We analyze a master equation formulation of stochastic neurodynamics for a 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 deterministic limit (N → ∞) we
recover standard activity-based or voltage-based rate models. We derive the lowest order corrections
to these rate equations for large but finite N using two different approximation schemes, one based on
the Van Kampen system-size expansion and the other based on path integral methods. Both methods
yield the same series expansion of the moment equations, which at O(1/N) can be truncated to form
a closed system of equations for the first- and second-order moments. Taking a continuum limit of
the moment equations while keeping the system size N fixed generates a system of integrodifferential
equations for the mean and covariance of the corresponding stochastic neural field model. We also
show how the path integral approach can be used to study large deviation or rare event statistics
underlying escape from the basin of attraction of a stable fixed point of the mean-field dynamics;
such an analysis is not possible using the system-size expansion since the latter cannot accurately
determine exponentially small transitions.