An augmented moment method for stochastic ensembles with delayed couplings: I. Langevin model

Abstract

By employing a semi-analytical dynamical mean-field approximation theory previously proposed by the author [H. Hasegawa, Phys. Rev. E {\bf 67}, 041903 (2003)], we have developed an augmented moment method (AMM) in order to discuss dynamics of an $N$-unit ensemble described by linear and nonlinear Langevin equations with delays. In AMM, original $N$-dimensional {\it stochastic} delay differential equations (SDDEs) are transformed to infinite-dimensional {\it 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 (AMM$m$), 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 AMM6 are in good agreement with those obtained by direct simulations.

18 pages, 3 figures, changed the title with re-arranged figures, accepted in Phys. Rev. E with some changes

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Cited by in corpus (9)

• Dynamical mean-filed approximation to small-world networks of spiking neurons: From local to global, and/or from regular to random couplings
• An augmented moment method for stochastic ensembles with delayed couplings: II. FitzHugh-Nagumo model
• Mean field approximation for noisy delay coupled excitable neurons
• Mean field approximation of two coupled populations of excitable units
• Generalized Rate-Code Model for Neuron Ensembles with Finite Populations
• $N$-dependent Multiplicative-Noise Contributions in Finite $N$-unit Langevin Models: Augmented Moment Approach
• Stationary and dynamical properties of finite $N$-unit Langevin models subjected to multiplicative noises
• Persistence and failure of mean-field approximations adapted to a class of systems of delay-coupled excitable units
• Stochastic bursting in unidirectionally delay-coupled noisy excitable systems