Stochastic Spiking-Bursting Excitability and Transition to Chaos in a Discrete-Time Neuron Model

The randomly forced Rulkov neuron model with the discontinuous 2D-map is considered. We study the phenomena of the stochastic excitement: (i) noise-induced spiking in the parametric zone where the equilibrium is a single attractor; (ii) stochastic generation of the spiking in bistability zone; (iii) noise-induced bursting in the parametric zone where the deterministic model exhibits the tonic spiking. These stochastic effects are investigated numerically by means of probability density functions and mean values of interspike (interburst) intervals. For the parametric study of these noise-induced transformations, we suggest an analytical approach taking into account the stochastic sensitivity of attractors and peculiarities of deterministic phase portraits. In this analysis, we study the mutual arrangement of confidence domains and superthreshold zones near deterministic attractors. This approach gives a prediction of the onset of the noise-induced excitement in the form of the transitions quiescence-spiking or spiking-bursting. A relationship of these phenomena with the order-chaos transformations are discussed.