Self-consistent stochastic dynamics for finite-size networks of spiking neurons

Despite the huge number of neurons composing a brain network, ongoing activity of local cell assemblies composing cortical columns is intrinsically stochastic. Fluctuations in their instantaneous rate of spike firing $\nu(t)$ scale with the size of the assembly and persist in isolated network, i.e., in absence of external source of noise. Although deterministic chaos due to the quenched disorder of the synaptic couplings likely underlies this seemingly stochastic dynamics, an effective theory for the network dynamics of a finite ensemble of spiking neurons is lacking. Here, we fill this gap by extending the so-called population density approach including an activity- and size-dependent stochastic source in the Fokker-Planck equation for the membrane potential density. The finite-size noise embedded in this stochastic partial derivative equation is analytically characterized leading to a self-consistent and non-perturbative description of $\nu(t)$ valid for a wide class of spiking neuron networks. Its power spectra of $\nu(t)$ are found in excellent agreement with those from detailed simulations both in the linear regime and across a synchronization phase transition, when a size-dependent smearing of the critical dynamics emerges.