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 in neurons at desired times. We furthermore take the charge-balanced constraint into account because in practice undesirable side effects may occur due to the accumulation of electric charge resulting from external stimuli. Charge-balanced minimum-power controls are derived for a general phase model using the maximum principle, where the cases with unbounded and bounded control amplitude are examined. The latter is of practical importance since phase models are more accurate for weak forcing. The developed optimal control strategies are then applied to both mathematically ideal and experimentally observed phase models to demonstrate their applicability, including the phase model for the widely studied Hodgkin-Huxley equations.