Nonlinear Dynamics

, Volume 82, Issue 1–2, pp 919–932 | Cite as

Order and chaos in the stochastic Hindmarsh–Rose model of the neuron bursting

  • Irina Bashkirtseva
  • Lev Ryashko
  • Evdokia Slepukhina
Original Paper


We study the stochastic dynamics of the two-dimensional Hindmarsh–Rose model. In the deterministic case, this system demonstrates mono- and bi-stable dynamic regimes. In the parametric zone of the coexisting stable equilibrium and limit cycle, the phenomenon of noise-induced transitions between the attractors is studied. In another parametric region, where the deterministic system has the only stable equilibrium, the stochastic generation of high-amplitude oscillations is also observed. We show that under the random disturbances, the system demonstrates noise-induced bursting: an alternation of small fluctuations near the equilibrium and high-amplitude oscillations. For the quantitative analysis of noise-induced bursting, an approach combining stochastic sensitivity function technique and confidence domains method is suggested. Constructive abilities of this method for the estimation of critical values for noise intensity corresponding to the qualitative changes in stochastic dynamics are demonstrated and confirmed by the good agreement with the direct numerical simulation. An interplay between noise-induced bursting and mutual transformations “order–chaos” is discussed.


Hindmarsh–Rose model Excitability Stochastic sensitivity Noise-induced bursting Chaos 



This work was partially supported by the Ministry of Education and Science of the Russian Federation under the project N 315, and RFBR (14-01-00181).


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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Irina Bashkirtseva
    • 1
  • Lev Ryashko
    • 1
  • Evdokia Slepukhina
    • 1
  1. 1.Institute of Mathematics and Computer ScienceUral Federal UniversityYekaterinburgRussia

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