  We study antiferromagnetic, $S=1/2$ Heisenberg models with nearest and second
neighbor interactions on the one-fifth depleted square lattice which describes
the spin degrees of freedom in the spin-gap system CaV$_4$O$_9$. The
meta-plaquette expansion for the triplet excitation spectrum is extended to
fifth order, and the results are compared with experimental data on
CaV$_4$O$_9$. We attempt to locate the phase boundary between magnetically
ordered and gapped phases.
