Stochastic synchronization of neuronal populations with intrinsic and extrinsic noise.

We extend the theory of noise-induced phase synchronization to the case of a neural master equation describing the stochastic dynamics of an ensemble of uncoupled neuronal population oscillators with intrinsic and extrinsic noise. The master equation formulation of stochastic neurodynamics represents the state of each population by the number of currently active neurons, and the state transitions are chosen so that deterministic Wilson-Cowan rate equations are recovered in the mean-field limit. We apply phase reduction and averaging methods to a corresponding Langevin approximation of the master equation in order to determine how intrinsic noise disrupts synchronization of the population oscillators driven by a common extrinsic noise source. We illustrate our analysis by considering one of the simplest networks known to generate limit cycle oscillations at the population level, namely, a pair of mutually coupled excitatory (E) and inhibitory (I) subpopulations. We show how the combination of intrinsic independent noise and extrinsic common noise can lead to clustering of the population oscillators due to the multiplicative nature of both noise sources under the Langevin approximation. Finally, we show how a similar analysis can be carried out for another simple population model that exhibits limit cycle oscillations in the deterministic limit, namely, a recurrent excitatory network with synaptic depression; inclusion of synaptic depression into the neural master equation now generates a stochastic hybrid system.