Two-variable Reaction-diffusion Research Articles

Reordering and synchronization of electrical turbulence in cardiac tissue through global and partial optogenetical illumination.

Electrical turbulence in the heart is considered the culprit of cardiac disease, including the fatal ventricular fibrillation. Optogenetics is an emerging technology that has the capability to produce action potentials of cardiomyocytes to affect the electric wave propagation in cardiac tissue, thereby possessing the potential to control the turbulence, by shining a rotating spiral pattern onto the tissue. In this paper, we present a method to reorder and synchronize electrical turbulence through optogenetics. A generic two-variable reaction-diffusion model and a simplified three-variable ionic cardiac model are used. We discuss cases involving either global or partial illumination.

Behavior of Spiral Wave Spectra with a Rank-Deficient Diffusion Matrix

Spiral waves emerge in numerous pattern forming systems and are commonly modeled with reaction-diffusion systems. Some systems used to model biological processes, such as ion-channel models, fall under the reaction-diffusion category and often have one or more non-diffusing species which results in a rank-deficient diffusion matrix. Previous theoretical research focused on spiral spectra for strictly positive diffusion matrices. In this paper, we use a general two-variable reaction-diffusion system to compare the essential and absolute spectra of spiral waves for strictly positive and rank-deficient diffusion matrices. We show that the essential spectrum is not continuous in the limit of vanishing diffusion in one component. Moreover, we predict locations for the absolute spectrum in the case of a non-diffusing slow variable. Predictions are confirmed numerically for the Barkley and Karma models.

Analysis and pattern formation scenarios in the superdiffusive system of predation described with Caputo operator

The concept of a fractional derivative is introduced to the predator-prey system to describe the species anomalous superdiffusive process. To achieve this, a new class of predator-prey model with the Beddington-DeAngelis functional response is formulated in the sense of the Caputo fractional order operator. This work aims to give a mathematical basis for computational studies of a two-variable fractional reaction-diffusion system in one and two dimensions from biological and numerical perspectives. As a result, some details of the local and global dynamics of the reaction-diffusion system are provided by using the idea of the linear stability analysis and well-known dynamical systems theory to derive conditions on the parameters which can guarantee biologically meaningful equilibria also serve as a guide in ensuring the correct choice of parameters when numerically experimenting with the solutions of the full fractional reaction-diffusion model. The behavior of the new dynamical system is examined for both diffusive and non-diffusive systems at different instances of fractional order.

An analytical velocity field of spiral tips in reaction–diffusion systems

Spiral waves are ubiquitous in diverse physical, chemical, and biological systems. The tip (phase singularity) of a spiral wave is considered to represent its organizing center. Here, we derive an analytical velocity field of spiral tips based on the variables of a general two-variable reaction–diffusion (RD) equation. From this velocity field, we can predict the velocities of spiral tips at time t as long as the values of the variables are given at that time. Numerical simulations with two-variable RD models are in quantitative agreement with the analytical results. Furthermore, we also demonstrate the velocity field of spiral tips in the Luo–Rudy model for cardiac excitation.

Diffusive instabilities in hyperbolic reaction-diffusion equations.

We investigate two-variable reaction-diffusion systems of the hyperbolic type. A linear stability analysis is performed, and the conditions for diffusion-driven instabilities are derived. Two basic types of eigenvalues, real and complex, are described. Dispersion curves for both types of eigenvalues are plotted and their behavior is analyzed. The real case is related to the Turing instability, and the complex one corresponds to the wave instability. We emphasize the interesting feature that the wave instability in the hyperbolic equations occurs in two-variable systems, whereas in the parabolic case one needs three reaction-diffusion equations.

Amplitude equations for reaction-diffusion systems with cross diffusion

Using Taylor series expansion, multiscaling, and further expansion in powers of a small parameter, we develop general amplitude equations for two-variable reaction-diffusion systems with cross-diffusion terms in the cases of Hopf and Turing instabilities. We apply this analysis to the Oregonator and Brusselator models and find that inhibitor cross diffusion induced by the activator and activator cross diffusion induced by the inhibitor have opposite effects in the two models as a result of the different structure of their community matrices. Our analysis facilitates finding regions of supercritical and subcritical bifurcations, as well as wave and antiwave domains and domains of turbulent waves in the case of Hopf instability.

