Cilia and flagella are essential building blocks for biological fluid transport and locomotion at the micron scale. They often beat in synchrony and may transition between different synchronization modes in the same cell type. Here, we investigate the behavior of elastic microfilaments, protruding from a surface and driven at their base by a configuration-dependent torque. We consider full hydrodynamic interactions among and within filaments and no slip at the surface. Isolated filaments exhibit periodic deformations, with increasing waviness and frequency as the magnitude of the driving torque increases. Two nearby but independently-driven filaments synchronize their beating in-phase or anti-phase. This synchrony arises autonomously via the interplay between hydrodynamic coupling and filament elasticity. Importantly, in-phase and anti-phase synchronization modes are bistable and co-exist for a range of driving torques and separation distances. These findings are consistent with experimental observations of in-phase and anti-phase synchronization in the biflagellate textit{Chlamydomonas reinhardtii} and could have important implications on understanding the biophysical mechanisms underlying transitions between multiple synchronization modes.
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