Mean-field models of populations of quadratic integrate-and-fire neurons with noise on the basis of the circular cumulant approach.

We develop a circular cumulant representation for the recurrent network of quadratic integrate-and-fire neurons subject to noise. The synaptic coupling is global or macroscopically equivalent to it. We assume a Lorentzian distribution of the parameter controlling whether the isolated individual neuron is periodically spiking or excitable. For the infinite chain of circular cumulant equations, a hierarchy of smallness is identified; on the basis of it, we truncate the chain and suggest several two-cumulant neural mass models. These models allow one to go beyond the Ott-Antonsen Ansatz and describe the effect of noise on hysteretic transitions between macroscopic regimes of a population with inhibitory coupling. The accuracy of two-cumulant models is analyzed in detail.