We discuss the critical brain hypothesis and its relationship with intermittent renewal processes displaying power-law decay in the distribution of waiting times between two consecutive renewal events. In particular, studies on complex systems in a "critical" condition show that macroscopic variables, integrating the activities of many individual functional units, undergo fluctuations with an intermittent serial structure characterized by avalanches with inverse-power-law (scale-free) distribution densities of sizes and inter-event times. This condition, which is denoted as "fractal intermittency", was found in the electroencephalograms of subjects observed during a resting state wake condition. It remained unsolved whether fractal intermittency correlates with the stream of consciousness or with a non-task-driven default mode activity, also present in non-conscious states, like deep sleep. After reviewing a method of scaling analysis of intermittent systems based of eventdriven random walks, we show that during deep sleep fractal intermittency breaks down, and reestablishes during REM (Rapid Eye Movement) sleep, with essentially the same anomalous scaling of the pre-sleep wake condition. From the comparison of the pre-sleep wake, deep sleep and REM conditions we argue that the scaling features of intermittent brain events are related to the level of consciousness and, consequently, could be exploited as a possible indicator of consciousness in clinical applications.

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