  We study the asymptotic law of a network of interacting neurons when the
number of neurons becomes infinite. Given a completely connected network of
firing rate neurons in which the synaptic weights are Gaussian correlated
random variables, we describe the asymptotic law of the network when the number
of neurons goes to infinity. We introduce the process-level empirical measure
of the trajectories of the solutions to the equations of the finite network of
neurons and the averaged law (with respect to the synaptic weights) of the
trajectories of the solutions to the equations of the network of neurons. The
main result of this article is that the image law through the empirical measure
satisfies a large deviation principle with a good rate function which is shown
to have a unique global minimum. Our analysis of the rate function allows us
also to characterize the limit measure as the image of a stationary Gaussian
measure defined on a transformed set of trajectories. This is potentially very
useful for applications in neuroscience since the Gaussian measure can be
completely characterized by its mean and spectral density. It also facilitates
the assessment of the probability of finite-size effects.
