  We investigate both continuous (second-order) and discontinuous (first-order)
transitions to macroscopic synchronization within a single class of discrete,
stochastic (globally) phase-coupled oscillators. We provide analytical and
numerical evidence that the continuity of the transition depends on the
coupling coefficients and, in some nonuniform populations, on the degree of
quenched disorder. Hence, in a relatively simple setting this class of models
exhibits the qualitative behaviors characteristic of a variety of considerably
more complicated models. In addition, we study the microscopic basis of
synchronization above threshold and detail the counterintuitive subtleties
relating measurements of time averaged frequencies and mean field oscillations.
Most notably, we observe a state of suprathreshold partial synchronization in
which time-averaged frequency measurements from individual oscillators do not
correspond to the frequency of macroscopic oscillations observed in the
population.
