  It is now well established that nearby beating pairs of eukaryotic flagella
or cilia typically synchronize in phase. A substantial body of evidence
supports the hypothesis that hydrodynamic coupling between the active
filaments, combined with waveform compliance, provides a robust mechanism for
synchrony. This elastohydrodynamic mechanism has been incorporated into
`bead-spring' models in which flagella are represented by microspheres tethered
by radial springs as internal forces drive them about orbits. While these
low-dimensional models reproduce the phenomenon of synchrony, their parameters
are not readily relatable to those of flagella. More realistic models which
reflect the elasticity of the axonemes and active force generation take the
form of fourth-order nonlinear PDEs. While computational studies have shown
synchrony, the effects of hydrodynamic coupling between nearby filaments
governed by such models have been theoretically examined only in the regime of
interflagellar distances $d$ large compared to flagellar length $L$. Yet, in
many biological situations $d/L \ll 1$. Here, we first present an asymptotic
analysis of the hydrodynamic coupling between two filaments in the regime $d/L
\ll 1$, and find that the form of the coupling is independent of the details of
the internal forces that govern the motion of the filaments. The analysis is
like the localized induction approximation for vortex filament motion, extended
to the case of mutual induction. To understand how the coupling mechanism leads
to synchrony of extended objects, we introduce a heuristic model of flagellar
beating, a single fourth-order nonlinear PDE whose form is derived from
symmetry considerations, the physics of elasticity, and the overdamped nature
of the dynamics. Analytical and numerical studies of this model illustrate how
synchrony between two filaments is achieved through the asymptotic coupling.
