Dynamics and synchronization of Langevin ensembles with time-delayed couplings

By employing a semi-analytical dynamical mean-field approximation theory previously proposed by the author [H. Hasegawa, Phys. Rev. E 67, 041903 (2003)], we have developed an augmented moment method (AMM) in order to discuss dynamics of an N -unit ensemble described by delay Langevin equations. In AMM, original N -dimensional stochastic delay differential equations (SDDEs) are transformed to infinite-dimensional deterministic DEs for means and correlations of local as well as global variables. Infinite-order DEs arising from the non-Markovian property of SDDE, are terminated at the finite level m in the level-m AMM, which yields (3+m)-dimensional deterministic DEs. Model calculations have been made for linear and nonlinear Langevin models. The stationary solution of AMM for the linear Langevin model with N = 1 is nicely compared to the exact result. The synchronization induced by an applied single spike is shown to be enhanced in the nonlinear Langevin ensemble with model parameters locating at the transition between oscillating and non-oscillating states. Results calculated by the level-6 AMM are in good agreement with those obtained by direct simulations. PACS numbers 05.40.-a 02.30.Ks 02.50.Ey Typeset using REVTEX e-mail: [email protected] 1