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We study the asymptotic behaviour for asymmetric neuronal dynamics in anetwork of Hopfield neurons. The randomness in the network is modelled byrandom couplings which are centered Gaussian correlated random variables. Weprove that the annealed law of the empirical measure satisfies a largedeviation principle without any condition on time. We prove that the good ratefunction of this large deviation principle achieves its minimum value at aunique Gaussian measure which is not Markovian. This implies almost sureconvergence of the empirical measure under the quenched law. We prove that thelimit equations are expressed as an infinite countable set of linear nonMarkovian SDEs.

The meanfield limit of a network of Hopfield neurons with correlated synaptic weights