Minimum energy spike randomization for neurons

With inspiration from Arthur Winfree's idea of randomizing the phase of an oscillator by driving its state to a set in which the phase is not defined, i.e., the phaseless set, we employ a Hamilton-Jacobi-Bellman approach to design a minimum energy control law that effectively randomizes the next spiking time for a two-dimensional conductance-based model of noisy oscillatory neurons. The control is initially designed for the deterministic system through the numerical solution of the Hamilton-Jacobi-Bellman partial differential equation for the cost-to-go function, from which the minimum energy stimulus can be found; then its performance is investigated in the presence of noise. It is shown that such control causes a considerable amount of randomization in the timing of the neuron's next spike.