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 inter-spike interval distribution, the single-neuron power spectra, and intrinsic timescales.
Highlights
Neural dynamics in the cerebral cortex of awake behaving animals unfolds over multiple timescales, ranging from milliseconds up to seconds and more [1,2,3,4,5]
The recurrent inputs ηi(t ) can be approximated by independent Gaussian processes, which leads to a coarse-grained description of the dynamics: since all inputs are statistically equivalent, the neurons become statistically equivalent as well and the system reduces to N independent, identical stochastic equations
We developed a self-consistent theory for the second-order statistics, in particular the intrinsic timescales as defined by autocorrelation decay times, in block-structured random networks of spiking neurons in an asynchronous irregular state
Summary
Neural dynamics in the cerebral cortex of awake behaving animals unfolds over multiple timescales, ranging from milliseconds up to seconds and more [1,2,3,4,5]. Our approximation leads to integrals of which the computationally most involved ones can be solved analytically We use these results to explore the parameter space of a balanced random network of LIF neurons for long timescales, and apply the theory to a more elaborate model with population–specific connection probabilities that are constrained by biological data [43]. We start this manuscript with the derivation of the DMFT equations from the characteristic functional of the recurrent input. We use our theory to investigate the timescale in the respective network models
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