  An ensemble of pulse-coupled phase-oscillators is thoroughly analysed in the
presence of a mean-field coupling and a dispersion of their natural
frequencies. In spite of the analogies with the Kuramoto setup, a much richer
scenario is observed. The "synchronised phase", which emerges upon increasing
the coupling strength, is characterized by highly-irregular fluctuations: a
time-series analysis reveals that the dynamics of the order parameter is indeed
high-dimensional. The complex dynamics appears to be the result of the
non-perturbative action of a suitably shaped phase-response curve. Such
mechanism differs from the often invoked balance between excitation and
inhibition and might provide an alternative basis to account for the
self-sustained brain activity in the resting state. The potential interest of
this dynamical regime is further strengthened by its (microscopic) linear
stability, which makes it quite suited for computational tasks. The overall
study has been performed by combining analytical and numerical studies,
starting from the linear stability analysis of the asynchronous regime, to
include the Fourier analysis of the Kuramoto order parameter, the computation
of various types of Lyapunov exponents, and a microscopic study of the
inter-spike intervals.
