Extended Plefka Expansion for Stochastic Dynamics

TL;DR

This paper extends the Plefka expansion to stochastic differential equations with nonlinearities, providing a new mean field approximation that accounts for second moments and is applicable to complex biochemical networks.

Contribution

The authors develop a generalized Plefka expansion incorporating second moments, leading to effective single-variable equations with memory and colored noise, applicable to large stochastic networks.

Findings

The method applies to nonlinear stochastic dynamics.

Exactness is demonstrated for linear Gaussian systems.

The approach captures memory effects and correlations in complex networks.

Abstract

We propose an extension of the Plefka expansion, which is well known for the dynamics of discrete spins, to stochastic differential equations with continuous degrees of freedom and exhibiting generic nonlinearities. The scenario is sufficiently general to allow application to e.g. biochemical networks involved in metabolism and regulation. The main feature of our approach is to constrain in the Plefka expansion not just first moments akin to magnetizations, but also second moments, specifically two-time correlations and responses for each degree of freedom. The end result is an effective equation of motion for each single degree of freedom, where couplings to other variables appear as a self-coupling to the past (i.e. memory term) and a coloured noise. This constitutes a new mean field approximation that should become exact in the thermodynamic limit of a large network, for suitably long-ranged couplings. For the analytically tractable case of linear dynamics we establish this exactness explicitly by appeal to spectral methods of Random Matrix Theory, for Gaussian couplings with arbitrary degree of symmetry.