Synchronization, phase locking, and metachronal wave formation in ciliary chains

Synchronization and wave formation in one-dimensional ciliary arrays are studied analytically and numerically. We develop a simple model for ciliary motion that is complex enough to describe well the behavior of beating cilia but simple enough to study collective effects analytically. Beating cilia are described as phase oscillators moving on circular trajectories with a variable radius. This radial degree of freedom turns out to be essential for the occurrence of hydrodynamically induced synchronization of ciliary beating between neighboring cilia. The transitions to the synchronized and phase-locked state of two cilia and the formation of metachronal waves in ciliary chains with different boundary conditions are discussed.