Evdokia Slepukhina

Noise-induced torus bursting in the stochastic Hindmarsh-Rose neuron model

We study the phenomenon of noise-induced torus bursting on the base of the three-dimensional Hindmarsh-Rose neuron model forced by additive noise. We show that in the parametric zone close to the Neimark-Sacker bifurcation, where the deterministic system exhibits rapid tonic spiking oscillations, random disturbances can turn tonic spiking into bursting, which is characterized by the formation of a peculiar dynamical structure resembling that of a torus. This phenomenon is confirmed by the changes in dispersion of random trajectories as well as the power spectral density and interspike intervals statistics. In particular, we show that as noise increases, the system undergoes P and D bifurcations, transitioning from order to chaos. We ultimately characterize the transition from stochastic (tonic) spiking to bursting by stochastic sensitivity functions.

Noise-induced bursting in Rulkov model

A problem of mathematical modeling and analysis of the stochastic phenomena in neuronal activity is considered. As a basic example, we use the nonlinear Rulkov map-based neuron model with random disturbances. In deterministic case, this one-dimensional model demonstrates quiescence, tonic and chaotic spiking regimes. We show that due to presence of random disturbances, a new regime of noise-induced bursting is generated not only in bistability zones, but also in monostability zones. To estimate noise intensity corresponding to the onset of bursting, the stochastic sensitivity technique and confidence domains method are applied. An effciency of our approach is confirmed by the statistics of interspike intervals.

Stochastic Bifurcations and Noise-Induced Chaos in 3D Neuron Model

The stochastically forced three-dimensional Hindmarsh–Rose model of neural activity is considered. We study the effect of random disturbances in parametric zones where the deterministic model exhibits mono- and bistable dynamic regimes with period-adding bifurcations of oscillatory modes. It is shown that in both cases the phenomenon of noise-induced bursting is observed. In the monostable zone, where the only attractor of the system is a stable equilibrium, this effect is connected with a stochastic generation of large-amplitude oscillations due to the high excitability of the model. In a parametric zone of coexisting stable equilibria and limit cycles, bursts appear due to noise-induced transitions between the attractors. For a quantitative analysis of the noise-induced bursting and corresponding stochastic bifurcations, an approach based on the stochastic sensitivity function (SSF) technique is applied. Our estimations of the strength of noise that generates such qualitative changes in stochastic dynamics are in a good agreement with the direct numerical simulation. A relationship of the noise-induced generation of bursts with transitions from order to chaos is discussed.

Methods of Stochastic Analysis of Complex Regimes in the 3D Hindmarsh–Rose Neuron Model

A problem of the stochastic nonlinear analysis of neuronal activity is studied by the example of the Hindmarsh–Rose (HR) model. For the parametric region of tonic spiking oscillations, it is shown that random noise transforms the spiking dynamic regime into the bursting one. This stochastic phenomenon is specified by qualitative changes in distributions of random trajectories and interspike intervals (ISIs). For a quantitative analysis of the noise-induced bursting, we suggest a constructive semi-analytical approach based on the stochastic sensitivity function (SSF) technique and the method of confidence domains that allows us to describe geometrically a distribution of random states around the deterministic attractors. Using this approach, we develop a new algorithm for estimation of critical values for the noise intensity corresponding to the qualitative changes in stochastic dynamics. We show that the obtained estimations are in good agreement with the numerical results. An interplay between noise-indu...

Stochastic dynamics and chaos in the 3D Hindmarsh-Rose model

We study the effect of random disturbances on the three-dimensional Hindmarsh-Rose model of neural activity. In a parametric zone, where the only attractor of the system is a stable equilibrium, a stochastic generation of bursting oscillations is observed. For a sufficiently small noise, random states concentrate near the equilibrium. With an increase of the noise intensity, along with small-amplitude oscillations around the equilibrium, bursts are observed. The relationship of the noise-induced generation of bursts with system transitions from order to chaos is discussed. For a quantitative analysis of these stochastic phenomena, an approach based on the stochastic sensitivity function technique is suggested.

Analysis of stochastic phenomena in 2D Hindmarsh-Rose neuron model

In mathematical research of neuronal activity, conceptual models play an important role. We consider 2D Hindmarsh-Rose model, which exhibits the fundamental property of neuron, the excitability. We study how random disturbances affect this property. The effects of noise are analysed in the parametric zone where the deterministic model is characterized by the coexistence of two stable equilibria. We show that under random disturbances, noise-induced transitions between the attractors occur, forming a new complex dynamic regime of stochastic bursting. It is confirmed by changes of distribution of random trajectories and interspike intervals. For the analysis of this noise-induced phenomenon, we apply the stochastic sensitivity technique and confidence domains method. We suggest a method for estimation of threshold noise intensity corresponding to the onset of noise-induced bursting. We show that the obtained values are in a good agreement with direct numerical simulations.

Noise-induced quasi-periodic oscillations in Hindmarsh-Rose neuron model

We study the phenomenon of noise-induced quasi-periodic oscillations in the stochastic Hindmarsh-Rose neuron model. We show that with the increase of the noise intensity the quiescent regime in this model transforms into the quasi-periodic (bursting) one with the formation of the stochastic torus. This phenomenon is confirmed by changes in the probability distribution of random trajectories and by the interspike intervals statistics. We show that the emergence of the torus bursting oscillations is related to the peculiarities of the geometrical arrangement of deterministic trajectories near the equilibrium and its stochastic sensitivity.

Stochastic oscillations near the “blue sky catastrophe” bifurcation in neuron model

We study the stochastic Hindmarsh-Rose neuron model near the “blue sky catastrophe” bifurcation. This specific bifurcation describes a particular type of transition between tonic spiking and bursting oscillations in the considered model. We show that in the parameter zone of tonic spiking regime, the increase of the noise intensity can lead to the stochastic generation of bursting oscillations. This noise-induced phenomenon is studied using power spectral density and interspike intervals statistics.We study the stochastic Hindmarsh-Rose neuron model near the “blue sky catastrophe” bifurcation. This specific bifurcation describes a particular type of transition between tonic spiking and bursting oscillations in the considered model. We show that in the parameter zone of tonic spiking regime, the increase of the noise intensity can lead to the stochastic generation of bursting oscillations. This noise-induced phenomenon is studied using power spectral density and interspike intervals statistics.