Collision-stable waves in excitable reaction-diffusion systems

Dissipative solitons stabilized by nonlinear gradient terms: Time-dependent behavior and generic properties

We study the time-dependent behavior of dissipative solitons (DSs) stabilized by nonlinear gradient terms. Two cases are investigated: first, the case of the presence of a Raman term, and second, the simultaneous presence of two nonlinear gradient terms, the Raman term and the dispersion of nonlinear gain. As possible types of time-dependence, we find a number of different possibilities including periodic behavior, quasi-periodic behavior, and also chaos. These different types of time-dependence are found to form quite frequently from a window structure of alternating behavior, for example, of periodic and quasi-periodic behaviors. To analyze the time dependence, we exploit extensively time series and Fourier transforms. We discuss in detail quantitatively the question whether all the DSs found for the cubic complex Ginzburg-Landau equation with nonlinear gradient terms are generic, meaning whether they are stable against structural perturbations, for example, to the additions of a small quintic perturbation as it arises naturally in an envelope equation framework. Finally, we examine to what extent it is possible to have different types of DSs for fixed parameter values in the equation by just varying the initial conditions, for example, by using narrow and high vs broad and low amplitudes. These results indicate an overlapping multi-basin structure in parameter space.

Interaction of dissipative solitons stabilized by nonlinear gradient terms

We study the interaction of stable dissipative solitons of the cubic complex Ginzburg-Landau equation which are stabilized only by nonlinear gradient terms. In this paper we focus for the interactions in particular on the influence of the nonlinear gradient term associated with the Raman effect. Depending on its magnitude, we find up to seven possible outcomes of theses collisions: Stationary bound states, oscillatory bound states, meandering oscillatory bound states, bound states with large-amplitude oscillations, partial annihilation, complete annihilation, and interpenetration. Detailed results and their analysis are presented for one value of the corresponding nonlinear gradient term, while the results for two other values are just mentioned briefly. We compare our results with those obtained for coupled cubic-quintic complex Ginzburg-Landau equations and with the cubic-quintic complex Swift-Hohenberg equation. It turns out that both meandering oscillatory bound states as well as bound states with large-amplitude oscillations appear to be specific for coupled cubic complex Ginzburg-Landau equations with a stabilizing cubic nonlinear gradient term. Remarkably, we find for the large-amplitude oscillations a linear relationship between oscillation amplitude and period.

Excitable solitons: Annihilation, crossover, and nucleation of pulses in mass-conserving activator-inhibitor media

Excitable pulses are among the most widespread dynamical patterns that occur in many different systems, ranging from biological cells to chemical reactions and ecological populations. Traditionally, the mutual annihilation of two colliding pulses is regarded as their prototypical signature. Here we show that colliding excitable pulses may exhibit solitonlike crossover and pulse nucleation if the system obeys a mass conservation constraint. In contrast to previous observations in systems without mass conservation, these alternative collision scenarios are robustly observed over a wide range of parameters. We demonstrate our findings using a model of intracellular actin waves since, on time scales of wave propagations over the cell scale, cells obey conservation of actin monomers. The results provide a key concept to understand the ubiquitous occurrence of actin waves in cells, suggesting why they are so common, and why their dynamics is robust and long-lived.

Controlling excitable wave behaviors through the tuning of three parameters

Excitable systems are a class of dynamical systems that can generate self-sustaining waves of activity. These waves are known to manifest differently under diverse conditions, whereas some travel as planar or radial waves, and others evolve into rotating spirals. Excitable systems can also form stationary stable patterns through standing waves. Under certain conditions, these waves are also known to be reflected at no-flux boundaries. Here, we review the basic characteristics of these four entities: traveling, rotating, standing and reflected waves. By studying their mechanisms of formation, we show how through manipulation of three critical parameters: time-scale separation, space-scale separation and threshold, we can interchangeably control the formation of all the aforementioned wave types.

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.

