Dynamical mean-field approximation to coupled active rotator networks subject to white noises

A semi-analytical dynamical mean-field approximation (DMA) has been developed for large but finite $N$-unit active rotator (AR) networks subject to individual white noises. Assuming weak noises and the Gaussian distribution of state variables, we have derived equations of motions for moments of local and global variables up to the {\it infinite} order. In DMA, the original $N$-dimensional {\it stochastic} differential equations (DEs) are replaced by three-dimensional {\it deterministic} DEs while the conventional moment method yields $(1/2)N(N+3)$ deterministic DEs for moments of local variables. We have discussed the characters of the stationary state, the time-periodic state and the random, disordered state, which are realized in excitable AR networks, depending on the model parameters. It has been demonstrated that although fluctuations of global variable vary as $1/\sqrt{N}$ when $N$ is increased, those of local variables remain finite even for $N \to \infty$. Results calculated with the use of our DMA are compared to those obtained by direct simulations and by the Fokker-Planck equation which is applicable to the $N=\infty$ AR model. The advantage and disadvantage of DMA are also discussed.