https://doi.org/10.1140/epjb/s10051-026-01173-8
Research - Statistical and Nonlinear Physics
Nonlinear collective dynamics and Hamiltonian energy in time-delayed Morris–Lecar small-world networks
1 School of Mathematics and Physics, Lanzhou Jiaotong University, 730070, Lanzhou, China
2 College of Mathematics and Statistics, Chongqing University, 400030, Chongqing, China
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Received:
26
August
2025
Accepted:
14
April
2026
Published online: 21 June 2026
Abstract
The collective dynamics of biological neural systems are profoundly influenced by the interplay between nonlinearity and network topology. In this study, a time-delayed bilayer neuronal network model is developed to investigate the mechanisms underlying synchronization and dynamic transitions. The network is constructed using Morris–Lecar (M–L) neurons within a small-world topology, incorporating electrical, chemical, and field coupling to replicate the complex feedback mechanism in neural information processing. Synchronization stability is rigorously evaluated using the master stability function, and the system’s Hamiltonian energy function is derived based on the Helmholtz theorem. Bifurcation analysis, Lyapunov exponents, and time series inspection reveal a rich repertoire of nonlinear dynamical modes induced by varying coupling parameters, including transitions from periodic spiking to chaotic bursting. Specifically, numerical simulations illustrate that the interplay of coupling strength and time delay drives the system through distinct dynamical regimes—ranging from asynchrony and chimera states to complete synchronization. Moreover, the results indicate that hybrid coupling mechanisms can effectively modulate the energy balance, suppressing chaos and enhancing global synchrony. This work not only elucidates the specific regulatory roles of delay and field effects but also contributes to the broader theoretical understanding of nonlinear dynamics in complex networks, providing references for designing efficient bio-inspired systems.
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© The Author(s), under exclusive licence to EDP Sciences, SIF and 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.
