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We analyze the macroscopic behavior of multi-populations randomly connectedneural networks with interaction delays. Similar to cases occurring in spinglasses, we show that the sequences of empirical measures satisfy a largedeviation principle, and converge towards a self-consistent non-Markovianprocess. The proof differs in that we are working in infinite-dimensionalspaces (interaction delays), non-centered interactions and multiple cell types.The limit equation is qualitatively analyzed, and we identify a number of phasetransitions in such systems upon changes in delays, connectivity patterns anddispersion, particularly focusing on the emergence of non-equilibrium statesinvolving synchronized oscillations.

Large deviations, dynamics and phase transitions in large stochastic heterogeneous neural networks