  Theoretical studies of synchronization are usually based on models of coupled
phase oscillators which, when isolated, have constant angular frequency.
Stochastic discrete versions of these uniform oscillators have also appeared in
the literature, with equal transition rates among the states. Here we start
from the model recently introduced by Wood et al. [Phys. Rev. Lett. 96}, 145701
(2006)], which has a collectively synchronized phase, and parametrically modify
the phase-coupled oscillators to render them (stochastically) nonuniform. We
show that, depending on the nonuniformity parameter $0\leq \alpha \leq 1$, a
mean field analysis predicts the occurrence of several phase transitions. In
particular, the phase with collective oscillations is stable for the complete
graph only for $\alpha \leq \alpha^\prime < 1$. At $\alpha=1$ the oscillators
become excitable elements and the system has an absorbing state. In the
excitable regime, no collective oscillations were found in the model.
