  Noise can induce coherent oscillations in excitable systems without periodic
orbits. Here, we establish a method to derive a hybrid system approximating the
noise-induced coherent oscillations in excitable systems and further perform
phase reduction of the hybrid system to derive an effective,
dimensionality-reduced phase equation. We apply the reduced phase model to a
periodically forced excitable system and two-coupled excitable systems, both
undergoing noise-induced oscillations. The reduced phase model can
quantitatively predict the entrainment of a single system to the periodic force
and the mutual synchronization of two coupled systems, including the phase
slipping behavior due to noise, as verified by Monte Carlo simulations. The
derived phase model gives a simple and efficient description of noise-induced
oscillations and can be applied to the analysis of more general cases.
