  Neural firing is often subject to negative feedback by adaptation currents.
These currents can induce strong correlations among the time intervals between
spikes. Here we study analytically the interval correlations of a broad class
of noisy neural oscillators with spike-triggered adaptation of arbitrary
strength and time scale. Our weak-noise theory provides a general relation
between the correlations and the phase-response curve (PRC) of the oscillator,
proves anti-correlations between neighboring intervals for adapting neurons
with type I PRC and identifies a single order parameter that determines the
qualitative pattern of correlations. Monotonically decaying or oscillating
correlation structures can be related to qualitatively different voltage traces
after spiking, which can be explained by the phase plane geometry. At high
firing rates, the long-term variability of the spike train associated with the
cumulative interval correlations becomes small, independent of model details.
Our results are verified by comparison with stochastic simulations of the
exponential, leaky, and generalized integrate-and-fire models with adaptation.
