Stochastic Neural Field Theory

We survey recent work on extending neural field theory to take into account synaptic noise. We begin by showing how mean field theory can be used to represent the macroscopic dynamics of a local population of N spiking neurons, which are driven by Poisson inputs, as a rate equation in the thermodynamic limit N → ∞. Finite-size effects are then used to motivate the construction of stochastic rate-based models that in the continuum limit reduce to stochastic neural fields. The remainder of the chapter illustrates how methods from the analysis of stochastic partial differential equations can be adapted in order to analyze the dynamics of stochastic neural fields. First, we consider the effects of extrinsic noise on front propagation in an excitatory neural field. Using a separation of time scales, it is shown how the fluctuating front Front(s) can be described in terms of a diffusive–like displacement (wandering) of the front from its uniformly translating position at long time scales, and fluctuations in the front profile around its instantaneous position at short time scales. Second, we investigate rare noise-driven transitions in a neural field with an absorbing state, which signals the extinction of all activity. In this case, the most probable path to extinction can be obtained by solving the classical equations of motion that dominate a path integral Path integral representation of the stochastic neural field in the weak noise limit. These equations take the form of nonlocal Hamilton equations Hamilton equations in an infinite–dimensional phase space.