  In a series of two papers, we investigate the large deviations and asymptotic
behavior of stochastic models of brain neural networks with random interaction
coefficients. In this first paper, we take into account the spatial structure
of the brain and consider (i) the presence of interaction delays that depend on
the distance between cells and (ii) Gaussian random interaction amplitude whose
mean and variance depend on the neurons positions and scale as the inverse of
the network size. We show that the empirical measure satisfies a
large-deviation principle with good rate function reaching its minimum at a
unique spatially extended probability measure. This result implies averaged
convergence of the empirical measure and propagation of chaos. The limit is
characterized through complex non-Markovian implicit equation in which the
network interaction term is replaced by a non-local Gaussian process whose
statistics depend on the solution over the whole neural field.
