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

TL;DR

This paper introduces an augmented moment method (AMM) that transforms complex stochastic delay differential equations into deterministic equations for analyzing the dynamics of Langevin ensembles, showing good agreement with simulations.

Contribution

The paper develops an augmented moment method (AMM) that simplifies the analysis of stochastic ensembles with delays by converting SDDEs into finite-dimensional deterministic equations, extending previous mean-field approaches.

Findings

AMM accurately reproduces exact results for linear Langevin models.

Synchronization is enhanced near the transition between oscillating and non-oscillating states.

AMM6 results agree well with direct simulations.

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.