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Despite the huge number of neurons composing a brain network, ongoingactivity of local cell assemblies composing cortical columns is intrinsicallystochastic. Fluctuations in their instantaneous rate of spike firing $\nu(t)$scale with the size of the assembly and persist in isolated network, i.e., inabsence of external source of noise. Although deterministic chaos due to thequenched disorder of the synaptic couplings likely underlies this seeminglystochastic dynamics, an effective theory for the network dynamics of a finiteensemble of spiking neurons is lacking. Here, we fill this gap by extending theso-called population density approach including an activity- and size-dependentstochastic source in the Fokker-Planck equation for the membrane potentialdensity. The finite-size noise embedded in this stochastic partial derivativeequation is analytically characterized leading to a self-consistent andnon-perturbative description of $\nu(t)$ valid for a wide class of spikingneuron networks. Its power spectra of $\nu(t)$ are found in excellent agreementwith those from detailed simulations both in the linear regime and across asynchronization phase transition, when a size-dependent smearing of thecritical dynamics emerges.

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