Emergence of polar order and cooperativity in hydrodynamically coupled model cilia

As a model of ciliary beat, we use two-state oscillators that have a defined direction of oscillation and have strong synchronization properties. By allowing the direction of oscillation to vary according to the interaction with the fluid, with a timescale longer than the timescale of synchronization, we show in simulations that several oscillators can align in a direction set by the geometrical configuration of the system. In this system, the alignment depends on the state of synchronization of the system, and is therefore linked to the beat pattern of the model cilia. By testing various configurations from two to 64 oscillators, we deduce empirically that, when the synchronization state of neighbouring oscillators is in phase, the angles of the oscillators align in a configuration of high hydrodynamic coupling. In arrays of oscillators that break the planar symmetry, a global direction of alignment emerges reflecting this polarity. In symmetric configurations, where several directions are geometrically equivalent, the array still displays strong internal cooperative behaviour. It also appears that the shape of the array is more important than the lattice type and orientation in determining the preferred direction.