  Thermally pulsing asymptotic giant branch models of globular cluster stars
are calculated using a synthetic model with the goal of reproducing the
chemical composition, core masses and other observational parameters of the
four known globular cluster planetary nebulae as well as roughly matching the
overall cluster properties. The evolution of stars with an enhanced helium
abundance ($Y$) and blue stragglers are modeled. New pre-thermally pulsing
asymptotic giant branch mass-losses for red giant branch and early asymptotic
giant branch stars are calculated from the Padova stellar evolution models
\citep{berta,bertb}. The new mass-losses are calculated to get the relative
differences in mass-losses due to enhanced helium abundances.
  The global properties of the globular cluster planetary nebula are reproduced
with these models. The metallicity, mass of the central star, overall
metallicities, helium abundance and the nebular mass are matched to the
observational values. Globular cluster planetary nebulae JaFu 1 and JaFu 2 are
reproduced {\it by assuming progenitor stars} with masses near the typical main
sequence turn-offs of globular clusters and with enhanced helium abundances
very similar to the enhancements inferred from fitting isochrones to globular
cluster colour-magnitude diagrams. The globular cluster PN GJJC-1 can be
roughly fit by a progenitor star with very extreme helium enhancement
($Y\approx0.40$) near the turn-off producing a central star with the same mass
as inferred by observations and a very low nebular mass. The abundances and
core mass of planetary nebula Ps 1 and its central star (K648) are reproduced
by a blue straggler model. However, it turned out to be impossible to reproduce
its nebular mass and it is concluded some kind of binary scenario may be needed
to explain K648.