Collisions of non-explosive dissipative solitons can induce explosions

We investigate the interaction of stationary and oscillatory dissipative solitons in the framework of two coupled cubic-quintic complex Ginzburg-Landau equation for counter-propagating waves. We analyze the case of a stabilizing as well as a destabilizing cubic cross-coupling between the counter-propagating dissipative solitons. The three types of interacting localized solutions investigated are stationary, oscillatory with one frequency, and oscillatory with two frequencies. We show that there is a large number of different outcomes as a result of these collisions including stationary as well as oscillatory bound states and compound states with one and two frequencies. The two most remarkable results are (a) the occurrence of bound states and compound states of exploding dissipative solitons as outcome of the collisions of stationary and oscillatory pulses; and (b) spatiotemporal disorder due to the creation, interaction, and annihilation of dissipative solitons for colliding oscillatory dissipative solitons as initial conditions.

Waves spontaneously generated by heterogeneity in oscillatory media

Wave propagation is an important characteristic for pattern formation and pattern dynamics. To date, various waves in homogeneous media have been investigated extensively and have been understood to a great extent. However, the wave behaviors in heterogeneous media have been studied and understood much less. In this work, we investigate waves that are spontaneously generated in one-dimensional heterogeneous oscillatory media governed by complex Ginzburg-Landau equations; the heterogeneity is modeled by multiple interacting homogeneous media with different system control parameters. Rich behaviors can be observed by varying the control parameters of the systems, whereas the behavior is incomparably simple in the homogeneous cases. These diverse behaviors can be fully understood and physically explained well based on three aspects: dispersion relation curves, driving-response relations, and wave competition rules in homogeneous systems. Possible applications of heterogeneity-generated waves are anticipated.

Non-unique results of collisions of quasi-one-dimensional dissipative solitons

We investigate collisions of quasi-one-dimensional dissipative solitons (DSs) for a large class of initial conditions, which are not temporally asymptotic quasi-one-dimensional DSs. For the case of sufficiently small approach velocity and sufficiently large values of the dissipative cross-coupling between the counter-propagating DSs, we find non-unique results for the outcome of collisions. We demonstrate that these non-unique results are intrinsically related to a modulation instability along the crest of the quasi-one-dimensional objects. As a model, we use coupled cubic-quintic complex Ginzburg-Landau equations. Among the final results found are stationary and oscillatory compound states as well as more complex assemblies consisting of quasi-one-dimensional and localized states. We analyse to what extent the final results can be described by the solutions of one cubic-quintic complex Ginzburg-Landau equation with effective parameters.

Actin and PIP3 waves in giant cells reveal the inherent length scale of an excited state

The membrane and actin cortex of a motile cell can autonomously differentiate into two states, one typical of the front, the other of the tail. On the substrate-attached surface of Dictyostelium discoideum cells, dynamic patterns of front-like and tail-like states are generated that are well suited to monitor transitions between these states. To image large-scale pattern dynamics independently of boundary effects, we produced giant cells by electric-pulse-induced cell fusion. In these cells, actin waves are coupled to the front and back of phosphatidylinositol (3,4,5)-trisphosphate (PIP3)-rich bands that have a finite width. These composite waves propagate across the plasma membrane of the giant cells with undiminished velocity. After any disturbance, the bands of PIP3 return to their intrinsic width. Upon collision, the waves locally annihilate each other and change direction; at the cell border they are either extinguished or reflected. Accordingly, expanding areas of progressing PIP3 synthesis become unstable beyond a critical radius, their center switching from a front-like to a tail-like state. Our data suggest that PIP3 patterns in normal-sized cells are segments of the self-organizing patterns that evolve in giant cells.

Class of compound dissipative solitons as a result of collisions in one and two spatial dimensions

We study the interaction of quasi-one-dimensional (quasi-1D) dissipative solitons (DSs). Starting with quasi-1D solutions of the cubic-quintic complex Ginzburg-Landau (CGL) equation in their temporally asymptotic state as the initial condition, we find, as a function of the approach velocity and the real part of the cubic interaction of the two counterpropagating envelopes: interpenetration, one compound state made of both envelopes or two compound states. For the latter class both envelopes show DSs superposed at two different locations. The stability of this class of compound states is traced back to the quasilinear growth rate associated with the coupled system. We show that this mechanism also works for 1D coupled cubic-quintic CGL equations. For quasi-1D states that are not in their asymptotic state before the collision, a breakup along the crest can be observed, leading to nonunique results after the collision of quasi-1D states.

