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

The excitatory-inhibitory branching process: a parsimonious view of cortical asynchronous states, excitability, and criticality