Hamiltonian dynamics and extrinsic electric field indirect coupling as a tool for management of neuronal activity in a generalized two dimensional Hindmarsh–Rose model

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.