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Periodically kicked feedforward chains of simple excitable FitzHugh–Nagumo neurons

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Abstract

This article communicates results on regular depolarization cascades in periodically kicked feedforward chains of excitable two-dimensional FitzHugh–Nagumo systems driven by sufficiently strong excitatory forcing at the front node. The study documents a parameter exploration by way of changes to the forcing period, upon which the dynamics undergoes a transition from simple depolarization to more complex behavior, including the emergence of mixed-mode oscillations. Both rigorous studies and careful numerical observations are presented. In particular, we provide rigorous proofs for existence and stability of periodic traveling waves of depolarization, as well as the existence and propagation of a simple mixed-mode oscillation that features depolarization and refraction in alternating fashion. Detailed numerical investigation reveals a mechanism for the emergence of complex mixed-mode oscillations featuring a potentially high number of large amplitude voltage spikes interspersed by an occasional small amplitude reset that fails to cross threshold. Further careful numerical investigation provides insights into the propagation of this complex phenomenology in the downstream, where we see an effective filtration property of the network; the latter amounts to a successive reduction in the complexity of mixed-mode oscillations down the chain.

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Notes

  1. Of interest for the feedforward chain network studied here, how do MMOs propagate downstream (do they regularize or become more irregular?), and are there regimes in which wave propagation/signal transmission downstream is altogether blocked?

  2. See our discussion of the state of affairs for \(\alpha \) only slightly smaller than \(\alpha _2\) in Remark 11.

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Acknowledgements

The research of Benjamin Ambrosio (BA) on Neuroscience inspired modeling is partially funded by CNRS (IEA00134). More generally BA wants to thank Le Havre Normandie University, LMAH, Région Normandie, ISCN, the Courant Institute of Mathematical Sciences and New York University, the Hudson School of Mathematics, New York, USA, for material and/or financial support for research in Dynamical Systems, Complex Systems and its applications.

Funding

The research of BA has been partially funded by CNRS (IEA00134), LMAH, Région Normandie and the Hudson School of Mathematics.

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Author notes

  1. Stanislav M. Mintchev contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, UNIHAVRE, LMAH, FR CNRS 3335, ISCN, Normandie Univ, 25 Rue Philippe Lebon, 76600, Le Havre, Normandie, France

    Benjamin Ambrosio

  2. The Hudson School of Mathematics, 244 Fifth Avenue, Suite Q224, New York, NY, 10001, USA

    Benjamin Ambrosio

  3. Department of Mathematics, The Cooper Union for the Advancement of Science and Art, 41 Cooper Sq., New York, NY, 10003, USA

    Stanislav M. Mintchev

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  1. Benjamin Ambrosio
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Correspondence to Benjamin Ambrosio.

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Ambrosio, B., Mintchev, S.M. Periodically kicked feedforward chains of simple excitable FitzHugh–Nagumo neurons. Nonlinear Dyn 110, 2805–2829 (2022). https://doi.org/10.1007/s11071-022-07757-0

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