Excitable Reaction-diffusion Systems Research Articles

The proneural wave in the Drosophila optic lobe is driven by an excitable reaction-diffusion mechanism.

In living organisms, self-organised waves of signalling activity propagate spatiotemporal information within tissues. During the development of the largest component of the visual processing centre of the Drosophila brain, a travelling wave of proneural gene expression initiates neurogenesis in the larval optic lobe primordium and drives the sequential transition of neuroepithelial cells into neuroblasts. Here, we propose that this 'proneural wave' is driven by an excitable reaction-diffusion system involving epidermal growth factor receptor (EGFR) signalling interacting with the proneural gene l'sc. Within this framework, a propagating transition zone emerges from molecular feedback and diffusion. Ectopic activation of EGFR signalling in clones within the neuroepithelium demonstrates that a transition wave can be excited anywhere in the tissue by inducing signalling activity, consistent with a key prediction of the model. Our model illuminates the physical and molecular underpinnings of proneural wave progression and suggests a generic mechanism for regulating the sequential differentiation of tissues.

Resonance in an ensemble of excitable reaction–diffusion systems under spatially periodic force

Abstract In this paper, we investigate the resonance collective behavior in an ensemble of excitable reaction–diffusion systems subjected to the weak signal and spatially periodic force. It is demonstrated that the resonance behavior is optimized by intermediate values of the spatial force’s amplitude and frequency, which is termed spatially periodic-force-induced resonance. Moreover, we study that how the diffusion coefficient and modulation period influence the response of the system to the external weak signal, and present the mechanism of this resonance phenomenon. These findings show that spatially periodic force as intrinsic diversity might have a constructive role and shed light on our understanding of the collective behaviors of nonlinear systems driven by spatially periodic force in response to the weak signal.

Hysteresis and drift of spiral waves near heterogeneities: From chemical experiments to cardiac simulations

Dissipative patterns in excitable reaction-diffusion systems can be strongly affected by spatial heterogeneities. Using the photosensitive Belousov-Zhabotinsky reaction, we show a hysteresis effect in the transition between free and pinned spiral rotation. The latter state involves the rotation around a disk-shaped obstacle with an impermeable and inert boundary. The transition is controlled by changes in light intensity. For permeable heterogeneities of higher excitability, we observe spiral drift along both linear and circular boundaries. Our results confirm recent theoretical predictions and, in the case of spiral drift, are further reproduced by numerical simulations with a modified Oregonator model. Additional simulations with a cardiac model show that orbital motion can also exist in anisotropic and three-dimensional systems.

Pinned chemical waves in the presence of Stokes flow.

Excitable reaction-diffusion systems form a wealth of dissipative concentration patterns that exist not only in chemical systems but also control or disrupt biological functions. An important example are rotating spiral waves in the autocatalytic Belousov-Zhabotinsky reaction. We show that the viscosity of this system can be increased by the addition of the polymer xanthan gum. In the resulting system, we pin spiral waves to a thin glass rod and then reposition the vortex centers by a linear motion of the heterogeneity. The Stokes flow generated by this motion can be a weak perturbation to the wave pattern and follows a simple, analytical expression. Numerical simulations of a corresponding reaction-diffusion-flow model reproduce the experimental observations and show that the spatial extent of the flow field can vary widely around the characteristic wavelength of the spiral. We find that a sharp spatial decay of the flow pattern corresponds to our experimental observations, whereas more expansive flow fields surprisingly allow the repositioning of spiral tips at speeds faster than the wave velocity.

Stochastic simulations of pattern formation in excitable media.

We present a method for mesoscopic, dynamic Monte Carlo simulations of pattern formation in excitable reaction–diffusion systems. Using a two-level parallelization approach, our simulations cover the whole range of the parameter space, from the noise-dominated low-particle number regime to the quasi-deterministic high-particle number limit. Three qualitatively different case studies are performed that stand exemplary for the wide variety of excitable systems. We present mesoscopic stochastic simulations of the Gray-Scott model, of a simplified model for intracellular Ca oscillations and, for the first time, of the Oregonator model. We achieve simulations with up to particles. The software and the model files are freely available and researchers can use the models to reproduce our results or adapt and refine them for further exploration.

Deformable self-propelled domain in an excitable reaction-diffusion system in three dimensions

We derive a set of equations of motion for an isolated domain in an excitable reaction-diffusion system in three dimensions. In the singular limit where the interface is infinitesimally thin, the motion of the center of mass coupled with deformation is investigated near the drift bifurcation where a motionless domain becomes unstable and undergoes migration. This is an extension of our previous theory in two dimensions. We show that there are three basic motions of a domain, straight motion, rotating motion, and helical motion. The last one is a characteristic of three dimensions. The phase diagram of these three solutions is given in the parameter space of the original reaction-diffusion equations.

