Stochastic Phenomena in One-Dimensional Rulkov Model of Neuronal Dynamics

Irina Bashkirtseva

doi:10.1155/2015/495417

Abstract

We study the nonlinear Rulkov map-based neuron model forced by random disturbances. For this model, an overview of the variety of stochastic regimes is given. For the parametric analysis of these regimes, the stochastic sensitivity functions technique is used. In a period-doubling zone, we analyze backward stochastic bifurcations modelling changes of modality of noisy neuron spiking. Noise-induced transitions in a zone of bistability are considered. It is shown how such random transitions can generate a new neuronal regime of the stochastic bursting and transfer the system from order to chaos. A transient zone of values of noise intensity corresponding to the onset of noise-induced bursting and chaotization is localized by the stochastic sensitivity functions technique.

Highlights

Since the pioneering work of Hodgkin and Huxley [1], neuron models attract attention of many researchers [2]
We study the nonlinear Rulkov map-based neuron model forced by random disturbances
Continuoustime systems for modeling of the excitable neuron systems are widely studied. Various regimes such as spiking, bursting, fast-slow dynamics, and Canard oscillations are subject to the active investigation on the base of continuous-time models of Hindmarsh-Rose, FitzHugh-Nagumo, Morris-Lecar, and others

Summary

Introduction

Since the pioneering work of Hodgkin and Huxley [1], neuron models attract attention of many researchers [2]. Continuoustime systems for modeling of the excitable neuron systems are widely studied Various regimes such as spiking, bursting, fast-slow dynamics, and Canard oscillations are subject to the active investigation on the base of continuous-time models of Hindmarsh-Rose, FitzHugh-Nagumo, Morris-Lecar, and others. Rulkov model exhibits transitions from quiescence to spiking regime, both tonic (periodic) and chaotic. This deterministic system still cannot model the neuronal bursting. In current paper for in Rulkov model, we study stochastic deformations of the spiking regime and suggest a constructive method for parametric analysis of the noise-induced bursting. In the stochastic Rulkov model, noise-induced transitions and stochastic bifurcations are observed It is shown how random transitions between stable equilibrium and chaotic attractor cause noise-induced bursting. Backward bifurcations of reducing of the multiplicity of stochastic cycles, and noise-induced bursting with transitions from order to chaos are studied

Stochastic Phenomena in Rulkov Model

SSF Analysis of Stochastic Bifurcations

Noise-Induced Bursting and Transition to Chaos

SSF Technique