  The branching process is the minimal model for propagation dynamics,
avalanches and criticality, broadly used in neuroscience. A simple extension of
it, adding inhibitory nodes, induces a much-richer phenomenology, including, an
intermediate phase, between quiescence and saturation, that exhibits the key
features of "asynchronous states" in cortical networks. Remarkably, in the
inhibition-dominated case, it exhibits an extremely-rich phase diagram, that
captures a wealth of non-trivial features of spontaneous brain activity, such
as collective excitability, hysteresis, tilted avalanche shapes, and partial
synchronization, allowing us to rationalize striking empirical findings within
a common and parsimonious framework.
