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Dynamics of a modified Hindmarsh–Rose neural model with random perturbations: Moment analysis and firing activities

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  • Mondal, Argha
  • Upadhyay, Ranjit Kumar

Abstract

In this paper, an attempt has been made to understand the activity of mean membrane voltage and subsidiary system variables with moment equations (i.e., mean, variance and covariance’s) under noisy environment. We consider a biophysically plausible modified Hindmarsh–Rose (H–R) neural system injected by an applied current exhibiting spiking–bursting phenomenon. The effects of predominant parameters on the dynamical behavior of a modified H–R system are investigated. Numerically, it exhibits period-doubling, period halving bifurcation and chaos phenomena. Further, a nonlinear system has been analyzed for the first and second order moments with additive stochastic perturbations. It has been solved using fourth order Runge–Kutta method and noisy systems by Euler’s scheme. It has been demonstrated that the firing properties of neurons to evoke an action potential in a certain parameter space of the large exact systems can be estimated using an approximated model. Strong stimulation can cause a change in increase or decrease of the firing patterns. Corresponding to a fixed set of parameter values, the firing behavior and dynamical differences of the collective variables of a large, exact and approximated systems are investigated.

Suggested Citation

  • Mondal, Argha & Upadhyay, Ranjit Kumar, 2017. "Dynamics of a modified Hindmarsh–Rose neural model with random perturbations: Moment analysis and firing activities," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 486(C), pages 144-160.
    • Handle: RePEc:eee:phsmap:v:486:y:2017:i:c:p:144-160
    DOI: 10.1016/j.physa.2017.05.086
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    References listed on IDEAS

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    1. Wang, Qingyun & Zheng, Yanhong & Ma, Jun, 2013. "Cooperative dynamics in neuronal networks," Chaos, Solitons & Fractals, Elsevier, vol. 56(C), pages 19-27.
    2. Tuckwell, Henry C. & Jost, Jürgen, 2009. "Moment analysis of the Hodgkin–Huxley system with additive noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(19), pages 4115-4125.
    3. Tuckwell, Henry C. & Jost, Jürgen, 2012. "Analysis of inverse stochastic resonance and the long-term firing of Hodgkin–Huxley neurons with Gaussian white noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(22), pages 5311-5325.
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    Cited by:

    1. Liu, Yaru & Liu, Shenquan & Zhan, Feibiao & Zhang, Xiaohan, 2020. "Firing patterns of the modified Hodgkin–Huxley models subject to Taylor ’s formula," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 547(C).
    2. Das, Parthasakha & Das, Pritha & Mukherjee, Sayan, 2020. "Stochastic dynamics of Michaelis–Menten kinetics based tumor-immune interactions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 541(C).

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