  Using a stochastic nonlinear phase oscillator model, we study the effect of
event-triggered feedback on the statistics of interevent intervals. Events are
associated with the entering of a new cycle. The feedback is modeled by an
instantaneous increase (positive feedback) or decrease (negative feedback) of
the oscillators frequency, whenever an event occurs followed by an exponential
decay on a slow timescale. In contrast to previous works, we also consider
positive feedback that leads to various novel effects. For instance, besides
the known excitable and oscillatory regime, that are separated by a saddle-node
on invariant circle bifurcation, positive feedback can lead to bistable
dynamics and a change of the system's excitability. The feedback has also a
strong effect on noise-induced phenomena like coherence resonance or
anti-coherence resonance. Both positive and negative feedback can lead to more
regular output for particular noise strengths. Finally, we investigate serial
correlation in the sequence of interevent intervals that occur due to the
additional slow dynamics. We derive approximations for the serial correlation
coefficient and show that positive feedback results in extended positive
interval correlations whereas negative feedback yields short-ranging negative
correlations. Investigating the interplay of feedback and the nonlinear phase
dynamics close to the bifurcation, we find that correlations are most
pronounced for an optimal feedback strengths.