Numerical simulation of reaction–diffusion equations on spherical domains

The two-variable reaction diffusion equations on the spherical domain is considered and simulated, using the semi-implicit Euler finite difference method. It is shown that the method keeps the kinetics from overshooting the stable branches when a large time step is used in the simulation.

Pacemakers in a Reaction-Diffusion Mechanics System

Non-linear waves of excitation are found in various biological, physical and chemical systems and are often accompanied by deformations of the medium. In this paper, we numerically study wave propagation in a deforming excitable medium using a twovariable reaction-diffusion system coupled with equations of continuum mechanics. We study the appearance and dynamics of different excitation patterns organized by pacemakers that occur in the medium as a result of deformation. We also study the interaction of several pacemakers with each other and the characteristics of pacemakers in the presence of heterogeneities in the medium. We found that mechanical deformation not only induces pacemakers, but also has a pronounced effect on spatial organization of various excitation patterns. We show how these effects are modulated by the size of the medium, the location of the initial stimulus, and the properties of the reactiondiffusion-mechanics feedback.

Oscillatory Turing Patterns in a Simple Reaction-Diffusion System

Turing suggested that, under certain conditions, chemicals can react and diffuse in such a way as to produce steady-state inhomogeneous spatial patterns of chemical concentrations. We consider a simple two-variable reaction-diffusion system and find there is a spatio-temporally oscillating solution (STOS) in parameter regions where linear analysis predicts a pure Turing instability and no Hopf instability. We compute the boundary of the STOS and spatially non-uniform solution (SSNS) regions and investigate what features control its behavior.

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.

Nonlinear fingering dynamics of reaction-diffusion acidity fronts: Self-similar scaling and influence of differential diffusion

Nonlinear interactions between chemical reactions and buoyancy-driven Rayleigh-Taylor instability of reaction-diffusion acidity fronts of the chlorite-tetrathionate (CT) reaction are studied theoretically in a vertical Hele-Shaw cell or a porous medium. To do so, we perform a numerical integration of a two-variable reaction-diffusion model of the CT system coupled through an advection term to Darcy's law ruling the evolution of the velocity field of the fluid. The fingering dynamics of these chemical fronts is characterized by the appearance of several fingers at onset. These fingers then undergo coarsening and eventually merge to form one single symmetric finger. We study this asymptotic dynamics as a function of the three dimensionless parameters of the problem, i.e., the Damkohler number Da, the diffusivity ratio delta of the two chemical species, and the Rayleigh number Ra constructed here on the basis of the width L(y) of the system. For moderate values of Ra, the asymptotic single finger is shown to have self-similar scaling properties while above a given value of Ra, which depends on the other values of the parameters, tip splitting comes into play. Increasing the difference of diffusivities of the two chemical species (i.e., increasing delta) leads to more efficient coarsening and smaller asymptotic fingers. Experimental procedures to verify our predictions are proposed.

Critical properties of autowaves propagating on deformed cylindrical surfaces

Propagation of ring-shaped autowaves on deformed cylindrical surfaces is studied theoretically within a kinematical model. Theoretical results predict the existence of a critical deformation of the surface above which propagation of ring-shaped fronts is impossible and the structure of stable autowaves becomes similar to the so-called V-patterns with free ends. This prediction is confirmed by numerical simulations of a two-variable reaction–diffusion excitable medium on a torus.

Scroll waves in isotropic excitable media: linear instabilities, bifurcations, and restabilized states.

Scroll waves are three-dimensional analogs of spiral waves. The linear stability spectrum of untwisted and twisted scroll waves is computed for a two-variable reaction-diffusion model of an excitable medium. Different bands of modes are seen to be unstable in different regions of parameter space. The corresponding bifurcations and bifurcated states are characterized by performing direct numerical simulations. In addition, computations of the adjoint linear stability operator eigenmodes are also performed and serve to obtain a number of matrix elements characterizing the long-wavelength deformations of scroll waves.

Analysis of front interaction and control in stationary patterns of reaction-diffusion systems.

