a b s t r a c t While several mental functions are characterized by parallel computation performed by moduli in the cortex, consciousness is sustained by a serial global integration: a single scene at a time takes place. Studies on complex systems show that macroscopic variables, integrating many components activities, undergo fluctuations with an intermittent serial structure when the system is in a state called ''criticality'', characterized by avalanches with inverse-power-law (scale-free) distribution densities of sizes and inter-event times. Criticality has been established in human brain dynamics during wakefulness. Here we review how the critical hypothesis is able to explain many recent studies on brain complex dynamics. We focus, in particular, on the global, serial, intermittent behavior that can be assessed via high-density electroencephalograms, studying transitions between metasta-ble states. Established as it is during wakefulness, it remained unsolved whether this global intermittent dynamics correlates with consciousness or with a non-task-driven default mode, also present in non-conscious states, like deep (NREM) sleep. Here we show that in NREM sleep seriality breaks down, and re-establishes during REM sleep (dreams), with unaltered spacial structure, in terms of complex branching of avalanches. We conjecture that this connectivity is exploited in NREM sleep by neural bistability, resetting and ''par-allelizing'' portions of the cortex.