Reaction-diffusion Medium Research Articles

Dynamics of spiral pairs induced by unidirectional propagating pulses

The dynamics of unidirectionally propagating pulses in a two-dimensional uniform excitable reaction-diffusion medium is investigated. It is shown that under weak diffusion coupling between medium points such a pulse can evolve into a pair of counter-rotating spirals (spiral pair). We analyze the drift of such a pair and examine the collisions between several drifting pairs. It is demonstrated that collisions can result in a special type of reflection or, alternatively, in new types of complex stationary spiral structures. A possible application of these findings for the diagnosis of cardiac arrhythmias is suggested.

Spiral turbulence developed through the formation of superimposed target waves in an oscillatory reaction-diffusion medium

An approach leading to the development of spiral turbulence is reported here in an oscillatory reaction-diffusion medium, which is through the spontaneous formation of targetlike waves near the core of a spiral wave. The newly formed target wave emerges with its own characteristic frequency and propagates on top of the original spiral wave, which eventually leads to the breakup of the spiral at a location far from the spiral center. The radius of the surviving spiral segment decreases rapidly with the bifurcation control parameter. Calculation of power spectra suggests that the meandering of the spiral tip is responsible for the onset of the superimposed target and the phase desynchronization of the superimposed target waves.

Chaotic wave trains in an oscillatory/excitable medium

We study the chaotic dynamics of a heterogeneous reaction–diffusion medium composed of two uniform regions: one oscillatory, and the other excitable. It is shown that, by altering the diffusion coefficient, local chaotic oscillations can be induced at the interface between regions, which in turn, generate different chaotic sequences of pulses traveling in the excitable region. We analyze the properties of the local chaotic driver, as well as the diffusion-induced transitions. A procedure based on the abnormal frequency-locking phenomenon is proposed for controlling such sequences. Relevance of the obtained results to cardiac dynamics is briefly discussed.

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Spontaneous scroll ring creation and scroll instability in oscillatory medium with gradients

Scroll waves in a quasi-three-dimensional reaction-diffusion medium with a gradient in the third dimension are studied by numerical simulations using the Fitzhugh-Nagumo model. Under a simple initial condition with only one straight filament, we show a spontaneous creation of twisted scroll rings surrounding the initial filament when the control parameter is increased across a threshold. We find that due to the presence of the gradient, the difference of oscillation frequencies in the third dimension is the underlying cause of the phenomenon. Further increase of the control parameter will lead to the spiral breakups, as a result of the interaction of filaments. Observations in the experiments conducted in the same type of medium with the Belousov-Zhabotinsiky reaction qualitatively agree with our simulation results.

The investigation of the minimum size of the domain supporting a spiral wave in oscillatory media

We investigate the effects of the domain size on the dynamics of the spiral wave drifting in reaction–diffusion media such as complex Ginzburg–Landau equation and Barkley model. We find that the boundary of the domain becomes the only attractor for the drifting spiral wave when the domain size is small enough. The minimum size of the domain supporting a sustained spiral wave can be determined in this way.

Inwards propagating waves in heterogeneous limit cycle media

We investigate the dynamics of an oscillatory reaction–diffusion medium composed of two adjacent uniform regions where inwards propagating (IP) waves are generated in at least one of them. As was shown, IP waves display unusual “backward” motion directed towards their origin. Here pseudo-reflection of the IP waves at the interface between regions is discussed. Such an interface can drive pseudo-reflected IP waves at a frequency which, surprisingly, exceeds the natural frequencies of both regions. Simple models of two unidirectionally coupled oscillators , and a single oscillator with a stepwise constant bias, are analyzed to help understand the functioning of the interface.

Accumulations of T-points in a model for solitary pulses in an excitable reaction–diffusion medium

We consider a family of differential equations that describes traveling waves in a reaction–diffusion equation modeling oxidation of carbon oxide on a platinum surface, near the onset of spatio-temporal chaos. The organizing bifurcation for the bifurcation structure with small carbon oxide pressures, turns out to be a codimension 3 bifurcation involving a homoclinic orbit to an equilibrium undergoing a transcritical bifurcation. We show how infinitely many T-point bifurcations of multi loop heteroclinic cycles occur in the unfolding.

