Specifically, we consider a one-parameter set of oscillators, which are characterized by a phase-resetting curve (PRC), given by the linear combination of two prototypical examples: the theta model PRC, typical of Type I excitable neurons, and the PRC near a Hopf bifurcation for a Type II excitable neuron. We examine the correlation coefficient between spike counts over a time window T. For very long time (T -> infinity) we use linear response theory for renewal processes to write this quantity as the ratio of integrals related to exit time moments.
We find that correlation transfer over long time scales exhibits striking differences from the short-time (synchrony) correlation levels computed by Marella and Ermentrout (PRE 2008); they also differ qualitatively from the results on the linear integrate-and-fire neuron presented in de la Rocha et al. (Nature 2007, PRL 2008). We find that correlation transfer for neural oscillators is nearly independent of both input statistics (mean variance of afferent currents) and output statistics (firing rate and CV). Moreover, Type I neurons maintain a positive limiting correlation coefficient; correlation in the Type II case decays to near zero.