Stochastic Generation of Bursting Oscillations in the Three-dimensional Hindmarsh–Rose Model

Abstract

We study the effect of random disturbances on the dynamics of the three-dimensional Hindmarsh–Rose model of neuronal activity. Due to the strong nonlinearity, even the original deterministic system exhibits diverse and complex dynamic regimes (various types of periodic oscillations, oscillations zones with period doubling and adding, coexistence of several attractors, chaos). In this paper, we consider a parametric zone where a stable equilibrium is the only attractor. We show that even in this zone with simple deterministic dynamics, under the random disturbances, such complex effect as the stochastic generation of bursting oscillations can occur. For a small noise, random states concentrate near the equilibrium. With the increase of the noise intensity, random trajectories can go far from the stable equilibrium, and along with small-amplitude oscillations around the equilibrium, bursts are observed. This phenomenon is analysed using the mathematical methods based on the stochastic sensitivity function technique. An algorithm of estimation of critical values for noise intensity is proposed.