Homogenization and flame propagation in periodic excitable media: The asymptotic speed of propagation

We study the effect of homogenization on flame propagation in periodic excitable media when the width of the flame is much smaller than the characteristic size of the heterogeneities. © 2005 Wiley Periodicals, Inc.     

Models of unidirectional propagation in heterogeneous excitable media

Two differential equation models of excitable media (threshold and recovery kinetics) with solutions that exhibit unidirectional propagation are presented. It is shown that unidirectional propagation in heterogeneous excitable media with non-oscillatory kinetics can be initiated from homogeneous initial data. Simulations on a reaction-diffusion model with FitzHugh-Nagumo kinetics and spatially heterogeneous parameters yields a rotating wave on a one-dimensional circular spatial domain. An ordinary differential equation model with four semi-coupled excitable cells and heterogeneous parameters is analyzed to determine a critical parameter region over which unidirectional propagation may occur.     

Front propagation for discrete periodic monostable equations

This paper deals with front propagation for discrete periodic monostable equations. We show that there is a minimal wave speed such that a pulsating traveling front solution exists if and only if the wave speed is above this minimal speed. Moreover, in comparing with the continuous case, we prove the convergence of discretized minimal wave speeds to the continuous minimal wave speed.     

Spiral waves in excitable media due to noise and periodic forcing

We investigate the jointly driven effects of external periodic forcing and Gaussian white noise on meandering spiral waves in excitable media with FitzHugh-Nagumo local dynamics. Interesting phenomena resulted from various forcing periods are found, for example, piece-wise line drift, intermittent straight-line drift and so on. We also observe new type of breakup of spiral wave between entrainment bands with 1:1 and 2:1. It is believed that the occurrence of the new type is relevant to the appearance of local bidirectional propagation window. There exist optimized noise intensities which can induce the broadest entrainments and Arnold tongues. Such a phenomenon is referred to as stochastic resonance. It is also observed that the noise makes significant effects on the spiral wave with straight-line drift. Via the tip Fourier spectrum, the varying of tip motion with external periods on the resonance band is interpreted.     

Computational biology of propagation in excitable media models of cardiac tissue

Biophysically detailed models of the electrical activity of single cardiac cells are modular, stiff, high order, differential systems that are continually being updated by incorporating new formulations for ionic fluxes, binding and sequestration. They are validated by their representation of the ionic flux and concentration data they summarise, and by their ability to reproduce cell action potentials, their stability to perturbations, structural stability and robustness. They can be used to construct discrete or continuous, one- (1D), two- (2D) or three-dimensional (3D) virtual cardiac tissues, with heterogeneities, anisotropy and realistic cardiac geometry. These virtual cardiac tissues are being applied to understand the propagation of excitation in the heart, provide insights into the generation and nature of arrhythmias, aid the interpretation of electrical signs of arrhythmia, to develop defibrillation and antiarrhythmic strategies, and to prescreen potential antiarrhythmic agents.     

Pulsating type entire solutions of monostable reaction-advection-diffusion equations in periodic excitable media

We establish the existence of pulsating type entire solutions of reaction-advection-diffusion equations with monostable nonlinearities in a periodic framework. Here the nonlinearities include the classic KPP case. The pulsating type entire solutions are defined in the whole space and for all time t∈R. By studying a pulsating traveling front connecting a constant unstable stationary state to a stable stationary state which is allowed to be a positive function, we proved that there exist pulsating type entire solutions behaving as two pulsating traveling fronts coming from both directions, and approaching each other. The key techniques are to characterize the asymptotic behavior of the solutions as t→−∞ in terms of appropriate subsolutions and supersolutions.     

Traveling waves in excitable media

We consider a two-component reaction-diffusion model that describes the oxygenation of CO molecules on the surface of platinum in the one-dimensional case. The partial differential equations of the model are reduced to a system of ordinary differential equations. We show that the system of partial differential equations with fixed parameter values has a family of autowave solutions running along the spatial axis at various velocities. These solutions are described by some singular attractors and limit cycles of the corresponding period in the system of ordinary differential equations.     

Controlling nonlinear waves in excitable media

A new feedback control method is proposed to control the spatio-temporal dynamics in excitable media. Applying suitable external forcing to the system’s slow variable, successful suppression and control of propagating pulses as well as spiral waves can be obtained. The proposed controller is composed by an observer to infer uncertain terms such as diffusive transport and kinetic rates, and an inverse-dynamics feedback function. Numerical simulations shown the effectiveness of the proposed feedback control approach.     

