Maximizing relaxation time in oscillator networks with implications for neurostimulation.

High frequency deep brain stimulation (HF-DBS) is a pervasive clinical neurostimulation paradigm in which rapid (> 100Hz) pulses of electrical current are invasively delivered to the brain. Here, we use dynamical systems analysis to provide hypotheses regarding the frequency-specificity of the therapeutic effects of HF-DBS. Using phase oscillator-based models, we study the relaxation time of a synchronized network following impulsive stimulation. In particular, by approximating a standard DBS pulse by a finite-energy (Dirac) delta function, we show the existence of a minimum bound on the frequency of stimulation necessary to keep the network in a desynchronized regime. If, as evidence suggests, pathological synchronization is central to the pathology in DBS-responsive disorders, then the analysis gives conceptual insight into why lower frequency and/or randomized stimulation therapy is less effective, and provides a way to study alternative design strategies.