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We study the asymptotic behavior for asymmetric neuronal dynamics in anetwork of linear Hopfield neurons. The interaction between the neurons ismodeled by random couplings which are centered i.i.d. random variables withfinite moments of all orders. We prove that if the initial condition of thenetwork is a set of i.i.d. random variables and independent of the synapticweights, each component of the limit system is described as the sum of thecorresponding coordinate of the initial condition with a centered Gaussianprocess whose covariance function can be described in terms of a modifiedBessel function. This process is not Markovian. The convergence is in lawalmost surely with respect to the random weights. Our method is essentiallybased on the method of moments to obtain a Central Limit Theorem.

Asymptotic behavior of a network of neurons with random linear interactions