Wave reflection in a reaction-diffusion system: breathing patterns and attenuation of the echo

Formation and interaction of the one-dimensional excitation waves in a reaction-diffusion system with the piecewise linear reaction functions of the Tonnelier-Gerstner type are studied. We show that there exists a parameter region where the established regime of wave propagation depends on initial conditions. Wave phenomena with a complex behavior are found: (i) the reflection of waves at a growing distance (the remote reflection) upon their collision with each other or with no-flux boundaries and (ii) the periodic transformation of waves with the jumping from one regime of wave propagation to another (the periodic trigger wave).

Exploding dissipative solitons in reaction-diffusion systems

We show that exploding dissipative solitons can arise in a reaction-diffusion system for a range of parameters. As a function of a vorticity parameter, we observe a sequence of transitions from oscillatory localized states via meandering dissipative solitons to exploding dissipative solitons propagating in one direction for long times followed by the reverse cascade back to oscillatory localized states. While exploding dissipative solitons are known from the cubic-quintic complex Ginzburg-Landau (CGL) equation, propagating exploding dissipative solitons appear to require for their existence a system of lower symmetry such as the reaction-diffusion model studied here.

Wave-processing of long-scale information by neuronal chains

Investigation of mechanisms of information handling in neural assemblies involved in computational and cognitive tasks is a challenging problem. Synergetic cooperation of neurons in time domain, through synchronization of firing of multiple spatially distant neurons, has been widely spread as the main paradigm. Complementary, the brain may also employ information coding and processing in spatial dimension. Then, the result of computation depends also on the spatial distribution of long-scale information. The latter bi-dimensional alternative is notably less explored in the literature. Here, we propose and theoretically illustrate a concept of spatiotemporal representation and processing of long-scale information in laminar neural structures. We argue that relevant information may be hidden in self-sustained traveling waves of neuronal activity and then their nonlinear interaction yields efficient wave-processing of spatiotemporal information. Using as a testbed a chain of FitzHugh-Nagumo neurons, we show that the wave-processing can be achieved by incorporating into the single-neuron dynamics an additional voltage-gated membrane current. This local mechanism provides a chain of such neurons with new emergent network properties. In particular, nonlinear waves as a carrier of long-scale information exhibit a variety of functionally different regimes of interaction: from complete or asymmetric annihilation to transparent crossing. Thus neuronal chains can work as computational units performing different operations over spatiotemporal information. Exploiting complexity resonance these composite units can discard stimuli of too high or too low frequencies, while selectively compress those in the natural frequency range. We also show how neuronal chains can contextually interpret raw wave information. The same stimulus can be processed differently or identically according to the context set by a periodic wave train injected at the opposite end of the chain.

Influence of Dirichlet boundary conditions on dissipative solitons in the cubic-quintic complex Ginzburg-Landau equation

We investigate the influence of Dirichlet boundary conditions on various types of localized solutions of the cubic-quintic complex Ginzburg-Landau equation as it arises as an envelope equation near the weakly inverted onset of traveling waves. We find that various types of nonmoving pulses and holes can accommodate Dirichlet boundary conditions by having, for holes, two halves of a pi hole at each end of the box. Moving pulses of fixed shape as they arise for periodic boundary conditions are replaced by a nonmoving asymmetric pulse, which has half a pi hole at the end of the box in the original moving direction to guarantee that Dirichlet boundary conditions are met. Moving breathing pulses as they arise for periodic boundary conditions propagate toward one end of the container and stop moving while the breathing persists indefinitely. Finally breathing and moving holes are replaced by two (nonbreathing) half pi holes at each end of the container and one hump in the bulk.

Noise induces partial annihilation of colliding dissipative solitons

Partial annihilation of two counterpropagating dissipative solitons, with only one pulse surviving the collision, has been widely observed in different experimental contexts, over a large range of parameters, from hydrodynamics to chemical reactions. However, a generic picture accounting for partial annihilation is missing. Based on our results for coupled complex cubic-quintic Ginzburg-Landau equations as well as for the FitzHugh-Nagumo equation we conjecture that noise induces partial annihilation of colliding dissipative solitons in many systems.

