Coherence Resonance in a Noise-Driven Excitable System

Abstract

We study the dynamics of the excitable Fitz Hugh ‐ Nagumo system under external noisy driving. Noise activates the system producing a sequence of pulses. The coherence of these noise-induced oscillations is shown to be maximal for a certain noise amplitude. This new effect of coherence resonance is explained by different noise dependencies of the activation and the excursion times. A simple one-dimensional model based on the Langevin dynamics is proposed for the quantitative description of this phenomenon. [S0031-9007(97)02349-1] The response of dynamical systems to noise has attracted large attention recently. There are many examples demonstrating that noise can lead to more order in the dynamics. To be mentioned here are the effects of noiseinduced order in chaotic dynamics [1], synchronization by external noise [2], and stochastic resonance [3‐5]. Also, noise has been shown to play a stabilizing role in ensembles of coupled oscillators and maps [6]. Especially interesting is the phenomenon of stochastic resonance, which appears when a nonlinear system is simultaneously driven by noise and a periodic signal. At a certain noise amplitude the periodic response is maximal; this has been confirmed by numerous experimental studies (cf. [7,8]). In this paper we study the effect of noise on the autonomous excitable oscillator—the famous Fitz Hugh ‐ Nagumo system. We demonstrate that a characteristic correlation time of the noise-excited oscillations has a maximum for a certain noise amplitude, and present a theory of this effect. Contrary to the usual setup of stochastic resonance, no external periodic driving is assumed, so the coherence appears as a nonlinear response to purely noisy excitation. The phenomenon considered is also different from stochastic resonance without periodic force reported recently in Ref. [9], where the effect of noise on a limit cycle at a bifurcation point was studied. The Fitz Hugh‐Nagumo model is a simple but representative example of excitable systems that occur in different fields of application ranging from kinetics of chemical reactions and solid-state physics to biological processes [10]. Originally it was suggested for the description of nerve pulses [11]; it was also widely used for modeling of spiral waves in a two-dimensional excitable medium. Different aspects of the dynamics of this and similar excitable models in the presence of noise have been discussed in Refs. [12‐16]. The equations of motion are