Deformation of a self-propelled domain in an excitable reaction-diffusion system

We formulate a theory for a self-propelled domain in an excitable reaction-diffusion system in two dimensions where the domain deforms from a circular shape when the propagation velocity is increased. In the singular limit where the width of the domain boundary is infinitesimally thin, we derive a set of equations of motion for the center of gravity and two fundamental deformation modes. The deformed shapes of a steadily propagating domain are obtained. The set of time-evolution equations exhibits a bifurcation from a straight motion to a circular motion by changing the system parameters.

Stationary Periodical Structure Emitting an Infinite Number of Traveling Impulses in a Model of a One-Dimensional Infinite Excitable Reaction−Diffusion System

A two-variable model of a one-dimensional infinite excitable reaction-diffusion system describing an expanding stationary periodical structure emitting traveling impulses is presented. The model is based on two coupled catalytic (enzymatic) reactions. The chemical scheme consists of mono- and bimolecular reactions.

Coherent wave patterns sustained by a localized inhomogeneity in an excitable medium

The influence of a localized inhomogeneity (oscillatory or stationary) on spatiotemporal chaotic state in an excitable reaction-diffusion system is investigated. We find that various coherent wave patterns, such as spiral waves (including multiarmed) and target wave patterns are able to be created by the inhomogeneity from the chaotic state. Due to the growth of these coherent wave patterns, the previously existing turbulent waves in the absence of inhomogeneity are suppressed. At last, the whole system is entrained by the coherent wave patterns. Closer investigations indicate that the possible mechanisms underlying the inhomogeneity sustained coherent wave patterns seem quite different for oscillatory and stationary inhomogeneities.

Bounds on the Hausdorff dimension of a renormalisation map arising from an excitable reaction-diffusion system on a fractal lattice

A renormalisation approach to investigate travelling wave solutions of an excitable reaction-diffusion system on a deterministic fractal structure has recently been derived. The dynamics of a particular class of solutions which are governed by a two-dimensional subspace of these renormalisation recursion relationships are discussed in this paper. The bifurcations of this mapping are discussed with reference to the discontinuities which arise at the singularities. The map is chaotic for a bounded region in parameter space and bounds on the Hausdorff dimension of the associated invariant hyperbolic set are calculated.

Wavetrain response of an excitable medium to local stochastic forcing

We consider the effect of stochastic forcing on a one-dimensional excitable reaction–diffusion system, where the noise only acts on a small subdomain of the medium. We show that there are distinguished scaling limits in which the stochastic forcing gives rise to a mean response of a periodic wavetrain of pulses which are deterministic in the limit. Thus, the system responds coherently and periodically to stochastic forcing. This is an instance of self-induced stochastic resonance, which has previously only been analysed for finite-dimensional systems.

Propagation failures, breathing pulses, and backfiring in an excitable reaction-diffusion system

We report results from experiments with a pseudo-one-dimensional Belousov-Zhabotinsky reaction that employs 1,4-cyclohexanedione as its organic substrate. This excitable system shows traveling oxidation pulses and pulse trains that can undergo complex sequences of propagation failures. Moreover, we present examples for (i) breathing pulses that undergo periodic changes in speed and size and (ii) backfiring pulses that near their back repeatedly generate new pulses propagating in opposite direction.

Breaking of translational symmetry of a traveling planar impulse in a two-dimensional two-variable reaction-diffusion model

The stability of a planar impulse in rectangular spatial domains for a two-variable excitable reaction-diffusion system is numerically studied. The dependence of the stability on the size of the domain perpendicular to the direction of the propagation of the impulse is shown. The instability results in asymptotic stable curved impulses or an asymptotic spatiotemporal structure, which is generated similarly to the one-dimensional backfiring phenomenon.

Periodical survival or decay of traveling impulse in a model of a one-dimensional reaction-diffusion system

A two-variable model of a one-dimensional (1D), open, excitable reaction-diffusion system describing space-time evolutions of traveling impulses is investigated. It is shown that depending on the size of the system, the traveling impulse can survive or decay. Continuous increase of the size of the system causes periodical repetitions of surviving and decay of the impulse. The qualitative properties of the model, which allow us to expect the phenomenon, are described. Numerical solutions confirm this expectation. The chemical reaction scheme is realistic and may be a stimulus for seeking the phenomenon in experiments.

Statistical pulse dynamics in a reaction–diffusion system

We study a nonlinear response to random external stimuli in an excitable reaction–diffusion system. Numerical simulations are carried out in one dimension to investigate formation of excited domains (pulses) in response to stimuli and pair-annihilate upon collision. Our main concern is how the area of excited domains in the steady state depends on the strength of stimuli. We have found three different stimuli-response behaviors: a power law dependence for sufficiently weak stimuli, a logarithmic dependence for intermediate strength and an oscillatory behavior for strong stimuli. The power law behavior can be understood by a theoretical analysis.

