  We study the asymptotic behaviour for asymmetric neuronal dynamics in a
network of Hopfield neurons. The randomness in the network is modelled by
random couplings which are centered Gaussian correlated random variables. We
prove that the annealed law of the empirical measure satisfies a large
deviation principle without any condition on time. We prove that the good rate
function of this large deviation principle achieves its minimum value at a
unique Gaussian measure which is not Markovian. This implies almost sure
convergence of the empirical measure under the quenched law. We prove that the
limit equations are expressed as an infinite countable set of linear non
Markovian SDEs.
