Spatiotemporal Stochastic Resonance in Excitable Media

Abstract

We present first numerical evidence that in an excitable medium the synchronization of spatiotemporal patterns with external excitatory waves shows a sharp peak at a finite, well-defined noise level independent of the system size. This effect can be understood as a generalization of the concept of stochastic resonance to spatially extended systems. We further show the impact of spatiotemporal stochastic resonance for the spreading of spiral waves, where the noise level controls the scale and size of the spiral. 05.40.+j, 47.54.+r Pattern formation far from equilibrium has been studied very extensively in the last years (for a recent review, see [1,2]). Representative examples are Rayleigh-Benard convection rolls, Taylor-Couette flow, and spiral waves in the Belousov-Zhabotinsky reaction. Typically a pattern starts to build up when the control parameter (the temperature difference in case of the Rayleigh-Benard system) becomes larger than a critical value. Noise makes the bifurcation smooth by triggering the onset of the pattern even below threshold r ( r, . The role of fluctuations for the onset and selection of patterns has been studied in some detail and is reported on in a number of articles in [3,4] and [5]. In this paper, we discuss the role of noise for the formation of patterns in two-dimensional excitable media from a different perspective.It has been shown that a certain amount of noise can amplify temporal patterns by increasing the system's sensitivity via stochastic resonance [6] (for recent reviews, see [7,8]). This effect has been shown first for symmetric bistable systems. The external forcing (the temporal pattern) tilts the bistable potential weakly back and force (weak enough that the potential remains bistable), thereby modulating the barrier height for noise-induced hopping between the stable domains. The synchronization of the hopping with the external forcing shows a bell-shaped curve as a function of the noise strength -the fingerprint of stochastic resonance. Only recently, stochastic resonance has been demonstrated in much simpler systems, namely in threshold devices [9 -11]. Here, Gaussian noise and a periodic signal is applied to a threshold device which responds with a spike if the sum of the noise and the signal is crossing the threshold from below. The intensity of the peak at the signal frequency in the power spectrum of the outgoing spike train shows a bell-shaped curve as a function of the variance of the noise. At the maximum, the variance of the noise matches half the square of the threshold -a result which we will make heavy use of in this paper. The basic question we study in this paper is in how far stochastic resonance can also be observed in spatially extended pattern forming systems. As a working model, we use a two-dimensional equidistant square array (lattice constant a) of N X N noisy threshold devices [9] d;, : x;, (t)~s;, (t) (i = 1, ... , N and j = 1, ... , N). The operation of a threshold device is defined as follows: If the input x;, (t) is below the threshold...

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