Speed of traveling fronts in a sigmoidal reaction-diffusion system.

We study a sigmoidal version of the FitzHugh-Nagumo reaction-diffusion system based on an analytic description using piecewise linear approximations of the reaction kinetics. We completely describe the dynamics of wave fronts and discuss the properties of the speed equation. The speed diagrams show front bifurcations between branches with one, three, or five fronts that differ significantly from the classical FitzHugh-Nagumo model. We examine how the number of fronts and their speed vary with the model parameters. We also investigate numerically the stability of the front solutions in a case when five fronts exist.

[Traveling waves in a piecewise-linear reaction-diffusion model of excitable medium].

One-dimensional autowaves (traveling waves) in excitable medium described by a piecewise-linear reaction-diffusion system have been investigated. Two main types of waves have been considered: a single impulse and a periodic sequence of impulses (wave trains). In a two-component system, oscillations appear due to the presence of the second component in the reaction-diffusion system. In a one-component system, oscillations appear owing to external periodic excitation (forcing). Using semianalytical solutions for the wave profile, the shape and velocity of autowaves have been found. It is shown that the dispersion relation for oscillating sequences of impulses has an anomalous character.

Wavy fronts and speed bifurcation in excitable systems with cross diffusion.

A bifurcation of excitation fronts induced by cross diffusion in two-component bistable reaction-diffusion systems of activator-inhibitor type is discovered. This bifurcation is similar to the nonequilibrium Ising-Bloch bifurcation. A different type of fronts, whose spatial profiles are characterized by oscillating tails, are associated with this bifurcation. These fronts are described using exact analytical solutions of piecewise linear reaction-diffusion equations.

Wavy front dynamics in a three-component reaction-diffusion system with one activator and two inhibitors.

An analytic description for traveling waves in a one-dimensional reaction-diffusion system with one activator and two inhibitors and with equal diffusion constants is developed using a piecewise linear approximation for the nonlinear activator reaction term. The case of front waves is examined in more detail, the monotonic and oscillating fronts being separately considered. The corresponding wave profiles are constructed, and the speed equation is obtained and discussed. It is found that the fronts in the three-component model propagate faster than the fronts in the two-component system. The front interaction is studied using numerical calculations. The results show that at head-on collisions two oscillating fronts produce a wavy domain, which spreads in space with time.

Front bifurcation in a tristable reaction-diffusion system under periodic forcing.

A piecewise linear tristable reaction-diffusion equation under external forcing of periodic type is considered. A special feature of the forcing is that the force moves together with the traveling wave. Front velocity equations are obtained analytically using matching procedures for the front solutions. It is noted that there is a restriction in building of null-cline. For each choice of outer branches of null-cline the middle interfacial zone should not exceed some critical value. When this zone is larger the front does not exist. It is found that in the presence of forcing there exists a set of front solutions with different phases (matching point coordinates). The periodic forcing produces a change in the velocity-versus-phase diagram. For a specific choice of wave number, there is a bubble formation which corresponds to additional solutions when the velocity bifurcates to form three fronts.

Stabilization of reaction-diffusion pulses by external forcing.

A pulse velocity equation under forcing is derived and the conditions for stationary waves (pinning conditions) are considered. It is found that there are two types of stationary pulses with symmetric and asymmetric parameter sets in the phase-amplitude diagram. The pulses with a symmetric set are always unstable, whereas the pulses with an asymmetric set may be stable. The stability criteria are presented.

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).

Oscillatory pulses in FitzHugh-Nagumo type systems with cross-diffusion.

We study FitzHugh-Nagumo type reaction-diffusion systems with linear cross-diffusion terms. Based on an analytical description using piecewise linear approximations of the reaction functions, we completely describe the occurrence and properties of wavy pulses, patterns of relevance in several biological contexts, in two prototypical systems. The pulse wave profiles arising in this treatment contain oscillatory tails similar to those in travelling fronts. We find a fundamental, intrinsic feature of pulse dynamics in cross-diffusive systems--the appearance of pulses in the bistable regime when two fixed points exist.

Wave propagation in a FitzHugh-Nagumo-type model with modified excitability.

We examine a generalized FitzHugh-Nagumo (FHN) type model with modified excitability derived from the diffusive Morris-Lecar equations for neuronal activity. We obtain exact analytic solutions in the form of traveling waves using a piecewise linear approximation for the activator and inhibitor reaction terms. We study the existence and stability of waves and find that the inhibitor species exhibits different types of wave forms (fronts and pulses), while the activator wave maintains the usual kink (front) shape. The nonequilibrium Ising-Bloch bifurcation for the wave speed that occurs in the FHN model, where the control parameter is the ratio of inhibitor to activator time scales, persists when the strength of the inhibitor nonlinearity is taken as the bifurcation parameter.