In this talk I will introduce the main mathematical questions arising in the modeling of large-scale neuronal networks involved at functional scales in the brain. Such networks are composed of multiple populations (different neuronal types), in which each neuron has a stochastic dynamics and operate in a random environment. Understanding the collective dynamics of such neuronal assemblies involves mathematical tools developed in statistical physics, and most cortical activity regimes are out-of-equilibrium, related to periodic or chaotic solutions in law. I will specifically present two recent works on the subject. The first one deals with mesoscopic limits of spatially extended neural fields and the resulting spatio-temporal pattern formation, in particular the presence of transitions from stationary to synchronized periodic activity induced by noise or heterogeneity, and the second one will analyze in depth a phase transition between stationary and dynamical chaotic activity in relationship with the topological complexity of the network.