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Canard Mechanism and Rhythm Dynamics of Neuron Models

Author

Listed:
  • Feibiao Zhan

    (Department of Applied Mathematics, Nanjing Audit University, Nanjing 211815, China)

  • Yingteng Zhang

    (Department of Mathematics, Taizhou University, Taizhou 225300, China)

  • Jian Song

    (School of Mathematics, South China University of Technology, Guangzhou 510640, China
    School of Mathematical and Computational Sciences, Massey University, Auckland 4442, New Zealand)

  • Shenquan Liu

    (School of Mathematics, South China University of Technology, Guangzhou 510640, China)

Abstract

Canards are a type of transient dynamics that occur in singularly perturbed systems, and they are specific types of solutions with varied dynamic behaviours at the boundary region. This paper introduces the emergence and development of canard phenomena in a neuron model. The singular perturbation system of a general neuron model is investigated, and the link between the transient transition from a neuron model to a canard is summarised. First, the relationship between the folded saddle-type canard and the parabolic burster, as well as the firing-threshold manifold, is established. Moreover, the association between the mixed-mode oscillation and the folded node type is unique. Furthermore, the connection between the mixed-mode oscillation and the limit-cycle canard (singular Hopf bifurcation) is stated. In addition, the link between the torus canard and the transition from tonic spiking to bursting is illustrated. Finally, the specific manifestations of these canard phenomena in the neuron model are demonstrated, such as the singular Hopf bifurcation, the folded-node canard, the torus canard, and the “blue sky catastrophe”. The summary and outlook of this paper point to the realistic possibility of canards, which have not yet been discovered in the neuron model.

Suggested Citation

  • Feibiao Zhan & Yingteng Zhang & Jian Song & Shenquan Liu, 2023. "Canard Mechanism and Rhythm Dynamics of Neuron Models," Mathematics, MDPI, vol. 11(13), pages 1-22, June.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:13:p:2874-:d:1180295
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    References listed on IDEAS

    as
    1. Richard B. Sowers, 2008. "Random Perturbations of Canards," Journal of Theoretical Probability, Springer, vol. 21(4), pages 824-889, December.
    2. Brøns, Morten & Kaasen, Rune, 2010. "Canards and mixed-mode oscillations in a forest pest model," Theoretical Population Biology, Elsevier, vol. 77(4), pages 238-242.
    3. Zhan, Feibiao & Su, Jianzhong & Liu, Shenquan, 2023. "Canards dynamics to explore the rhythm transition under electromagnetic induction," Chaos, Solitons & Fractals, Elsevier, vol. 169(C).
    4. Morten Brøns & Mathieu Desroches & Martin Krupa, 2015. "Mixed-Mode Oscillations Due to a Singular Hopf Bifurcation in a Forest Pest Model," Mathematical Population Studies, Taylor & Francis Journals, vol. 22(2), pages 71-79, June.
    Full references (including those not matched with items on IDEAS)

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