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.

Since the pioneering work of Hodgkin and Huxley [

Discrete-time dynamical neuron models began to be studied only recently [

Random noise is an inevitable attribute of any living system. In presence of stochastic disturbances, nonlinear system can exhibit unexpected regimes of dynamics which have no analogue in the deterministic case [

Full description of the dynamics of the probabilistic distribution is given by Fokker-Planck-Kolmogorov equation [

In the present paper, we consider one-dimensional Rulkov model of the neuronal dynamics [

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 Section

For the parametric analysis of these phenomena, the stochastic sensitivity functions technique and confidence domains method are used in Section

Consider a stochastic Rulkov system:

The deterministic system (

In Figure

Attractors of the deterministic Rulkov model.

For

So, the deterministic Rulkov map-based system (

Under stochastic disturbances the solutions of system (

Random states of stochastic Rulkov model for (a)

First, branches of multiple cycles merge. For example, for

Second, in the zone of bistability, noise-induced transitions between attractors occur. In this zone, the lower attractor (stable equilibrium) coexists with the upper attractor (regular or chaotic). In spite of the fact that the initial state

It is worth noting that such one-way transitions downwards occur only in the left interval

Time series for Rulkov model with

Stochastic phenomena, presented here on the base of direct numerical simulation, are studied in the next section with the help of the stochastic sensitivity function technique.

For the analysis of the dispersions of the random states, the approximations (see Appendix for details) based on the stochastic sensitivity functions technique are used.

In Figure

Stochastic sensitivity functions: (a) for equilibria

The branches of the stochastic sensitivity function for the points of 2 cycles and 4 cycles are plotted in Figures

Backward stochastic bifurcation for

Consider a variation of the form of these peaks under increasing noise. For weak noise (

Backward stochastic bifurcation for

It is worth noting that SSF technique allows us to analyze parametrically backward stochastic bifurcations for the multiple cycles. Such analysis is important for the understanding of the probabilistic mechanism of the stochastic deformation of the modality for the spiking regime.

In what follows, we show how this technique can be used for the study of noise-induced bursting.

Consider Figure

In Figure

Noise-induced bursting (a) and noise-induced chaotization (b) for

For small

Boundaries of the confidence intervals around the stable equilibrium

Consider now how these qualitative changes in the stochastic dynamics of the Rulkov model are connected with the changes in the largest Lyapunov exponent

In Figure

So, the noise-induced bursting in the Rulkov model is accompanied by the transition from order to chaos.

Consider a nonlinear stochastic discrete-time system:

For the approximation of probabilistic distribution of random states around deterministic attractors (equilibria or cycles), the stochastic sensitivity functions technique can be used.

It is supposed that system (

Let

It is supposed that the deterministic system (

A dynamics of the second moments

For

Probability density function

Stochastic sensitivity function (SSF) technique was elaborated for the analysis of randomly forced equilibria and limit cycles for both continuous [

Confidence domains are sufficiently simple and evident geometrical models for the spatial description of random states of the stochastic system. The SSF technique enables to construct confidence intervals, ellipses, bands, and tori for the probabilistic analysis of various stochastic attractors. Constructive potentialities of the confidence regions method have been demonstrated in the analysis of stochastic phenomena for population [

The author declares that there is no conflict of interests regarding the publication of this paper.

This work was partially supported by the RFBR (14-01-00181).