Global dynamics in networks of 1d elements with time delayed mean field coupling subject to noise: synchronization threshold, multistability and hysteresis

We determine the boundary of the synchronization domain of a large number of one-dimensional continuous stochastic elements with time delayed nonhomogeneous mean-field coupling. The exact location of the synchronization threshold is shown to be a solution of the boundary value problem (BVP) which was derived from the linearized Fokker-Planck equation. Here the synchronization threshold is found by solving this BVP using the continuation technique (AUTO). Approximate analytics is obtained using expansion into eigenfunctions of the stationary Fokker-Planck operator. Multistability and hysteresis are demonstrated for the case of bistable elements with a polynomial potential