Résumé: Since the seminal works of Sompolinsky, Crisanti and Sommers 25 years ago, where was shown a phase transition as a

function of the variance of the synaptic weights, the question of the link between the spectrum of random connectivity matrix and

macroscopic behaviors of random neural networks has remained largely open. In this talk, I will expose some recent works on the

analysis of the dynamics of large scale randomly coupled neural networks, and the link with the spectral properties of the underlying

random matrix. We will explore what happens at the phase transition in the classical Sompolinsky model, but also the dynamics of

networks with specific bio-inspired topologies both satisfying Dale's principle and balance conditions. I will conclude on the

characterization of rare events in the spectrum of the Ginibre ensemble motivated by the understanding of the index of the trivial state

in the random neural network. These topics are the result of a collaboration with Gilles Wainrib, Khashayar Pakdaman, Romain Allez

and Luis-Carlos Garcia del Molino.