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Motile cilia beat in an asymmetric fashion in order to propel the surroundingfluid. When many cilia are located on a surface, their beating can synchronisesuch that their phases form metachronal waves. Here, we computationally study amodel where each cilium is represented as a spherical particle, moving along atilted trajectory with a position-dependent active driving force and aposition-dependent internal drag coefficient. The model thus takes into accountall the essential broken symmetries of the ciliary beat. We show that takinginto account the near-field hydrodynamic interactions, the effective couplingbetween cilia can become nonreciprocal: the phase of a cilium is more stronglyaffected by an adjacent cilium on one side than by a cilium at the samedistance in the opposite direction. As a result, synchronisation starts from aseed at the edge of a group of cilia and propagates rapidly across the system,leading to a synchronisation time that scales proportionally to the lineardimension of the system. We show that a ciliary carpet is characterised bythree different velocities: the velocity of fluid transport, the phase velocityof metachronal waves and the group velocity of order propagation. Unlike insystems with reciprocal coupling, boundary effects are not detrimental forsynchronisation, but rather enable the formation of the initial seed.

Nonreciprocal interactions give rise to fast cilium synchronisation in finite systems