Strongly nonlinear age-structured equation, time-elapsed model and large delays

The time-elapsed model for neural assemblies is a nonlinear age-structured equation where the renewal term describes the network activity and influences the discharge rate, possibly with a delay due to the length of connections. We first solve a long standing question, namely that an inhibitory network without delay can promote desynchronization and stabilizes network activity by proving rigorously that the solution converges to a unique steady state. Our approach is based on the observation that a non-expansion property holds. However a non-degeneracy condition is needed and, besides the standard one, we introduce a new condition based on strict nonlinearity. When a delay is included, following previous works for Fokker-Planck models, we prove that the network can generate periodic solutions, both in inhibitory and excitatory networks. To this end, we introduce a new formalism to establish rigorously this property for large delays. Moreover, the fundamental contraction property can extend to other age-structured equations and systems.