This paper investigates the issue of prescribed-time mean-square synchronization (PTMSS) for stochastic coupled inertial neural networks (SCINNs), encompassing both scenarios without time delays (TDs) and with TDs. By constructing appropriate Lyapunov functionals and utilizing the prescribed-time stability (PTS) approach, combined with the theory of directed spanning trees, several sufficient conditions are rigorously derived to realize the prescribed-time mean-square synchronization within a given time $t_p$ for SCINNs with and without TDs. Different from most existing results that primarily address deterministic systems or neglect stochastic disturbances and time delays, this work explicit... More >