Event-based feedback control of nonlinear oscillators using phase response curves

The phase response curve for a nonlinear oscillator describes the phase-shift of the oscillation due to an impulsive perturbation as a function of the phase at which the perturbation occurs. We propose a novel feedback control mechanism which allows one to control the phase of an oscillation, assuming only that the phase response curve is known and that a once-per-period marker event, such as the time at which a neuron fires, can be detected. The effectiveness of this control method is demonstrated through analytical and numerical results. This work represents a first 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 pathological neural synchronization.