Distributions of covariances as a window intothe operational regime of neuronal networks

Massively parallel recordings of spiking activity in cortical networks show that spike count covariances vary widely across pairs of neurons [Ecker et al., Science (2010)]. Their low average is well understood [Renart et al., Science (2010), Tetzlaff et al., PLoS CB (2012)], but an explanation for the wide distribution in relation to the static (quenched) disorder of the connectivity in recurrent random networks was so far elusive. Starting from spin-glass techniques [Sompolinsky and Zippelius, Phys. Rev. B (1982)] and a generating function representation for the joint probability distribution of the network activity [Chow and Buice, J. Math. Neurosci. (2015)], we derive a finite-size mean-field theory that reduces a disordered to a highly symmetric network with fluctuating auxiliary fields. The exposed analytical relation between the statistics of connections and the statistics of pairwise covariances shows that both, average and dispersion of the latter, diverge at a critical coupling. At this point, a network of nonlinear units transits from regular to chaotic dynamics. Applying these results to recordings from the mammalian brain suggests its operation close to this edge of criticality