Constrained charge-balanced minimum-power controls for spiking neuron oscillators

In this paper, we study the optimal control of phase models for spiking neuron oscillators. We focus on the design of minimum-power current stimuli that elicit spikes of neurons at specified times, subject to the charge-balance constraint. Such charge-balanced controls, in practice, are used to suppress undesirable side effects occurred due to the accumulation of electric charge resulting from the applied external stimuli. We derive charge-balanced minimum-power controls for a general phase model using the maximum principle, where the cases with unbounded and bounded control amplitude are examined. The latter is of fundamental importance because phase models are valid under weak forcing. The developed optimal control strategies are applied to both mathematically ideal and experimentally observed phase models to demonstrate their applicability, including the widely studied Hodgkin-Huxley model.