MATHEMATICS

Stochastic generation of high amplitude oscillations in two-dimensional Hindmarsh–Rose model

L. B. Ryashko, E. S. Slepukhina

Department of Mathematical Physics, Institute of Mathematics and Computer Science, Ural Federal University, pr. Lenina, 51, Yekaterinburg, 620083, Russia

Abstract: We study the stochastic dynamics of the two-dimensional Hindmarsh–Rose model. In the deterministic Hindmarsh–Rose model the parametric zones of coexistence of different stable attractors (equilibria and limit cycles) are possible. The emergence of high amplitude oscillations under the influence of random disturbances on the system in these zones is due to the presence of a limit cycle. However, the stochastic generation of high amplitude oscillations is possible in a parametric zone where the deterministic system has the only stable equilibrium. This article discusses this case. For a sufficiently low noise intensity values, random states concentrate near the stable equilibrium. With the increasing of the noise intensity, trajectories go far from the equilibrium making high amplitude oscillations in the neighborhood of the unstable equilibrium. This phenomenon is confirmed by changing of the probability distribution of random trajectories. This effect is analyzed using the stochastic sensitivity function technique. A method of estimation of critical values for noise intensity is proposed.

Keywords: Hindmarsh–Rose model, excitability, stochastic sensitivity, stochastic generation of high amplitude oscillations.

UDC: 531.36

MSC: 37H20, 60H10

Received: 25.04.2014

DOI: 10.20537/vm140205