Spike timing control of oscillatory neuron models using impulsive and quasi-impulsive charge-balanced inputs

We propose a method to control the spike timing of a Type II oscillatory neuron to match the phase of a given reference oscillator. The control method is inspired by the impulsive character of neural communication in nature, and leads to a simple mathematical solution. We show that the phase response curve, which describes the phase-shift of the oscillation due to an impulsive perturbation as a function of the phase at which the perturbation occurs, contains sufficient information to design a charge-balanced control law that provides global monotonic convergence of oscillator phase to the reference phase. This feedback law requires only the knowledge of the dynamics gained through the phase reduction, and the ability to detect a once-per-period marker event, such as the time at which a neuron fires. The effectiveness of this control law is demonstrated through analytical and numerical results, including application to the full-dimensional conductance-based neuron model from which the phase-reduced model was derived. This work represents a step toward a closed-loop form of electrical deep brain stimulation, a treatment for neuromotor disorders such as Parkinson's disease, with symptoms characterized by pathologically synchronized neural firing.