  We study the phase diagram of a generalized Winfree model. The modification
is such that the coupling depends on the fraction of synchronized oscillators,
a situation which has been noted in some experiments on coupled Josephson
junctions and mechanical systems. We let the global coupling k be a function of
the Kuramoto order parameter r through an exponent z such that z=1 corresponds
to the standard Winfree model, z<1 strengthens the coupling at low r (low
amount of synchronization) and, at z>1, the coupling is weakened for low r.
Using both analytical and numerical approaches, we find that z controls the
size of the incoherent phase region, and one may make the incoherent behavior
less typical by choosing z<1. We also find that the original Winfree model is a
rather special case, indeed the partial locked behavior disappears for z>1. At
fixed k and varying gamma, the stability boundary of the locked phase
corresponds to a transition that is continuous for z<1 and first-order for z>1.
This change in the nature of the transition is in accordance with a previous
study on a similarly modified Kuramoto model.
