A new paper in Physical Review E is now available on line here. In this paper Michael Buice and I show how you can derive an effective stochastic differential (Langevin) equation for a single element (e.g. neuron) embedded in a network by averaging over the unknown dynamics of the other elements. This then implies that given measurements from a single neuron, one might be able to infer properties of the network that it lives in. We hope to show this in the future. In this paper, we perform the calculation explicitly for the Kuramoto model of coupled oscillators (e.g. see here) but it can be generalized to any network of coupled elements. The calculation relies on the path or functional integral formalism Michael developed in his thesis and generalized at the NIH. It is a nice application of what is called “effective field theory”, where new dynamics (i.e. action) are obtained by marginalizing or integrating out unwanted degrees of freedom. The path integral formalism gives a nice platform to perform this averaging. The resulting Langevin equation has a noise term that is nonwhite, non-Gaussian and multiplicative. It is probably not something you would have guessed a priori.
| Michael A. Buice1,2 and Carson C. Chow11Laboratory of Biological Modeling, NIDDK, NIH, Bethesda, Maryland 20892, USA
2Center for Learning and Memory, University of Texas at Austin, Austin, Texas, USA
Received 25 July 2011; revised 12 September 2011; published 17 November 2011
Complex systems are generally analytically intractable and difficult to simulate. We introduce a method for deriving an effective stochastic equation for a high-dimensional deterministic dynamical system for which some portion of the configuration is not precisely specified. We use a response function path integral to construct an equivalent distribution for the stochastic dynamics from the distribution of the incomplete information. We apply this method to the Kuramoto model of coupled oscillators to derive an effective stochastic equation for a single oscillator interacting with a bath of oscillators and also outline the procedure for other systems.
Published by the American Physical Society