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Noise in spiking neurons is commonly modeled by a noisy input current or bygenerating output spikes stochastically with a voltage-dependent hazard rate("escape noise"). While input noise lends itself to modeling biophysical noiseprocesses, the phenomenological escape noise is mathematically more tractable.Using the level-crossing theory for differentiable Gaussian processes, wederive an approximate mapping between colored input noise and escape noise inleaky integrate-and-fire neurons. This mapping requires the first-passage-time(FPT) density of an overdamped Brownian particle driven by colored noise withrespect to an arbitrarily moving boundary. Starting from the Wiener-Rice seriesfor the FPT density, we apply the second-order decoupling approximation ofStratonovich to the case of moving boundaries and derive a simplifiedhazard-rate representation that is local in time and numerically efficient.This simplification requires the calculation of the non-stationaryauto-correlation function of the level-crossing process: For exponentiallycorrelated input noise (Ornstein-Uhlenbeck process), we obtain an exact formulafor the zero-lag auto-correlation as a function of noise parameters, meanmembrane potential and its speed, as well as an exponential approximation ofthe full auto-correlation function. The theory well predicts the FPT andinterspike interval densities as well as the population activities obtainedfrom simulations with time-dependent stimulus or boundary. The agreement withsimulations is strongly enhanced compared to a first-order decouplingapproximation that neglects correlations between level crossings. Thesecond-order approximation also improves upon a previously proposed theory inthe subthreshold regime. Depending on a simplicity-accuracy trade-off, allconsidered approximations represent useful mappings from colored input noise toescape noise.

Mapping Input Noise to Escape Noise in Integrate-and-fire neurons: A Level-Crossing Approach