Collective dynamics of globally coupled phase oscillators under multisite delayed feedback stimulation

Recently, multisite delayed feedback stimulation was proposed as a novel method for mild and effective deep brain stimulation in neurological diseases characterized by pathological cerebral synchronization. To develop the mathematical background of this technique we propose a continuous medium model of globally coupled phase oscillators in the form of a delayed integro-differential equation. This model provides a convenient description of the spatial phase patterns depending on the stimulation topology. We describe physically relevant phase-locked solutions and analyze their linear stability. Considering different spatial stimulation profiles as well as delays and polarities, we demonstrate that multisite delayed feedback is a robust, soft and effective method for the control of spatio-temporal synchronization patterns. Our results can be used for the optimization of parameters in applications of multisite delayed feedback stimulation to the therapy of neurological diseases with pathological synchronization and the therapy of defective central pattern generators.