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Extended Plefka expansion for stochastic dynamics
File(s)
JPhysA-104544_Proof_hi.pdf (1.65 MB)
Accepted version
Author(s)
Bravi, B
Sollich, P
Opper, M
Type
Journal Article
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.
Date Issued
2016-05-13
Date Acceptance
2015-12-15
Citation
Journal of Physics A: Mathematical and Theoretical, 2016, 49 (19), pp.1-39
URI
URL
DOI
ISSN
1751-8113
Publisher
IOP Publishing
Start Page
1
End Page
39
Journal / Book Title
Journal of Physics A: Mathematical and Theoretical
Volume
49
Issue
19
Copyright Statement
© 2016 IOP Publishing Ltd. This is an author-created, un-copyedited version of an article accepted for publication in Journal of Physics A: Mathematical and Theoretical. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The definitive publisher authenticated version is available online at https://iopscience.iop.org/article/10.1088/1751-8113/49/19/194003
Identifier
https://iopscience.iop.org/article/10.1088/1751-8113/49/19/194003
Subjects
Mathematical Physics
01 Mathematical Sciences
02 Physical Sciences
Publication Status
Published
Date Publish Online
2016-04-07