Anomalous pulse interaction in dissipative media

We review a number of phenomena occurring in one-dimensional excitable media due to modified decay behind propagating pulses. Those phenomena can be grouped in two categories depending on whether the wake of a solitary pulse is oscillatory or not. Oscillatory decay leads to nonannihilative head-on collision of pulses and oscillatory dispersion relation of periodic pulse trains. Stronger wake oscillations can even result in a bistable dispersion relation. Those effects are illustrated with the help of the Oregonator and FitzHugh-Nagumo models for excitable media. For a monotonic wake, we show that it is possible to induce bound states of solitary pulses and anomalous dispersion of periodic pulse trains by introducing nonlocal spatial coupling to the excitable medium.

Boundary-induced spatiotemporal complex patterns in excitable systems

We show that inhomogeneous boundary conditions (BCs) in a distributed reaction-diffusion excitable system are a natural source of permanent perturbations that can induce wave trains, which can be characterized as mixed-mode temporal oscillations and, when a parameter is varied, admit a period-adding bifurcation. To that end we analyze: a pair of coupled excitable and oscillatory cells, a distributed FitzHugh-Nagumo model, and a distributed five-variable model that describes catalytic oxidation. The obtained results account for the recently reported experimental observations of mixed-mode oscillations showing a period-adding bifurcation during oxidation on a disk-shaped catalytic cloth with imposed cold temperature BC.

Periodic spatiotemporal patterns in a two-dimensional two-variable reaction-diffusion model

Periodic spatiotemporal two-dimensional (2D) asymptotic patterns in an excitable two-variable thermochemical (reaction-diffusion) system are shown. In a one-dimensional system the traveling impulse which reflects from impermeable boundaries is a stable asymptotic solution if the diffusion coefficient of the reactant is greater than the thermal diffusivity of the system. Periodic patterns of two symmetries are presented in the 2D system: the impulse of excitation propagating along the diagonal of a square spatial domain and a structure consisting of curved impulses which propagate in the direction perpendicular to one side of a rectangular domain.

Stable stationary and breathing holes at the onset of a weakly inverted instability

We show numerically different stable localized structures including stationary holes, moving holes, breathing holes, stationary and moving pulses in the one-dimensional subcritical complex Ginzburg-Landau equation with periodic boundary conditions, and using two classes of initial conditions. The coexistence between different types of stable solutions is summarized in a phase diagram. Stable breathing moving holes as well as breathing nonmoving holes have not been described before for dissipative pattern-forming systems including reaction-diffusion systems.

Static, oscillating modulus, and moving pulses in the one-dimensional quintic complex Ginzburg-Landau equation: an analytical approach

By means of a matching approach we study analytically the appearance of static and oscillating-modulus pulses in the one-dimensional quintic complex Ginzburg-Landau equation without nonlinear gradient terms. When considering nonlinear gradient terms the method enables us to calculate the velocities of the stable and unstable moving pulses. We focus on this equation since it represents a prototype envelope equation associated with the onset of an oscillatory instability near a weakly inverted bifurcation. The results obtained using the analytic approximation scheme are in good agreement with direct numerical simulations. The method is also useful in studying other localized structures like holes.

“Black spots” in a surfactant-rich Belousov–Zhabotinsky reaction dispersed in a water-in-oil microemulsion system

The Belousov-Zhabotinsky (BZ) reaction dispersed in water-in-oil aerosol OT (AOT) microemulsion has been studied at small radius R(d) of water nanodroplets (R(d) (nm) congruent with0.17omega,omega = [H(2)O][AOT] = 9). Stationary spotlike and labyrinthine Turing patterns are found close to the fully oxidized state. These patterns, islands of high concentration of the reduced state of the Ru(bpy)(3) (2+) catalyst, can coexist either with "black" reduction waves or, under other conditions, with the "white" oxidation waves usually observed in the BZ reaction. The experimental observations are analyzed with the aid of a new Oregonator-like model and qualitatively reproduced in computer simulations.