Pulse waves in hydrogen peroxide-sulfite-thiosulfate-perchlorate acid system

The spatiotemporal dynamics of the hydrogen peroxide-thiosulfate-sulfite-perchlorate acid system was investigated in reaction-diffusion medium. In the Petri dish, pulse waves were discovered, which had the maximum life-time and the maximum traveling distance when the initial concentration of thiosulfate was increased. In addition, pulse waves contracted back before disappearing, and the pulse waves with red spot were observed when the initial concentration of hydrogen peroxide was higher. The experimental results were analyzed and explained according to the reaction mechanism of positive and negative feedback in this system.

Refraction of autowaves: Tangent rule

The refraction of autowaves at the interface between homogeneous regions of a reaction-diffusion medium with different diffusion coefficients obeys the tangent rule.

Abnormal frequency locking and the function of the cardiac pacemaker

A heterogeneous reaction-diffusion medium consisting of two adjoining uniform regions is analyzed. The first region is a purely oscillatory one, while the second is bistable (oscillatory/excitable). We show that such a construction allows an abnormal domination of the low natural frequency of the oscillatory regime over the whole medium (abnormal frequency locking). Bifurcations leading to the appearance of the bistable regime are discussed as well as the specific dynamics of the bistable oscillations. The abnormal frequency-locking phenomenon could explain some dynamical properties of the cardiac pacemaker.

Propagation of autowaves in excitable media with chiral anisotropy

We study properties of autowave patterns in reaction–diffusion media with non-uniform chiral anisotropy of the diffusion tensor. The curvature–velocity relation for circular wave fronts in such media is derived. This relation is confirmed by numerical simulations. Strong dependence of spiral waves angular velocity on the angle of chirality is predicted theoretically and confirmed numerically. Long-ranged interaction between spiral waves is predicted and observed in computer experiments.

Pseudoreflection from interface between two oscillatory media: extended driver.

The dynamics of a reaction-diffusion medium composed of two uniform self-oscillating regions is considered. We analyze the phenomenon of pseudoreflection of waves at the region's interface. The reflected waves show an unusual change of wavelength, amplitude, and period. In contrast to our previous results, here this behavior can be perceived as an action of a spatially extended higher-frequency "driver." Observed also are the interesting phenomena of the appearance of narrow transient zones near the interface and of diffusion-induced bifurcations. Furthermore, the pseudoreflection is shown to be a possible mechanism of spiral and "target" waves generation. The relevance of the obtained results to the dynamics of the cardiac sinus node is discussed.

A transition from propagating fronts to target patterns in the hydrogen peroxide–sulfite–thiosulfate medium

In this study we explored nonlinear spatial–temporal behavior in the oxidation of sulfite by hydrogen peroxide, in which thiosulfate was added to implement a negative feedback, which consequently led to a transition from propagating fronts to target patterns. Our investigation in the homogeneous system showed that increasing the concentration of hydrogen peroxide or decreasing the concentration of sulfite accelerated both the positive and negative feedbacks and therefore could facilitate the transition from front waves to target patterns. On the other hand, increasing the initial concentration of hydrogen ions or decreasing the concentration of thiosulfate mainly accelerated the autocatalytic production of hydrogen ions. The experimental observations were qualitatively reproduced with a model developed recently by Rabai and Hanazaki. In reaction–diffusion media, spontaneous formation of propagating pH fronts was obtained within a narrow range of sulfite and thiosulfate concentrations. The velocity of these traveling fronts was found to decrease with respect to increasing concentrations of thiosulfate and sulfite or decreasing concentrations of hydrogen ion and hydrogen peroxide.

