  Much progress has been made in uncovering the computational capabilities of
spiking neural networks. However, spiking neurons will always be more expensive
to simulate compared to rate neurons because of the inherent disparity in time
scales - the spike duration time is much shorter than the inter-spike time,
which is much shorter than any learning time scale. In numerical analysis, this
is a classic stiff problem. Spiking neurons are also much more difficult to
study analytically. One possible approach to making spiking networks more
tractable is to augment mean field activity models with some information about
spiking correlations. For example, such a generalized activity model could
carry information about spiking rates and correlations between spikes
self-consistently. Here, we will show how this can be accomplished by
constructing a complete formal probabilistic description of the network and
then expanding around a small parameter such as the inverse of the number of
neurons in the network. The mean field theory of the system gives a rate-like
description. The first order terms in the perturbation expansion keep track of
covariances.