Subcritical wave instability in reaction-diffusion systems

We report an example of subcritical wave instability in a model of a reaction-diffusion system and discuss the potential implications for localized patterns found in experiments on the Belousov-Zhabotinsky reaction in a microemulsion.

Coexistence of stable particle and hole solutions for fixed parameter values in a simple reaction diffusion system

We present a simple autocatalytic reaction-diffusion model for two variables, which shows for fixed parameter values the simultaneous stable coexistence of particle solutions as well of two types of hole solutions. The associated spatially homogeneous system is characterized by the coexistence of one stable fixed point and a stable limit cycle solution. We compare our results to other dissipative systems which have for fixed parameters either stable particle or stable hole solutions including the quintic complex Ginzburg-Landau equation and the envelope equation for optical bistability as well as other reaction-diffusion models.

Analytical approach to localized structures in a simple reaction-diffusion system

We study from an analytical point of view a simple reaction-diffusion model, which admits stable oscillating localized structures as a consequence of the coexistence between a stable limit cycle and a stable fixed point. Using a generalized matching approach we are able to find approximate analytical expressions for localized oscillating structures in this reaction-diffusion model capturing all the essential ingredients of these breathing particlelike solutions.

Quasisoliton interaction of pursuit-evasion waves in a predator-prey system

We consider a system of partial differential equations describing two spatially distributed populations in a "predator-prey" interaction with each other. The spatial evolution is governed by three processes: positive taxis of predators up the gradient of prey (pursuit), negative taxis of prey down the gradient of predators (evasion), and diffusion resulting from random motion of both species. We demonstrate a new type of propagating wave in this system. The mechanism of propagation of these waves essentially depends on the taxis and is entirely different from waves in a reaction-diffusion system. Unlike typical reaction-diffusion waves, which annihilate on collision, these "taxis" waves can often penetrate through each other and reflect from impermeable boundaries.

Pattern selection in a cooperative biochemical in vitro amplification system: the role of parasites

Previously, numerical simulations have shown that evolving systems can be stabilized against emerging parasites by pattern formation in spatially extended flow reactors. Hence, it can be argued that pattern formation is a prerequisite for any experimental investigation of the biochemical evolution of cooperative function. Here, we study a model of an experimental biochemical system for the cooperative in vitro amplification of DNA strands and show that emerging parasites can induce a complex pattern formation even when no pattern formation occurs without parasites. In an adiabatic approximation where the cooperative amplification reaction is assumed to adapt fast to slowly emerging parasites, the parasite concentration itself acts as a Steuer parameter for the selection of various complex patterns. Without such an adiabatic approximation only transient patterns emerge. As any species can grow for very low concentrations, the parasite is able to infect the entire reactor and the system is finally diluted out. In the experimental biochemical system, however, the species are individual molecules and the growth of spatially separated, non-infected regions becomes feasible. Hence a cutoff threshold for the minimal concentration is applied. In these simulations the otherwise lethal infection by parasites induces the formation of spatiotemporal spirals, and this spatial structure help the host and parasitoid species to survive together. These theoretical results describe an inherent property of cooperative reactions and have an important impact on experimental investigations on the molecular evolution and complex function in spatially extended reactors. Since the formation of the complex pattern is restricted either to a rather large cutoff value or a special choice of the kinetic parameters, we, however, conclude that the persistence of evolving cooperative amplification is not possible in a simple reaction-diffusion reactor. Experimental set-ups with patchy environments, e.g. biomolecular amplification in coupled microstructured flow chambers or in microemulsion, are eligible candidates for the observation of such a self-organized pattern selection.

Influence of spatiotemporally correlated noise on structure formation in excitable media

We discuss the influence of additive, spatiotemporally correlated (i.e., colored) noise on pattern formation in a two-dimensional network of excitable systems. The signature of spatiotemporal stochastic resonance (STSR) is analyzed using cross-correlation and information theoretic measures. It is found that the STSR behavior is affected by both the spatial and temporal correlations of the noise due to an interplay with the length scales of the deterministic network. Increasing the spatiotemporal noise correlation shifts the occurrence of STSR to smaller values of the noise variance. Additionally, if the spatial correlation of the noise exceeds that of the network, the excitation patterns disappear in favor of cloudy structures, directly rendering the underlying spatial noise field.

