Synchronization transition in ensemble of coupled phase oscillators with coherence-induced phase correction

We study synchronization phenomenon in a self-correcting population of noisy phase oscillators with randomly distributed natural frequencies. In our model each oscillator stochastically switches its phase to the ensemble-averaged value $\psi$ at a typical rate which is proportional to the degree of phase coherence $r$. The system exhibits a continuous phase transition to collective synchronization similar to classical Kuramoto model. Based on the self-consistent arguments and on the linear stability analysis of an incoherent state we derive analytically the threshold value $k_c$ of coupling constant corresponding to the onset of a partially synchronized state. Just above the transition point the linear scaling law $r\propto k-k_c$ is found. We also show that nonlinear relation between rate of phase correction and order parameter leads to non-trivial transition between incoherence and synchrony. To illustrate our analytical results, numerical simulations have been performed for a large population of phase oscillators with proposed type of coupling. The results of this work could become useful in designing distributed networked systems capable of self-synchronization.