  Cellular appendages conferring motility, such as flagella or cilia, are known
to synchronise their periodic beats. The origin of synchronisation is a
combination of long-range hydrodynamic interactions with physical mechanisms
allowing the phases of these biological oscillators to evolve. Two of such
mechanisms have been identified by previous work, the elastic compliance of the
periodic orbit or oscillations driven by phase-dependent biological forcing. To
help uncover the physical mechanism for hydrodynamic synchronisation most
essential overall in biology, we theoretically investigate the effect of strong
confinement on the effectiveness of hydrodynamic synchronisation. We use
minimal models where appendages are modelled as rigid spheres forced to move
along circular trajectories near a rigid surface. Strong confinement is
modelled by adding a second nearby surface, parallel to the first one, where
the distance between the surfaces is much smaller than the typical distance
between the cilia. We calculate separately the impact of confinement on the
synchronisation dynamics of the elastic compliance and the force modulation
mechanisms and compare our results to the case with no confinement. Applying
our results to the biologically-relevant situation of nodal cilia, we show that
force modulation is a mechanism that leads to phase-locked states under strong
confinement that are very similar to those without confinement as a difference
with the elastic compliance mechanism. Our results point therefore to the
robustness of force modulation for synchronisation, an important feature for
biological dynamics that suggests it could be the most essential physical
mechanism overall in arrays of nodal cilia. We further examine the distinct
situation of primary cilia and show in that case that the difference in
robustness of the mechanisms is not as pronounced but still favours the force
modulation.
