Persistency of noise-induced spatial periodicity in excitable media

– We study effects of spatiotemporal additive noise in conjunction with subthreshold travelling waves on the spatial dynamics of excitable media. We show that solely additive noise is able to extract an inherent spatial periodicity of the media in a resonant manner, thus marking the existence of spatial coherence resonance in the studied system. Next, in addition to noise, we introduce to the media excitatory waves to investigate the possibility of spatial stochastic resonance. We find that the solely noise-induced inherent spatial periodicity of the media cannot be altered by the spatial frequency of the waves. This so-called persistency of inherent spatial periodicity is attributed to the noise-robust excursion time that is characteristic for the local excitable dynamics. In temporal systems, stochastic resonance [1] stands for the resonant noisy enhancement of a system’s response to external subthreshold periodic stimuli. Fascinatingly, constructive effects of noise on the temporal domain of dynamical systems can also be observed in the absence of any deterministic external inputs in systems with no explicit time scales [2, 3]. This striking phenomenon, on the other hand, has been termed coherence resonance [4]. In systems with spatial degrees of freedom the spatiotemporal stochastic resonance (STSR) has been first reported in [5] for excitable systems, while spatial coherence resonance (SCR) has been introduced in [6] for systems near pattern-forming instabilities. Moreover, there exist studies reporting noise-induced spiral growth and enhancement of spatiotemporal order [7– 12], noise-sustained coherence of space-time clusters and self-organized criticality [13], noiseenhanced and -induced excitability [14,15], noise-induced propagation of harmonic signals [16], as well as noise-sustained and -controlled synchronization [17] in space extended systems. Until now, however, little attention has been devoted to the explicit analysis of characteristic spatial frequencies of nonlinear media. Carrillo et al. [6] where the first to report that spatiotemporal noisy perturbations are able to resonantly evoke noisy precursors of a nearby supercritical pattern-forming Turing bifurcation with a characteristic spatial frequency. Since no additional deterministic inputs were introduced to the media, their study thus provides first evidence for SCR in systems near pattern-forming instabilities. c © EDP Sciences Article published by EDP Sciences and available at http://www.edpsciences.org/epl or http://dx.doi.org/10.1209/epl/i2005-10298-4 M. Perc: Persistency of spatial periodicity 713 The present study is aimed at extending the results published by Carrillo et al. [6] to excitable media, as well as reporting a novel phenomenon termed persistency of noise-induced spatial periodicity, which occurs when besides noise, also subthreshold travelling waves with a given spatial frequency are introduced to the system. First, we show that there exists an optimal intensity of additive spatiotemporal noise for which an inherent spatial frequency of the system is best pronounced, thus providing evidences for SCR in excitable media. Next, in addition to noise, subthreshold travelling waves with a given spatial frequency and propagation speed are introduced to the system to investigate the possibility of evidencing the direct spatial counterpart of classical temporal stochastic resonance, i.e. the so-called spatial stochastic resonance (SSR). Importantly, note that SSR is a different phenomenon from STSR reported in [5]. In particular, SSR would correspond to the resonant noisy enhancement of a system’s response to the excitatory waves, whereby the noise-induced spatial dynamics would be characterized by a spatial frequency that matches the spatial frequency of the waves. We find, however, that the solely noise-induced spatial frequency, marking SCR in the media, dominates the spatial dynamics even in the presence of excitatory waves, and thus SSR as defined above cannot be obtained. This so-called persistency of inherent spatial periodicity as well as SCR are explained by considering the different noise dependencies of the activation and excursion times that are characteristic for the local dynamics of excitable space elements. The excitable media under study is locally modelled by the FitzHugh-Nagumo equations [18,19]