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.
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.
| Original language | English |
|---|---|
| Journal | Journal of Physics A |
| Volume | 49 |
| Issue number | 194003 |
| Early online date | 6 Apr 2016 |
| DOIs | |
| Publication status | E-pub ahead of print - 6 Apr 2016 |
Keywords
- Plefka expansion
- Mean Field
- Dynamical Functional
- Biochemical Networks,
- Random Matrix Theory