Phase resetting and transient desynchronization in networks of globally coupled phase oscillators with inertia

Recently extensive work has been done towards developing methods for effective desynchronization of globally coupled phase oscillators (the Kuramoto model), by means of short stimulation pulses, or sequences of pulses. This is of great importance for the treatment of neurological disorders like Parkinson’s disease and essential tremor. As a progressive step towards the goal of being able to apply these desynchronization and phase-resetting techniques medically as a form of treatment, we demonstrate here how these ideas can be generalized and applied to a network of two-dimensional phase oscillators with inertia. This model has been previously presented as a simplification of a neuron with an axon and dendrite, and can be used to account for intrinsic transient behavior often seen experimentally. The stimulation techniques originally developed for the Kuramoto model work on a network of globally coupled inertial phase oscillators in a qualitatively similar way. In both cases desynchronization can be achieved when the stimulation causes nearby trajectories to diverge from each other. However, the mechanism by which this divergence of trajectories is achieved, is substantially different for the network of inertial oscillators. In particular, the addition of inertia results in a broad range of transient dynamics not present in the Kuramoto model. Nevertheless, the basic principles of phase resetting and desynchronization still apply. This suggests a robustness of these techniques which is of extreme importance to the medical applications.