Experimental implementation of collision-based gates in Belousov–Zhabotinsky medium

Abstract We experimentally demonstrate that excitation wave-fragments in a Belousov–Zhabotinsky (BZ) medium with immobilised catalyst can be used to build elementary logical gates and circuits. Following our previous theoretical constructions [Adamatzky A. Collision-based computing in Belousov Zhabotinsky medium. Chaos, Solitons & Fractals 2004;21:1259–64] on embedding logical schemes in BZ medium, we represent True/False values of logical variables by presence/absence of wave-fragments. We show that when wave-fragments collide with each other they may annihilate, fuse, split and change their velocity vectors. Thus the values of logical variables represented by the wave-fragments change and certain logical operations are implemented. In the paper we provide examples of experimental logical gates, and present pioneer results in dynamic, architectureless computing in excitable reaction–diffusion systems.

Dynamics of excitation pulses with attractive interaction: Kinematic analysis and chemical wave experiments

We present a theoretical analysis of stacking and destacking wave trains in excitable reaction-diffusion systems with anomalous velocity-wavelength dependence. For linearized dispersion relations, kinematic analysis yields an analytical function that rigorously describes front trajectories. The corresponding accelerations have exactly one extremum that slowly decays with increasing pulse number. For subsequent pulses these maxima occur with a lag time equal to the inverse slope of the linearized dispersion curve. These findings are reproduced in experiments with chemical waves in the 1,4-cyclohexanedione Belousov-Zhabotinsky reaction but should be also applicable to step bunching on crystal surfaces and certain traffic phenomena.

Dynamics of spiral waves under global feedback in excitable domains of different shapes.

It is found that the dynamics of spiral waves subjected to global feedback is extremely sensitive to the domain shape. Bifurcations in the velocity field which specifies the resonant drift of the spiral wave core induced by global feedback are analyzed. It is shown, for example, that smooth variation of the eccentricity of an elliptical domain induces a cascade of bifurcations that can dramatically change the spiral wave evolution. In a square domain a set of point attractors appears instead of the circular resonance attractor typical of a circular domain. These predictions are in good quantitative agreement with numerical integrations of an excitable reaction-diffusion system performed under global feedback.

Breakup of spiral waves caused by radial dynamics: Eckhaus and finite wavenumber instabilities

In this paper, we link numerical observations of spiral breakup to a stability analysis of simple rotating spirals. We review the phenomenology of spiral breakup, important applications in pattern formation and the state of the art in numerical stability analysis of spirals. A strategy for the latter procedure is suggested. Phenomenologically, spiral breakup can occur near the centre of rotation (‘core breakup’) or far away from it (‘far-field breakup’). It may be accompanied by instabilities of the spiral core in particular spiral meandering that affect also the stability of waves in the far-field, because an unstable core acts as a moving source and introduces a (nonlinear) Doppler effect. In general, breakup of non-meandering spirals is related to an absolute instability of the planar wavetrain with the same wave number. To simplify the stability problem, we consider a one-dimensional cut (‘1D spiral’) with a fixed core position in simulations and compare the results with a stability analysis of planar wavetrains. These 1D spirals approximate the radial dynamics of non-meandering 2D spirals. To fully account for instabilities of 1D spirals, it is not sufficient to compute the direction of propagation of the unstable modes of the wavetrains, one also needs to compute the so-called absolute spectrum of these wavetrains. This allows us to decide whether the instability is of convective or absolute nature. Only the latter case implies an instability of spirals in finite domains. We carry out this programme for the case of core breakup in an excitable reaction–diffusion system, the modified Barkley model. Our analysis yields that core breakup can result from the absolute variant of a novel finite wavenumber instability of the radial dynamics where the critical perturbations are transported towards the core. From these results, we can confirm that a simple spiral breaks up if the wavetrain in the far-field is absolutely unstable. The central new result is the discovery of respective convective and absolute instabilities of wavetrains connected with modes of finite wavenumber that propagate in the direction opposite to the wavetrain. Hence, in spirals, perturbations are moving inward and core breakup becomes possible even in the absence of meandering.

Excitability in a model with a saddle-node homoclinic bifurcation

In order to describe excitable reaction-diffusion systems, we derive a two-dimensional model with a Hopf and a semilocal saddle-node homoclinic bifurcation. This model gives the theoretical framework for the analysis of the saddle-node homoclinic bifurcation as observed in chemical experiments, and for the concepts of excitability and excitability threshold. We show that if diffusion drives an extended system across the excitability threshold then, depending on the initial conditions, wave trains, propagating solitary pulses and propagating pulse packets can exist in the same extended system. The extended model shows chemical turbulence for equal diffusion coefficients and presents all the known types of topologically distinct activity waves observed in chemical experiments. In particular, the approach presented here enables to design experiments in order to decide between excitable systems with sharp and finite width thresholds.