https://doi.org/10.1140/epjs/s11734-026-02224-y
Regular Article
Asymptotic return maps and geometric dynamics in multi-timescale neuronal systems: Decoding mixed-mode oscillations
1
Department of Mathematics, Jinan University, 510632, Guangzhou, China
2
College of Cyber Security, Jinan University, 510632, Guangzhou, China
3
School of Mathematics, South China University of Technology, 510640, Guangzhou, China
a
This email address is being protected from spambots. You need JavaScript enabled to view it.
Received:
3
September
2025
Accepted:
5
February
2026
Published online:
16
March
2026
Abstract
Mixed-mode oscillations (MMOs) represent a characteristic neuronal firing pattern, yet elucidating their generation mechanisms remains a complex and profound challenge. Our core goal in this paper is investigating the intrinsic dynamic properties of canards induced MMOs in proximity to singular folded curves across critical manifolds in multiscale systems. This study employs geometric singular perturbation theory (GSPT) and multiscale dynamic methods to analyze MMOs patterns in extended neuronal systems. For the two-timescale system, we derive canonical saddle-node bifurcation conditions and construct global asymptotic return map. However, since non-hyperbolicity emerges in the three-timescale system, we utilize geometric blow-up techniques to stabilize the return map asymptotically. Through systematic coordinate transformations based on asymptotic map formulations, we elucidate geometric bifurcation mechanisms underlying MMO transitions. Specifically, this investigation employs a parameterized model demonstrating rich dynamical phenomena including from MMOs to chaotic-MMOs transitions in multi-timescale neuronal systems. The proposed methodology demonstrates broad applicability for deciphering oscillation mechanisms in complex dynamical systems.
Copyright comment Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
© The Author(s), under exclusive licence to EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature 2026
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
