  The Kuramoto model has become a paradigm to describe the dynamics of
nonlinear oscillator under the influence of external perturbations, both
deterministic and stochastic. It is based on the idea to describe the
oscillator dynamics by a scalar differential equation, that defines the time
evolution for the phase of the oscillator. Starting from a phase and amplitude
description of noisy oscillators, we discuss the reduction to a phase
oscillator model, analogous to the Kuramoto model. The model derived shows that
the phase noise problem is a drift-diffusion process. Even in the case where
the expected amplitude remains unchanged, the unavoidable amplitude
fluctuations do change the expected frequency, and the frequency shift depends
on the amplitude variance. We discuss different degrees of approximation,
yielding increasingly accurate phase reduced descriptions of noisy oscillators.
