Finite-size effects for spiking neural networks with spatially dependent coupling

We study finite-size fluctuations in a network of spiking deterministic neurons coupled with nonuniform synaptic coupling. We generalize a previously developed theory of finite-size effects for globally coupled neurons with a uniform coupling function. In the uniform coupling 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 use a path-integral formalism to 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.