IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v123y2019icp383-399.html
   My bibliography  Save this article

Ergodicity and spike rate for stochastic FitzHugh–Nagumo neural model with periodic forcing

Author

Listed:
  • Uda, Kenneth

Abstract

We discuss ergodicity on a Poincaré section and estimate the average spike rate for a time periodically forced stochastic FitzHugh–Nagumo model with degenerate noise. Stochastic FitzHugh–Nagumo (SFHN) model is a prototype stochastic neural oscillator, describing the generation and propagation of action potentials or spikes in an excitable neuron at the intracellular level. Neuronal spikes play significant role in neural information coding of various nervous systems, they are described in terms of an infinitesimal probability that spikes occur, known as spike rate. Estimation of this spike rate is a subtle task for time continuous stochastic processes such as solutions of SFHN model and, in particular, time periodically forced SFHN model. Using the regularity of the ergodic periodic measure, we estimate the average spike rate in terms of the probability density of two-point motions of the membrane potential via Rice’s formula.

Suggested Citation

  • Uda, Kenneth, 2019. "Ergodicity and spike rate for stochastic FitzHugh–Nagumo neural model with periodic forcing," Chaos, Solitons & Fractals, Elsevier, vol. 123(C), pages 383-399.
    • Handle: RePEc:eee:chsofr:v:123:y:2019:i:c:p:383-399
    DOI: 10.1016/j.chaos.2019.04.014
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077919301249
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2019.04.014?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. D. Valenti & G. Augello & B. Spagnolo, 2008. "Dynamics of a FitzHugh-Nagumo system subjected to autocorrelated noise," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 65(3), pages 443-451, October.
    2. Mattingly, J. C. & Stuart, A. M. & Higham, D. J., 2002. "Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise," Stochastic Processes and their Applications, Elsevier, vol. 101(2), pages 185-232, October.
    3. Agudov, Nikolay V & Dubkov, Alexander A & Spagnolo, Bernardo, 2003. "Escape from a metastable state with fluctuating barrier," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 325(1), pages 144-151.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Guo, Yongfeng & Wang, Linjie & Wei, Fang & Tan, Jianguo, 2019. "Dynamical behavior of simplified FitzHugh-Nagumo neural system driven by Lévy noise and Gaussian white noise," Chaos, Solitons & Fractals, Elsevier, vol. 127(C), pages 118-126.
    2. Lemaire, Vincent, 2007. "An adaptive scheme for the approximation of dissipative systems," Stochastic Processes and their Applications, Elsevier, vol. 117(10), pages 1491-1518, October.
    3. Bao, Jianhai & Wang, Jian, 2022. "Coupling approach for exponential ergodicity of stochastic Hamiltonian systems with Lévy noises," Stochastic Processes and their Applications, Elsevier, vol. 146(C), pages 114-142.
    4. Bashkirtseva, Irina & Ryashko, Lev, 2022. "Stochastic generation and shifts of phantom attractors in the 2D Rulkov model," Chaos, Solitons & Fractals, Elsevier, vol. 159(C).
    5. Shu, Huisheng & Jiang, Ziwei & Zhang, Xuekang, 2023. "Parameter estimation for integrated Ornstein–Uhlenbeck processes with small Lévy noises," Statistics & Probability Letters, Elsevier, vol. 199(C).
    6. Jin, Yanfei & Wang, Haotian & Xu, Pengfei, 2023. "Noise-induced enhancement of stability and resonance in a tri-stable system with time-delayed feedback," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).
    7. dos Santos, Maike A.F. & Junior, Luiz Menon, 2021. "Random diffusivity models for scaled Brownian motion," Chaos, Solitons & Fractals, Elsevier, vol. 144(C).
    8. de la Cruz, H. & Jimenez, J. C, 2020. "Exact pathwise simulation of multi-dimensional Ornstein–Uhlenbeck processes," Applied Mathematics and Computation, Elsevier, vol. 366(C).
    9. Cai, Yongli & Kang, Yun & Wang, Weiming, 2017. "A stochastic SIRS epidemic model with nonlinear incidence rate," Applied Mathematics and Computation, Elsevier, vol. 305(C), pages 221-240.
    10. Ai, Hao & Yang, GuiJiang & Liu, Wei & Wang, Qiubao, 2023. "A fast search method for optimal parameters of stochastic resonance based on stochastic bifurcation and its application in fault diagnosis of rolling bearings," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).
    11. Bashkirtseva, Irina & Ryashko, Lev, 2023. "Transformations of spike and burst oscillations in the stochastic Rulkov model," Chaos, Solitons & Fractals, Elsevier, vol. 170(C).
    12. Susanne Ditlevsen & Adeline Samson, 2019. "Hypoelliptic diffusions: filtering and inference from complete and partial observations," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 81(2), pages 361-384, April.
    13. Guo, Shengli & Xu, Ying & Wang, Chunni & Jin, Wuyin & Hobiny, Aatef & Ma, Jun, 2017. "Collective response, synapse coupling and field coupling in neuronal network," Chaos, Solitons & Fractals, Elsevier, vol. 105(C), pages 120-127.
    14. Song, Renming & Xie, Longjie, 2020. "Well-posedness and long time behavior of singular Langevin stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 1879-1896.
    15. Birrell, Jeremiah & Herzog, David P. & Wehr, Jan, 2012. "The transition from ergodic to explosive behavior in a family of stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1519-1539.
    16. Li, Chen-Pu & Chen, Hong-Bin & Fan, Hong & Xie, Ge-Ying & Zheng, Zhi-Gang, 2018. "Cooperation and competition between two symmetry breakings in a coupled ratchet," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 494(C), pages 175-185.
    17. Dong, Yang & Wen, Shu-hui & Hu, Xiao-bing & Li, Jiang-Cheng, 2020. "Stochastic resonance of drawdown risk in energy market prices," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 540(C).
    18. Quentin Clairon & Adeline Samson, 2020. "Optimal control for estimation in partially observed elliptic and hypoelliptic linear stochastic differential equations," Statistical Inference for Stochastic Processes, Springer, vol. 23(1), pages 105-127, April.
    19. Lin, Lifeng & Lin, Tianzhen & Zhang, Ruoqi & Wang, Huiqi, 2023. "Generalized stochastic resonance in a time-delay fractional oscillator with damping fluctuation and signal-modulated noise," Chaos, Solitons & Fractals, Elsevier, vol. 170(C).
    20. Qiu Lin & Ruisheng Qi, 2023. "Optimal Weak Order and Approximation of the Invariant Measure with a Fully-Discrete Euler Scheme for Semilinear Stochastic Parabolic Equations with Additive Noise," Mathematics, MDPI, vol. 12(1), pages 1-29, December.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:123:y:2019:i:c:p:383-399. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.