Oscillations and Random Perturbations of a FitzHugh-Nagumo System

arXiv:0906.2671v1 [cs.DS] 15 Jun 2009 Oscillations and Random Perturbations of a FitzHugh-Nagumo System Catherine Doss∗ Mich`ele Thieullen† November 2, 2018 Abstract We consider a stochastic perturbation of a FitzHugh-Nagumo system. We show that it is possible to generate oscillations for values of pa- rameters which do not allow oscillations for the deterministic system. We also study the appearance of a new equilibrium point and new bifurcation parameters due to the noisy component. Keywords: FitzHugh-Nagumo system, fast-slow system, excitability, equi- librium points, bifurcation parameter, limit cycle, bistable system, random perturbation, large deviations, metastability, stochastic resonance ∗Laboratoire Jacques-Louis Lions, Boˆıte 189, Universit´e Pierre et Marie Curie-Paris 6, 4, Place Jussieu, 75252 Paris cedex 05, France; doss@ann.jussieu.fr †Laboratoire de Probabilit´es et Mod`eles Al´eatoires, Boˆıte 188, Universit´e Pierre et Marie Curie-Paris 6, 4, Place Jussieu, , 75252 Paris cedex 05, France; michele.thieullen@upmc.fr 1 1 Introduction. Let us consider the following family of deterministic systems indexed by the parameters a ∈R and δ > 0. δ ˙xt = −yt + f(xt), X0 = x (1.1) ˙yt = xt −a, Y0 = y (1.2) and their stochastic perturbation by a one dimensional Wiener process (Wt) as follows δdXt = (−Yt + f(Xt))dt + √ǫdWt, X0 = x (1.3) dYt = (Xt −a)dt, Y0 = y (1.4) The function f is a cubic polynomial: f(x) = −x(x −α)(x −β) with α < 0 < β. The parameter δ is small. The deterministic system (1.1)-(1.2) is an example of a slow-fast system: the two variables x, y have different time scales, xt evolves rapidly while yt evolves slowly. This system is one version of the so called FitzHugh-Nagumo system and plays an important role in neuronal modelling. In this context xt denotes the voltage or ac- tion potential of the membrane of a single neuron. It was first proposed by FitzHugh and Nagumo (cf. [3], [13]). One interest of this model is that it reproduces periodic oscillations observed experimentally. Indeed FitzHugh- Nagumo system finds its origin in the nonlinear oscillator model proposed by van der Pol. It is also a simplification of the Hodgkin-Huxley model which describes the coupled evolution of the membrane potential and the different ionic currents: existence of different time scales enable to pass from a four dimensional model to a two dimensional one. Oscillations can take place be- cause the deterministic system (1.1)-(1.2) exhibits bifurcations; more details will be given in section 3. Let us mention that oscillations in this system (1.1)-(1.2) can only occur when a ∈]a0, a1[ where a0 < a1 are two particular values of parameter a namely the bifurcation parameters. Our main interest in the present paper is to generate oscillations even for a < a0 (symmetrically a > a1) )by adding a stochastic perturbation to the deterministic system. What may be interpreted as some resonance type effect (cf. [9], [8]). We will therefore investigate possible oscillations for system (1.3)-(1.4). The presence of parameter ǫ introduces a third scale in the system and the relative strength of δ and ǫ measured by the ratio ǫ| log δ| δ will determine its evolution. Our study was inspired by reference [6] where 2 M. Freidlin considers a random perturbation of the second order equation δ d2yt dt2 = g( dyt dt , yt) and performs the study of its solution using the theory of large deviations (cf. [5]). See also [7] for the study of a more general situa- tion. In our case g( ˙y, y) = y −f( ˙y + a). Although our argument is close to M. Freidlin’s, the presence of parameter a leads to a richer behaviour. We prove the existence of equilibrium point and limit cycles different from the deterministic ones; as in [6], as well as a new bifurcation point which did not exist for the deterministic system. Our study relies on transitions between basins of attraction of stable equilibrium points due to noise. Relying on some estimation of a family of exit times(propositions 3.3 and 3.4) we study conditions on the parameters under which a convenient stochastic dynamic system approach its main state(proposition 3.5 ),(which corresponds to the equilibrium point exhibited in main theorem 2.2), or approach a metastable state (which corresponds to the limit cycle and to the new bifurcation pa- rameters exhibited in main theorem 2.1). A general study of slow-fast systems perturbed by noise can be found in [1]. Bursting oscillations in which a system alternates periodically between phases of quiescence and phases of repetitive spiking has been studied for stochastically perturbed systems in [10] and may be studied later in our stochastic setting. We recall that in the deterministic one a bursting-type behaviour has been generated in [2]. The paper is organized as follows. In section 2 we recall basic facts about (1.1)-(1.2) and we state the two main theorems (2.1) and (2.2). Section 3 is devoted to the application of large deviation theory to (1.3)-(1.4

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