Asymptotic behaviour of a network of neurons with random linear interactions
Résumé
We study the asymptotic behavior for asymmetric neuronal dynamics in a network
of linear Hopfield neurons.
The interaction between the neurons is modeled by random
couplings which are centered i.i.d. random variables with finite
moments of all orders. We prove that if the initial condition of the network is
a set of i.i.d. random variables
and independent of the synaptic weights, each component of the limit system is
described as the sum of the corresponding coordinate of the initial condition
with a centered Gaussian process whose covariance function can be described in
terms of a modified Bessel function. This process is not Markovian. The
convergence is in law almost surely with respect to the random weights. Our method is
essentially based on the method of moments to obtain a Central Limit Theorem.
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