  We present a phase-amplitude reduction framework for analyzing collective
oscillations in networked dynamical systems. The framework, which builds on the
phase reduction method, takes into account not only the collective dynamics on
the limit cycle but also deviations from it by introducing amplitude variables
and using them with the phase variable. The framework allows us to study how
networks react to applied inputs or coupling, including their synchronization
and phase-locking, while capturing the deviations of the network states from
the unperturbed dynamics. Numerical simulations are used to demonstrate the
effectiveness of the framework for networks composed of FitzHugh-Nagumo
elements. The resulting phase-amplitude equation can be used in deriving
optimal periodic waveforms or introducing feedback control for achieving fast
phase locking while stabilizing the collective oscillations.