Doppler Effect of Nonlinear Waves and Superspirals in Oscillatory Media

Nonlinear waves emitted from a moving source are studied. A meandering spiral in a reaction-diffusion medium provides an example in which waves originate from a source exhibiting a back-and-forth movement in a radial direction. The periodic motion of the source induces a Doppler effect that causes a modulation in wavelength and amplitude of the waves ("superspiral"). Using direct simulations as well as numerical nonlinear analysis within the complex Ginzburg-Landau equation, we show that waves subject to a convective Eckhaus instability can exhibit monotonic growth or decay as well as saturation of these modulations depending on the perturbation frequency. Our findings elucidate recent experimental observations concerning superspirals and their decay to spatiotemporal chaos.

Experimental Reaction–Diffusion Chemical Processors for Robot Path Planning

In this paper we discuss the experimental implementation of a chemical reaction–diffusion processor for robot motion planning in terms of finding the shortest collision-free path for a robot moving in an arena with obstacles. These reaction–diffusion chemical processors for robot navigation are not designed to compete with existing silicon-based controllers. These controllers are intended for the incorporation into future generations of soft-bodied robots built of electro- and chemo-active polymers. In this paper we consider the notion of processing as being implicit in the physical medium constituting the body of a ‘soft’ robot. This work therefore represents some early steps in the employment of excitable media controllers. An image of the arena in which the robot is to navigate is mapped onto a thin-layer chemical medium using a method that allows obstacles to be represented as local changes in the reactant concentrations. Disturbances created by the ‘objects’ generate diffusive and phase wave fronts. The spreading waves approximate to a repulsive field generated by the obstacles. This repulsive field is then inputted into a discrete model of an excitable reaction–diffusion medium, which computes a tree of shortest paths leading to a selected destination point. Two types of chemical processors are discussed: a disposable palladium processor, which executes arena mapping from a configuration of obstacles, given before an experiment and, a reusable Belousov–Zhabotinsky processor which allows for online path planning and adaptation for dynamically changing configurations of obstacles.

Knotted reaction—diffusion waves

The prevalence of toroidal scroll waves in experimental studies of excitable reaction–diffusion media motivates this study. To understand the dynamics of these structures in three dimensions we study their interactions with a toroidal manifold. Using an eikonal formulation we demonstrate the existence and stability of both rigidly rotating knotted–waveforms and stationary solutions. We also show that the full reaction–diffusion system supports knotted waves, with good quantitative agreement with the eikonal formulation.

Experimental logical gates in a reaction-diffusion medium: the XOR gate and beyond.

We exploit the particulars of diffusive wave front interactions in certain types of two-reactant reaction-diffusion medium to construct a laboratory prototype of an XOR gate. In the design, the values of the logic variables are represented by the presence or absence of a precipitate, "wires" are constructed of a substrate loaded gel, and the computation is based on diffusive wave dynamics. We also discuss implementation of AND gate and study a three-valued composition, derived from the gate dynamic, and discuss possible logics that could be derived from this composition.

Image processing using light-sensitive chemical waves

Basic principles of information processing by chemical light-sensitive reaction–diffusion media and dynamic modes of these media adequate to information processing operations are studied. Specialized experimental laboratory technique is elaborated optimum for image processing investigations. New modes of image evolution in the process of its transformation by reaction–diffusion medium are observed. Basic features of image processing by chemical reaction–diffusion media could be qualitatively explained using simple Field–Korosh–Noyes model.

The influence of concentration-dependent diffusivities on wave stability

Investigations on the onset of wave instability in excitable reaction–diffusion media have mainly focused on the effects of reaction kinetics and the relative diffusivity of activator and inhibitor. In this study, we characterize wave stability in a medium in which a diffusion coefficient depends on the local concentration of a reagent. Calculations with a simple CO oxidation model illustrate that the sensitivity of the diffusion coefficient to the concentration can behave like a dynamical parameter to induce transitions between “backfiring” and “regular” propagating pulses. The transition is due to the influence of the variable diffusion process on the refractory zone of wave activity.