https://doi.org/10.1140/epjs/s11734-025-01840-4
Regular Article
Heteroclinic cycles and switching activity in model of neuronal ensemble
1
Department of Control Theory and Dynamics of Systems, Lobachevsky State University of Nizhny Novgorod, 23, Gagarin Avenue, 603022, Nizhny Novgorod, Russia
2
Research and Education Mathematical Center “Mathematics for Future Technologies”, Lobachevsky State University of Nizhny Novgorod, 23, Gagarin Avenue, 603022, Nizhny Novgorod, Russia
Received:
30
March
2025
Accepted:
30
July
2025
Published online:
8
September
2025
The paper proposes dynamical system based on the generalized Lotka–Volterra model of three coupled excitable elements. It is shown that for some values of the coupling parameters in the phase space of the system, there are heteroclinic cycles containing three or six saddle equilibria and heteroclinic trajectories connecting them. There is a region of coexistence of three heteroclinic cycles: one stable and two unstable. Depending on initial states or external forces, there are three main types of behavior: (a) sequential switching activity of neurons resulting in regime of their silence (mathematical image is unstable heteroclinic cycle); (b) sequential switching activity of neurons resulting in regime of maximal activity of all of them (mathematical image is unstable heteroclinic cycle); (c) non-decaying sequential switching activity of single neuron, then couple of neurons, then another single neuron, then couple of other neurons, etc. (mathematical image is stable heteroclinic cycle). Last regime can be considered as some kind of self-organization. It is found that all observed regimes are not possible if all couplings are excitatory.
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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.
