  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.
