  We study finite-size fluctuations in a network of spiking deterministic
neurons coupled with non-uniform synaptic coupling. We generalize a previously
developed theory of finite size effects for uniform globally coupled neurons.
In the uniform case, mean field theory is well defined by averaging over the
network as the number of neurons in the network goes to infinity. However, for
nonuniform coupling it is no longer possible to average over the entire network
if we are interested in fluctuations at a particular location within the
network. We show that if the coupling function approaches a continuous function
in the infinite system size limit then an average over a local neighborhood can
be defined such that mean field theory is well defined for a spatially
dependent field. We then derive a perturbation expansion in the inverse system
size around the mean field limit for the covariance of the input to a neuron
(synaptic drive) and firing rate fluctuations due to dynamical deterministic
finite-size effects.
