IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v200y2025ip2s0960077925009063.html

Emergence of Shilnikov homoclinic bifurcation and orbit-induced mixed-mode oscillations in a neuronal dynamical system

Author

Listed:
  • Liu, Yaru
  • Xia, Yibo
  • Kurths, Jürgen

Abstract

Our paper investigates the dynamic relationship between homoclinic bifurcations and mixed-mode oscillations (MMOs) in a reduced three-dimensional singularly perturbed Hodgkin–Huxley (HH) liked model, developed to characterize complex neurodynamic oscillations in neural systems. Through invariant manifold tracking, we establish the coexistence of elusive homoclinic bifurcations and MMOs. Our manifold analysis reveals critical conditions for the emergence of Shilnikov homoclinic bifurcation and oscillatory stability parameter thresholds. Employing Fenichel’s theorem and Bogdanov–Takens (BT) bifurcation theory within a nonlinear multiscale framework, we construct a locally topologically equivalent system for singular dynamics. This system exhibits three canonical bifurcation curves that elucidate the mechanistic origins of MMOs induced by singular BT bifurcations. The proof strategy for Shilnikov homoclinic bifurcations relies on a systematic examination of invariant manifold intersections in multiscale systems. Application to neuronal models demonstrates rich dynamic phenomena including pseudo-plateau bursting, MMOs, and chaotic-MMOs. Numerical simulations validate theoretical predictions with remarkable consistency, particularly in the bifurcation parameter regions.

Suggested Citation

  • Liu, Yaru & Xia, Yibo & Kurths, Jürgen, 2025. "Emergence of Shilnikov homoclinic bifurcation and orbit-induced mixed-mode oscillations in a neuronal dynamical system," Chaos, Solitons & Fractals, Elsevier, vol. 200(P2).
    • Handle: RePEc:eee:chsofr:v:200:y:2025:i:p2:s0960077925009063
    DOI: 10.1016/j.chaos.2025.116893
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077925009063
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2025.116893?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

    for a different version of it.

    References listed on IDEAS

    as
    1. Yanhui, Li & Siming, Zhu, 2006. "n-Dimensional stable and unstable manifolds of hyperbolic singular point," Chaos, Solitons & Fractals, Elsevier, vol. 29(5), pages 1155-1164.
    2. Feng, Jingjing & Zhang, Qichang & Wang, Wei, 2012. "Chaos of several typical asymmetric systems," Chaos, Solitons & Fractals, Elsevier, vol. 45(7), pages 950-958.
    3. Nicholas Tolley & Pedro L C Rodrigues & Alexandre Gramfort & Stephanie R Jones, 2024. "Methods and considerations for estimating parameters in biophysically detailed neural models with simulation based inference," PLOS Computational Biology, Public Library of Science, vol. 20(2), pages 1-29, February.
    4. Wang, Lei & Li, Qingdu & Yang, Xiao-Song, 2021. "Periodic sinks and periodic saddle orbits induced by heteroclinic bifurcation in three-dimensional piecewise linear systems with two zones," Applied Mathematics and Computation, Elsevier, vol. 404(C).
    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. Bella, Giovanni, 2017. "Homoclinic bifurcation and the Belyakov degeneracy in a variant of the Romer model of endogenous growth," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 452-460.
    2. Liu, Yachong & Hu, Ankang & Han, Fenglei & Lu, Yu, 2015. "Stability analysis of nonlinear ship-roll dynamics under wind and wave," Chaos, Solitons & Fractals, Elsevier, vol. 76(C), pages 32-39.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    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:200:y:2025:i:p2:s0960077925009063. 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.