arXiv:cond-mat/9801119v1 13 Jan 1998 STOCHASTIC RESONANCE IN NONPOTENTIAL SYSTEMS T. Alarc´ on, A. P´ erez-Madrid, J.M. Rub´ ı Departament de F´ ısica Fonamental Facultat de F´ ısica Universitat de Barcelona Diagonal 647, 08028 Barcelona, Spain We propose a method to analytically show the possibility for the appearance of a maximum in the signal-to-noise ratio in nonpotential systems. We apply our results to the FitzHugh-Nagumo model under a periodic external forcing, showing that the model exhibits stochastic resonance. The procedure that we follow is based on the reduction to a one-dimensional dynamics in the adiabatic limit, and in the topology of the phase space of the systems under study. Its application to other nonpotential systems is also discussed. Pacs numbers: 05.40.+j, 87.10.+e I. INTRODUCTION Stochastic resonance ( SR ) [1]- [8] is a phenomenon in which an enhancement of the response of a non-linear system is observed when this system is yielded to an external forcing at some optimized nonzero noise level. Since the original proposition of SR as a possible explanation for periodic recurrences in the global climate dynamics, SR has become the object of copious theoretical and experimental research in a wide variety of systems in physics, biology and chemistry. In all these works the possibility of noise having beneficial effects in the dynamics of nonlinear systems has been pointed out. The original formulation of the problem, in terms of a bistable system and a periodic forcing has been extended to systems under the action of aperiodic forcing [9] and nondynamical systems [7], [10]. In the present work we focus our attention on nonpotential systems. Non-potential systems correspond to systems far from equilibrium for which the principle of detailed balance does not hold. There are abundant examples of such systems arising from biological, chemical and physical problems. Our contribution in this paper is to develop a formalism which allows us to analytically treat a wide class of nonpotential systems among which one can include excitable and threshold systems as well as, for example, symmetric double-well models [11]. In particular, we apply our approach to continue the work undertaken by some authors in studying the stochastic properties of the FHN model. This is a well known model with wide application in the field of neuronal research [13], [5]. Apart from several numerical simulations done on this subject, Collins et al. [9] have carried out some analytical work on this matter in the area of aperiodic stochastic resonance . Some experimental research has been performed to show the existence of SR in this model [24]. The results obtained were compared to the predictions of the theory of SR in nondynamical systems [10]. Our scheme allows to analytically approach this problem in a simple way by using a generalization of the kinetic equations approach used in the case of potential systems ( see [2], [14], [15]). All of the aforementioned models have a common feature; their dynamics exhibit three fixed points: an unstable point between two stable points. This feature established some resemblance between the processes described by these models and the hopping through a potential barrier. There are a great variety of systems that contains these features. The FitzHugh-Nagumo model, in its bistable regime, belongs to them. It is worth pointing out that this is not the regime in which this model is used to model neural activity. In this context the FHN model is taken to be in the excitable regime where only one globally attractor exists. As it is pointed out by Wiesenfeld et al. [6], a simple model of excitable system consists, among other things, of a threshold or potential barrier. Our theory provides a way to compute the escape rates from the attractors of a type of two-dimensional non-potential systems, and therefore it furnishes us with the main ingredient to apply the theory developed by Wiesenfeld et al.. In this fact you can find the relevance of our work to the field of excitable systems. Another example which fit the characteristics we are asking for is the class of symmetric double well systems [11]. The Sel’kov model [16] for autocatalytic systems gathers these features, too. This paper is organized as follows. In Section II we precisely define the range of applicability of our theory. We study the dynamics of the fluctuations and compute the kinetic equations. In Section III we introduce the FitzHugh- Nagumo model. We analyze its stochastic properties and show the existence of stochastic resonance. Finally in Section IV we discuss our main results. 1