  The human cortex is never at rest but in a state of sparse and noisy neural
activity that can be detected at broadly diverse resolution scales. It has been
conjectured that such a state is best described as a critical dynamical process
-- whose nature is still not fully understood -- where scale-free avalanches of
activity emerge at the edge of a phase transition. In particular, some works
suggest that this is most likely a synchronization transition, separating
synchronous from asynchronous phases. Here, by investigating a simplified model
of coupled excitable oscillators describing the cortex dynamics at a mesoscopic
level, we investigate the possible nature of such a synchronization phase
transition. Within our modeling approach, we conclude that -- in order to
reproduce all key empirical observations, such as scale-free avalanches and
bistability, on which fundamental functional advantages rely -- the transition
to collective oscillatory behavior needs to be of an unconventional hybrid
type, with mixed features of type-I and type-II excitability, opening the
possibility for a particularly rich dynamical repertoire.
