  Using a perturbative expansion for weak synaptic weights and weak sources of
randomness, we calculate the correlation structure of neural networks with
generic connectivity matrices. In detail, the perturbative parameters are the
mean and the standard deviation of the synaptic weights, together with the
standard deviations of the background noise of the membrane potentials and of
their initial conditions. We also show how to determine the correlation
structure of the system when the synaptic connections have a random topology.
This analysis is performed on rate neurons described by Wilson and Cowan
equations, since this allows us to find analytic results. Moreover, the
perturbative expansion can be developed at any order and for a generic
connectivity matrix. We finally show an example of application of this
technique for a particular case of biologically relevant topology of the
synaptic connections.
