Analysis of stochastic attractors under the stationary point-cycle bifurcation

Bashkirtseva, Irina; Perevalova, Tatyana

doi:10.1134/s0005117907100062

Cited by 4 publications

(4 citation statements)

References 2 publications

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“…For 2-D systems, can also be written in the form (Bashkirtseva and Perevalova 2007 ) and the Mahalanobis distance is given by Here, is the unique solution of the boundary problem with T -periodic coefficients q ( t ) is a normalized vector orthogonal to the velocity vector field and for Eq. ( 3 ) is given by We note that because our model is 2-dimensional, the hyperplane given by is a tangent line to the stable limit cycle solution which is normal to q ( t ) at .…”

Section: Stochastic Sensitivity Analysis and The Mahalanobis Metricmentioning

confidence: 99%

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Weak-noise-induced transitions with inhibition and modulation of neural oscillations

Yamakou

Jost

2018

Biol Cybern

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We analyze the effect of weak-noise-induced transitions on the dynamics of the FitzHugh–Nagumo neuron model in a bistable state consisting of a stable fixed point and a stable unforced limit cycle. Bifurcation and slow-fast analysis give conditions on the parameter space for the establishment of this bi-stability. In the parametric zone of bi-stability, weak-noise amplitudes may strongly inhibit the neuron’s spiking activity. Surprisingly, increasing the noise strength leads to a minimum in the spiking activity, after which the activity starts to increase monotonically with an increase in noise strength. We investigate this inhibition and modulation of neural oscillations by weak-noise amplitudes by looking at the variation of the mean number of spikes per unit time with the noise intensity. We show that this phenomenon always occurs when the initial conditions lie in the basin of attraction of the stable limit cycle. For initial conditions in the basin of attraction of the stable fixed point, the phenomenon, however, disappears, unless the timescale separation parameter of the model is bounded within some interval. We provide a theoretical explanation of this phenomenon in terms of the stochastic sensitivity functions of the attractors and their minimum Mahalanobis distances from the separatrix isolating the basins of attraction.

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Section: Stochastic Sensitivity Analysis and The Mahalanobis Metricmentioning

confidence: 99%

“…( 2), "+" means a pseudoinverse in this case. For 2-D systems, Θ ij (t) can also be written in the form [25]…”

Section: Stochastic Sensitivity Analysis and The Mahalanobis Metricmentioning

confidence: 99%

Weak-noise-induced transitions with inhibition and modulation of neural oscillations

Yamakou

Jost

2018

Biol Cybern

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“…Due to condition (1.8), matrix W (t) can be written as W (t) = μ(t)P (t), where μ(t) is a T -periodic scalar stochastic sensitivity function [17].…”

Section: Controlling Stochastic Cycles Of Two-dimensional Systemsmentioning

confidence: 99%

“…In the case of small noises, when random states are localized near a deterministic limit cycle, the necessary stationary density can be approximated by a suitable Gaussian distribution whose covariance matrix is given by the stochastic sensitivity function (SSF) [16,17]. In [18], the SSF method has been extended to cycles in discrete stochastic systems.…”

Section: Introduction Problem Settingmentioning

confidence: 99%

On controlling stochastic sensitivity of oscillatory systems

Bashkirtseva¹,

Nurmukhametova²,

Ryashko³

2013

Autom Remote Control

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Abstract-For a nonlinear oscillatory stochastic system, we study the control problem for the variance of random trajectories around a deterministic cycle. To describe the range of random trajectories, we use the method of stochastic sensitivity functions. We consider the problem of designing a given stochastic sensitivity function, discuss problems of controllability and reachability. Complete stochastic controllability is only possible when the control's dimension coincides with the system's dimension. Otherwise, the design problem becomes ill-posed. To solve it, we propose a regularization method that lets us produce a given stochastic sensitivity function with any given precision. The efficiency of the proposed approach is demonstrated with the example of controlling stochastic oscillations in a brusselator model.

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Analysis of Critical Phenomena in a Dynamic System Under the Influence of Random Perturbations

Firstova

2023

J Math Sci

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Analysis of stochastic attractors under the stationary point-cycle bifurcation

Cited by 4 publications

References 2 publications

Weak-noise-induced transitions with inhibition and modulation of neural oscillations

Weak-noise-induced transitions with inhibition and modulation of neural oscillations

On controlling stochastic sensitivity of oscillatory systems

Analysis of Critical Phenomena in a Dynamic System Under the Influence of Random Perturbations

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