Reaction-diffusion dynamics in an oscillatory medium of finite size: pseudoreflection of waves

Wave propagation in an oscillatory reaction-diffusion, one-dimensional domain of finite size with Dirichlet boundary conditions is analyzed. For sizes below a certain threshold length, the medium cannot sustain wave motion. Above this threshold we find that for a relatively small domain extent, a strong correlation exists between the dynamics of the system and its size. This correlation gradually disappears with increasing domain size. For still larger sizes, we observe an effect of wave pseudo reflection near the boundary. It is shown both numerically and analytically that pseudoreflected waves are periodically generated inside the medium by a fast, self-generated "source" near the boundary.

Self-replication of a pulse in excitable reaction-diffusion systems

We investigate self-replication of a pulse in Bonhoffer-van der Pol type reaction-diffusion systems in one dimension. The interface dynamics of front and back of a pulse developed for a bistable system is extended to a monostable case, which is useful to clarify the mechanism of the self-replication. We shall show that the threshold parameter for excitability plays the central role for self-replication. The present theory can be applied not only to a symmetric pulse, but also to a propagating asymmetric pulse.

Analytical and numerical studies of noise-induced synchronization of chaotic systems

We study the effect that the injection of a common source of noise has on the trajectories of chaotic systems, addressing some contradictory results present in the literature. We present particular examples of one-dimensional maps and the Lorenz system, both in the chaotic region, and give numerical evidence showing that the addition of a common noise to different trajectories, which start from different initial conditions, leads eventually to their perfect synchronization. When synchronization occurs, the largest Lyapunov exponent becomes negative. For a simple map we are able to show this phenomenon analytically. Finally, we analyze the structural stability of the phenomenon. (c) 2001 American Institute of Physics.

Self-replicating pulses and sierpinski gaskets in excitable media

In our previous papers, we have shown by computer simulations that a Sierpinski gasket pattern appears in a Bonhoeffer-van der Pol type reaction-diffusion system. In this paper, we show another class of regular self-similar structure which is found in four different excitable reaction-diffusion systems. This result strongly implies that the existence of the self-similar spatiotemporal evolution is universal in excitable reaction-diffusion media.

Lifetime enhancement of scroll rings by spatiotemporal fluctuations

The dynamics of three-dimensional scroll rings with spatiotemporal random excitability is investigated numerically using the FitzHugh-Nagumo model. Depending on the correlation time and length scales of the fluctuations, the lifetime of the ring filament is enlarged and a resonance effect between the time scale of the scroll ring and the time correlation of the noise is observed. Numerical results are interpreted in terms of a simplified stochastic model derived from the kinematical equations for three-dimensional excitable waves.

Noise-induced phase separation: mean-field results

We present a study of a phase-separation process induced by the presence of spatially correlated multiplicative noise. We develop a mean-field approach suitable for conserved-order-parameter systems and use it to obtain the phase diagram of the model. Mean-field results are compared with numerical simulations of the complete model in two dimensions. Additionally, a comparison between the noise-driven dynamics of conserved and nonconserved systems is made at the level of the mean-field approximation.

Soliton-like regimes and excitation pulse reflection (echo) in homogeneous cardiac purkinje fibers: results of numerical simulations

On the basis of numerical simulations of the partial McAllister-Noble-Tsien equations quantitatively describing the dynamics of electrical processes in conductive cardiac Purkinje fibers we reveal unusual - soliton-like - regimes of interaction of nonlinear excitation pulses governing the heart contraction rhythm: reflection of colliding pulses instead of their annihilation. The phenomenological mechanism of the reflection effects is that in a narrow (but finite) range of the system parameters the traveling pulse presents a doublet consisting of a high-amplitude leader followed by a low-amplitude subthreshold wave. Upon collisions of pulses the leaders are annihilated, but subthreshold waves summarize becoming superthreshold and initiating two novel echo-pulses traveling in opposite directions. The phenomenon revealed presents an analogy to the effect of reflection of colliding nerve pulses, predicted recently, and can be of use in getting insight into the mechanisms of heart rhythm disturbances.