Firing statistics and correlations in spiking neurons: A level-crossing approach

We present a time-dependent level-crossing theory for linear dynamical systems perturbed by colored Gaussian noise. We apply these results to approximate the firing statistics of conductance-based integrate-and-fire neurons receiving excitatory and inhibitory Poissonian inputs. Analytical expressions are obtained for three key quantities characterizing the neuronal response to time-varying inputs: the mean firing rate, the linear response to sinusoidally modulated inputs, and the pairwise spike correlation for neurons receiving correlated inputs. The theory yields tractable results that are shown to accurately match numerical simulations and provides useful tools for the analysis of interconnected neuronal populations.