media, we found a novel type of anomalous dispersion distinguished by bistable domains in the dependence of the propagation velocity on the wavelength. Within one medium alternative stable pulse trains can coexist having the same wavelength but different velocities. The phenomenon is related to oscillatory recovery of excitations, which causes small amplitude oscillations in the refractory tail of pulses. Crucial for the bistability is that the pulses in the trains are locked into one oscillation maximum in the tail of the preceding pulse in the train.