  In the absence of synaptic coupling, two or more neural oscillators may
become synchronized by virtue of the statistical correlations in their noisy
input streams. Recent work has shown that the degree of correlation transfer
from input currents to output spikes depends not only on intrinsic oscillator
dynamics, but also depends on the length of the observation window over which
the correlation is calculated. In this paper we use stochastic phase reduction
and regular perturbations to derive the correlation of the total phase elapsed
over long time scales, a quantity which provides a convenient proxy for the
spike count correlation. Over short time scales, we derive the spike count
correlation directly using straightforward probabilistic reasoning applied to
the density of the phase difference. Our approximations show that output
correlation scales with the autocorrelation of the phase resetting curve over
long time scales. We also find a concise expression for the influence of the
shape of the phase resetting curve on the initial slope of the output
correlation over short time scales. These analytic results together with
numerical simulations provide new intuitions for the recent counterintuitive
finding that type I oscillators transfer correlations more faithfully than do
type II over long time scales, while the reverse holds true for the better
understood case of short time scales.
