Microscopic theory of intrinsic timescales in spiking neural networks

Abstract

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 interspike-interval distribution, the single-neuron power spectra, and intrinsic timescales. With the help of our theory, we find parameter regimes where the intrinsic timescale significantly exceeds the membrane time constant, which indicates the influence of the recurrent dynamics. Although the resulting intrinsic timescales are on the same order for generalized linear model neurons and leaky integrate-and-fire neurons, the two systems differ fundamentally: for the former, the longer intrinsic timescale arises from an increased firing probability after a spike; for the latter, it is a consequence of a prolonged effective refractory period with a decreased firing probability. Furthermore, the intrinsic timescale attains a maximum at a critical synaptic strength for generalized linear model networks, in contrast to the minimum found for leaky integrate-and-fire networks.