Emergence of multiple synchronization modes in hydrodynamically-coupled cilia

Motile cilia and flagella exhibit different phase coordinations. For example, closely swimming spermatozoa are observed to synchronize together; bi-flagellates \textit{Chlamydomonas} regulate the flagella in a ``breast-stroke'' fashion; cilia on the surface of \textit{Paramecium} beat in a fixed phase lag in an orchestrated wave like fashion. Experimental evidence suggests that phase coordinations can be achieved solely via hydrodynamical interactions. However, the exact mechanisms behind it remain illusive. Here, adapting a ``geometric switch'' model, we observe different synchronization modes in pairs of hydrodynamically-coupled cilia by changing physical parameters such as the strength of the cilia internal motor and the separation distance between cilia. Interestingly, we find regions in the parameter space where the coupled cilia reach stable phase coordinations and regions where the phase coordinations are sensitive to perturbations. We also find that leaning into the fluid reduces the sensitivity to perturbations, and produces stable phase coordination that is neither in-phase nor anti-phase, which could explain the origin of metachronal waves in large cilia populations.