https://doi.org/10.1140/epjb/s10051-025-01044-8
Research - Statistical and Nonlinear Physics
Hamiltonian dynamics and extrinsic electric field indirect coupling as a tool for management of neuronal activity in a generalized two dimensional Hindmarsh–Rose model
1
Laboratory of Biophysics, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon
2
African Centre for Advanced Studies, P.O. Box 4477, Yaounde, Cameroon
3
Department of Mathematics and Physical Sciences, National Advanced School of Engineering of Yaounde, University of Yaounde, P.O. Box 8390, Yaounde, Cameroon
4
Department of Physics and Astronomy, Botswana University of Science and Technology, Private Bag 16, Palapye, Botswana
5
Laboratory of Mechanics, Department of Physics, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon
a
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Received:
27
August
2025
Accepted:
5
September
2025
Published online:
23
September
2025
Abstract
In this work, we investigate a generalized two dimensional Hindmarsh−Rose neuronal model which single out the distinct roles of the Hamiltonian and its time-derivative (instantaneous power) in shaping the neuronal activity of neurons coupled indirectly by an extrinsic electric field adjustable by an experimenter. By imposing a periodic extrinsic field, we demonstrate how neurons are driven into complete synchronization by the extrinsic electric field and we identify critical thresholds values of the amplitude and the frequency of the extrinsic electric field below which neuronal activity with amplitude death occurs. Strikingly, exceeding these thresholds values triggers a rebound of oscillatory activity. Analytical calculations using the Helmholtz’s theorem yield closed−form expressions for the Hamiltonian along with its instantaneous power reveal that the contributions of the extrinsic electric field add up in each neuron. Intensive numerical simulations confirm the existence of sharp amplitude and frequency boundaries separating regimes of neuronal silencing and revival of neuronal dynamics. Moreover, variations in neuronal radius are shown to modulate excitability through capacitance effects, with larger cells exhibiting suppressed oscillations. Our results highlight crucial mechanisms by which modulated extrinsic electric fields regulate neuronal behavior offering potential control strategies for neuromodulation applications.
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© The Author(s), under exclusive licence to EDP Sciences, SIF and Springer-Verlag GmbH Germany, part of Springer Nature 2025
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.
