Asymptotic behaviour of a network of neurons with random linear interactions

Olivier Faugeras (1, 2) , Emilie Soret (1, 2) , Etienne Tanré (2)

1 MATHNEURO - Mathématiques pour les Neurosciences
2 TOSCA - TO Simulate and CAlibrate stochastic models

We study the asymptotic behaviour for asymmetric neuronal dynamics in a network of linear Hopfield neurons. The randomness in the network is modelled 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 with finite moments of all orders 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 w.r.t. the random weights. Our method is essentially based on the CLT and the method of moments.

Domaines: Probabilités [math.PR]

Olivier Faugeras, Emilie Soret, Etienne Tanré. Asymptotic behaviour of a network of neurons with random linear interactions. 2019. ⟨hal-01986927v1⟩