Mean-field equations for stochastic firing-rate neural fields with delays: Derivation and noise-induced transitions

► Neural field equations for stochastic firing rate neuron models are derived. ► Solutions are Gaussian stochastic processes. ► The moments satisfy couple delayed (integro-) differential equations with noise as a parameter. ► Noise shapes the activity of finite-population networks, delays and noise induce cascades of Hopf bifurcations. ► Turing-Hopf bifurcations as a function of noise levels yield complex spatio-temporal pattern of activity.