We have analyzed the stability of one-dimensional patterns in one- or two-variable reaction-diffusion systems, by analyzing the interaction between adjacent fronts and between fronts and the boundaries in bounded systems. We have used model reduction to a presentation that follows the front positions while using approximate expressions for front velocities, in order to study various control modes in such systems. These results were corroborated by a few numerical experiments. A stationary single front or a pattern with n fronts is typically unstable due to the interaction between fronts. The two simplest control modes, global control and point-sensor control (pinning), will arrest a front in a single-variable problem since both control modes, in fact, respond to the front position. In a two-variable system incorporating a localized inhibitor, in the domain of bistable kinetics, global control may stabilize a single front only in short systems while point-sensor control can arrest such a front in any system size. Neither of these control modes can stabilize an n-front pattern, in either one- or two-variable systems, and that task calls for a distributed actuator. A single space-dependent actuator that is spatially qualitatively similar to the patterned setpoint, and which responds to the sum of deviations in front positions, may stabilize a pattern that approximates the desired state. The deviation between the two may be sufficiently small to render the obtained state satisfactory. An extension of these results to diffusion-convection-reaction systems are outlined.

Coexistence of large amplitude stationary structures in a model of reaction-diffusion system.

The two-variable reaction-diffusion model of a chemical system describing the spatiotemporal evolution to large amplitude stationary periodical structures in a one-dimensional open, continuous-flow, unstirred reactor is investigated. Numerical solutions show that the structures are generated by divisions of the traveling impulse and its stopping at the boundary of the system. Analyses of projections of numerical solutions on the phase plane of two variables elaborated in the present paper allow qualitative explanation of the results. The coexistence of the large amplitude stationary periodical structures is shown. A number of coexisting structures grows strongly with increasing length of the reactor and may be as large as one wishes. The relationship of these results to biological systems is stressed.

Electric field induced propagating structures in a model of spatio-temporal signalling

The conditions are established for the onset of convective instabilities in a general two-variable reaction–diffusion–advection system in which one of the reactant species is immobile. These results are applied in detail to predict the effects of applying electric fields in a model for cellular calcium wave propagation based on the calcium-induced calcium-release (CICR) mechanism. Extensive numerical simulations of this model are described and these show that spatial structures can be generated entirely through applying electric fields at the parameter values determined by the general theory. These structures take the form of propagating wave packets, and can be either relatively simple or have complex dynamics, depending on the kinetic parameters.

Noise Induced Pattern Transition and Spatiotemporal Stochastic Resonance

The influence of parametric noise on the spatiotemporal dynamics of a two-variable reaction-diffusion model, describing pattern formation in excitable media, is investigated. When a control parameter is perturbed by stochastic force with suitable intensity, noise induced pattern transition between ``single'' spiral wave and ``double'' spiral waves, which does not occur in the absence of noise, is observed. If this parameter also varies periodically in time around the bifurcation point, we find that the output signal-to-noise ratios show maxima with the variation of noise intensity, indicating the occurrence of spatiotemporal stochastic resonance.

Finding the optimal path with the aid of chemical wave

It is shown that the optimal path in a two-dimensional vector field is deduced by the use of chemical wave in the Belousov-Zhabotinsky reaction (BZ reaction). We also reproduced our experimental result in a numerical simulation based on a two-variable reaction-diffusion equation. The present result provides a simple model for the future application of excitable media to parallel computing, i.e., an excitable medium serves as a self-organized parallel processor.

Link between the effect of an electric field on wave propagation and the curvature-velocity relation

The link between the effect of an electric field in a 1D medium and the curvature-velocity relation is studied theoretically by means of a two-variable reaction-diffusion system and numerically in the framework of a two-variable Oregonator model. Theoretical results predict the existence of a relationship between critical electric field ( E cr) and critical curvature ( K cr), which is highlighted by means of the so-called V-shaped patterns.

Generic spatiotemporal dynamics near codimension-two Turing-Hopf bifurcations

The coupling of spatial and temporal symmetry breaking instabilities is studied in a two-variable reaction-diffusion model describing semiconductor transport. A variety of spatiotemporal patterns corresponding to pure Hopf and Turing modes, localized patterns, and mixed Turing-Hopf modes including subharmonic spatiotemporal spiking are found. By organizing the results in a time-scale versus space-scale diagram, and by comparing them with a chemical reaction-diffusion model, it is shown that such behavior is generic for a class of extended nonlinear dynamic systems near codimension-two Turing-Hopf bifurcations.