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J. Sib. Fed. Univ. Math. Phys., 2016, Volume 9, Issue 1, Pages 79–89 (Mi jsfu462)  

This article is cited in 1 scientific paper (total in 1 paper)

Stochastic generation of bursting oscillations in the three-dimensional Hindmarsh–Rose model

Lev B. Ryashko, Evdokia S. Slepukhina

Institute of Mathematics and Computer Science, Ural Federal University, Lenina, 51, Ekaterinburg, 620083, Russia

Abstract: We study the effect of random disturbances on the dynamics of the three-dimensional Hindmarsh–Rose model of neuronal activity. Due to the strong nonlinearity, even the original deterministic system exhibits diverse and complex dynamic regimes (various types of periodic oscillations, oscillations zones with period doubling and adding, coexistence of several attractors, chaos). In this paper, we consider a parametric zone where a stable equilibrium is the only attractor. We show that even in this zone with simple deterministic dynamics, under the random disturbances, such complex effect as the stochastic generation of bursting oscillations can occur. For a small noise, random states concentrate near the equilibrium. With the increase of the noise intensity, random trajectories can go far from the stable equilibrium, and along with small-amplitude oscillations around the equilibrium, bursts are observed. This phenomenon is analysed using the mathematical methods based on the stochastic sensitivity function technique. An algorithm of estimation of critical values for noise intensity is proposed.

Keywords: Hindmarsh–Rose model, neurodynamics, excitability, stochastic sensitivity, stochastic generation of bursting oscillations.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation 02.A03.21.0006
The work was supported by Government of the Russian Federation (Act 211, contract 02.A03.21.0006).


DOI: https://doi.org/10.17516/1997-1397-2016-9-1-79-89

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Bibliographic databases:

UDC: 517.917
Received: 15.11.2015
Received in revised form: 14.12.2015
Accepted: 12.01.2016
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Citation: Lev B. Ryashko, Evdokia S. Slepukhina, “Stochastic generation of bursting oscillations in the three-dimensional Hindmarsh–Rose model”, J. Sib. Fed. Univ. Math. Phys., 9:1 (2016), 79–89

Citation in format AMSBIB
\Bibitem{RyaSle16}
\by Lev~B.~Ryashko, Evdokia~S.~Slepukhina
\paper Stochastic generation of bursting oscillations in the three-dimensional Hindmarsh--Rose model
\jour J. Sib. Fed. Univ. Math. Phys.
\yr 2016
\vol 9
\issue 1
\pages 79--89
\mathnet{http://mi.mathnet.ru/jsfu462}
\crossref{https://doi.org/10.17516/1997-1397-2016-9-1-79-89}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000412005500009}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. L. Ryashko, E. Slepukhina, “Noise-induced quasi-periodic oscillations in Hindmarsh–Rose neuron model”, IV International Young Researchers' Conference: Physics, Technologies and Innovation (PTI-2017), AIP Conf. Proc., 1886, eds. A. Valeeva, V. Volkovich, Amer. Inst. Phys., 2017, UNSP 020084  crossref  mathscinet  isi  scopus
  • Журнал Сибирского федерального университета. Серия "Математика и физика"
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