  We formulate a theory for the collective phase description of globally
coupled noisy limit-cycle oscillators exhibiting macroscopic rhythms.
Collective phase equations describing such macroscopic rhythms are derived by
means of a two-step phase reduction. The collective phase sensitivity and
collective phase coupling functions, which quantitatively characterize the
macroscopic rhythms, are illustrated using three representative models of
limit-cycle oscillators. As an important result of the theory, we demonstrate
noise-induced anti-phase synchronization between macroscopic rhythms by direct
numerical simulations of the three models.