Asymptotic spreading in heterogeneous diffusive excitable media

We establish propagation and spreading properties for nonnegative solutions of nonhomogeneous reaction-diffusion equations of the type:

t_∂u−∇⋅(A(t,x)∇u)+q(t,x)⋅∇u=f(t,x,u)     

Resonances in periodic media

We study the propagation of linear acoustic waves (a) in an infinite string with a periodic material distribution, (b) in an infinite cylinder with a meterial distribution that is periodic in the longitudinal direction and does not depend on the transverse coordinates. We assume that the wave field is generated by a time-harmonic force distribution of frequency ω acting in a compact set. We show in both cases that resonances of order t^1/2 occur for a discrete set of frequencies and that the solution is bounded as t→∞ for the remaining frequencies. In case (a) ω is a resonance frequency if and only if ω² is a boundary point of one of the spectral bands of the corresponding spatial differential operator of Hill's type. A similar characterization of the resonance frequencies is given in case (b).     

Existence of pulses in excitable media with nonlocal coupling

We prove the existence of fast traveling pulse solutions in excitable media with non-local coupling. Existence results had been known, until now, in the case of local, diffusive coupling and in the case of a discrete medium, with finite-range, non-local coupling. Our approach replaces methods from geometric singular perturbation theory, that had been crucial in previous existence proofs, by a PDE oriented approach, relying on exponential weights, Fredholm theory, and commutator estimates.     

Planelike minimizers in periodic media

We show that given an elliptic integrand ?? in ?^d that is periodic under integer translations, and given any plane in ?^d, there is at least one minimizer of ?? that remains at a bounded distance from this plane. This distance can be bounded uniformly on the planes. We also show that, when folded back to ?^d/?^d, the minimizers we construct give rise to a lamination. One particular case of these results is minimal surfaces for metrics invariant under integer translations. The same results hold for other functionals that involve volume terms (small and average zero). In such a case the minimizers satisfy the prescribed mean curvature equation. A further generalization allows the formulation and proof of similar results in manifolds other than the torus provided that their fundamental group and universal cover satisfy some hypotheses. © 2001 John Wiley & Sons, Inc.     

Numerical simulation of stochastic PDEs for excitable media

We discuss the numerical solution of a number of stochastic perturbations of the Barkley model of excitable media, widely used in the study of spiral waves. Two numerical methods are considered for solving this equation, one based on Barkley's original formulation and one based on spectral methods. It is found to be beneficial to modify the nonlinearity describing the reaction kinetics. An efficient method of approximating the Wiener process is presented. The effectiveness of the methods depends on the stochastic PDE under consideration.     

Tile patterns in excitable media subject to non-solenoidal flow fields

The propagation of spiral waves in excitable media subject to a non-solenoidal advective field which satisfies the no-penetration condition on the boundaries of the domain is studied numerically, and it is shown that, depending on the amplitude and spatial frequencies of the velocity field, the spiral wave may be distorted highly, break up into a number of smaller spiral waves, or exhibit polygonal shapes or tile patterns. These patterns reflect the symmetry/asymmetry of the velocity field and are characterized by thick regions of high concentration at stagnation points where the velocity gradient is largest, and thin ones which are parallel to the velocity vector. It is also shown that the advective field distorts the spiral wave by decreasing its thickness where the velocity is largest due to the stretching of the wave, and by increasing it at the stagnation points where the curvature of the wave is largest.     

Traveling waves,impulses and diffusion chaos in excitable media

In the present work it is shown, that the FitzHugh–Nagumo type system of partial differential equations with fixed parameters can have an infinite number of different stable wave solutions, traveling along the space axis with arbitrary speeds, and also traveling impulses and an infinite number of different states of spatiotemporal (diffusion) chaos. Those solutions are generated by cascades of bifurcations of cycles and singular attractors according to the FSM theory (Feigenbaum–Sharkovskii–Magnitskii) in the three-dimensional system of ordinary differential equations (ODEs), to which the FitzHugh–Nagumo type system of equations with self-similar change of variables can be reduced.     

Wave propagation in nonlinear media

The behavior of nonlinear dispersive or dissipative waves is analyzed using the decomposition method.     

Lie symmetry analysis for classification on pattern in excitable media

The dynamics of wave front in two dimensional excitable media is described by the derivative Burgers’ equation. In this paper, we will carry out Lie symmetry analysis to the equation for constructing particular solutions associated with chemical patterns. The form of the variable coefficient in the reduced equation by using symmetries classifies the invariant solutions into three cases and the solutions include arc, circle, knee, spiral and double scroll patterns.