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Ornstein and Shields (Advances in Math., 10:143-146, 1973) proved that Brownian motion reflected on a bounded region is an infinite entropy Bernoulli flow and thus Ornstein theory yielded the existence of a measure-preserving isomorphism between any two such Brownian motions. For fixed h >0, we construct by elementary methods, isomorphisms with almost surely finite coding windows between Brownian motions reflected on the intervals [0, qh] for all positive rationals q., Published at https://doi.org/10.1214/19-AOP1412 in the Annals of Probability by the Institute of Mathematical Statistics

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. &nbsp

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 macro- scopic 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 in- termittency”, 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 event- driven random walks, we show that during deep sleep fractal intermittency breaks down, and re- establishes 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.