Mathematical Institute, University of Oxford, UK
Wednesday 2 December 2009
Seminar Room B10 (Basement)
Alexandra House, 17 Queen Square, London, WC1N 3AR
Stochastic neurodynamics, master equations and the system-size expansion
There are a growing number of examples where noise appears to play a role in generating spontaneous oscillatory and wave-like phenomena in large-scale cortical and subcortical networks. Examples include retinal waves and spontaneous oscillations in early cortical development, breathing rhythms, neuronal avalanches with power-law-like behaviour, and spontaneous waves in epileptiform tissue. In this talk we consider a master equation formulation of stochastic neurodynamics in a recurrent network of synaptically coupled homogeneous neuronal populations each consisting of N identical neurons. The state of the network is specified by the fraction of active or spiking neurons in each population, and transition rates are chosen so that in the thermodynamic or mean-field limit we recover standard rate-based models. We derive the lowest order corrections to these rate equations for large but finite N using the Van Kampen system-size expansion, and show how this is related to the path-integral approach of Buice and Cowan. We then briefly describe applications of the master equation approach to studying noise-induced oscillations and metastable states in recurrent networks.