  Population rate or activity equations are the foundation of a common approach
to modeling for neural networks. These equations provide mean field dynamics
for the firing rate or activity of neurons within a network given some
connectivity. The shortcoming of these equations is that they take into account
only the average firing rate while leaving out higher order statistics like
correlations between firing. A stochastic theory of neural networks which
includes statistics at all orders was recently formulated. We describe how this
theory yields a systematic extension to population rate equations by
introducing equations for correlations and appropriate coupling terms. Each
level of the approximation yields closed equations, i.e. they depend only upon
the mean and specific correlations of interest, without an {\it ad hoc}
criterion for doing so. We show in an example of an all-to-all connected
network how our system of generalized activity equations captures phenomena
missed by the mean fieldrate equations alone.
