  A complex interplay of single-neuron properties and the recurrent network
structure shapes the activity of cortical neurons. The single-neuron activity
statistics differ in general from the respective population statistics,
including spectra and, correspondingly, autocorrelation times. We develop a
theory for self-consistent second-order single-neuron statistics in
block-structured sparse random networks of spiking neurons. In particular, the
theory predicts the neuron-level autocorrelation times, also known as intrinsic
timescales, of the neuronal activity. The theory is based on an extension of
dynamic mean-field theory from rate networks to spiking networks, which is
validated via simulations. It accounts for both static variability, e.g. due to
a distributed number of incoming synapses per neuron, and temporal fluctuations
of the input. We apply the theory to balanced random networks of generalized
linear model neurons, balanced random networks of leaky integrate-and-fire
neurons, and a biologically constrained network of leaky integrate-and-fire
neurons. For the generalized linear model network with an error function
nonlinearity, a novel analytical solution of the colored noise problem allows
us to obtain self-consistent firing rate distributions, single-neuron power
spectra, and intrinsic timescales. For the leaky integrate-and-fire networks,
we derive an approximate analytical solution of the colored noise problem,
based on the Stratonovich approximation of the Wiener-Rice series and a novel
analytical solution for the free upcrossing statistics. Again closing the
system self-consistently, in the fluctuation-driven regime this approximation
yields reliable estimates of the mean firing rate and its variance across
neurons, the inter-spike interval distribution, the single-neuron power
spectra, and intrinsic timescales.
