  We consider the Kuramoto model of globally coupled phase oscillators with
time-delayed interactions, that is subject to the Ornstein-Uhlenbeck (Gaussian)
colored or the non-Gaussian colored noise. We investigate numerically the
interplay between the influences of the finite correlation time of noise $\tau$
and the time delay $\tau_{d}$ on the onset of the synchronization process. Both
cases for identical and nonidentical oscillators had been considered. Among the
obtained results for identical oscillators is a large increase of the
synchronization threshold as a function of time delay for the colored
non-Gaussian noise compared to the case of the colored Gaussian noise at low
noise correlation time $\tau$. However, the difference reduces remarkably for
large noise correlation times. For the case of nonidentical oscillators, the
incoherent state may become unstable around the maximum value of the threshold
(as a function of time delay) even at lower coupling strength values in the
presence of colored noise as compared to the noiseless case. We had studied the
dependence of the critical value of the coupling strength (the threshold of
synchronization) on given parameters of the stochastic Kuramoto model in great
details and presented results for possible cases of colored Gaussian and
non-Gaussian noises.
