On the meaning of Hurst entropy applied to EEG data series

Abstract

The Hurst exponent estimates the degree of self-similarity and predictability of a time series, which, under this nonlinear statistical model, can adopt two opposing tendencies with respect to the way these data series are mobilized over time. Persistent and anti-persistent series are thus described, depending on whether the Hurst value describing them is greater or less than 0.5 in a range of 0 to 1. Those series that show "Hurst effect" (persistent series with H>0.5) are potentially predictable in the short or medium term, which makes them especially interesting for any predictive interest in science or economics. In the brain, when the oscillatory electrical activity of an EEG is analyzed, the time series that make up a complete recording include a variable number of sources, electrodes or channels, which detect the electrical activity of millions of pyramidal neurons at the level of the cerebral cortex. When this signal is decomposed into its frequency components, its spectrum ranges from the slowest waves, or Delta (0.1-4Hz), to the fastest, Gamma (>30 Hz). In the intermediate range, the Theta (4-8Hz); Alpha (8-12Hz); and Beta (13-30Hz) oscillations complete the range of the EEG spectrum. When estimating the Hurst exponent by filtered intervals of the total frequency range, only for the Delta band, the H values exceed H=0.5. The Hurst values for Delta were close to H = 0.85 and all the rest of the spectrum falls to values less than 0.5, in a range from 1.5<H<3.5.In the present work, we extend the interpretation of the Hurst exponent applied to electroencephalographic time series, and we propose to pay special attention to the so-called anti-persistent series (H<0.5) which, overshadowed by the predictive promises of the Hurst effect of persistent series, we believe may hide the very long-term persistence of complex homeostatic processes, stabilized in the human brain over millions of years of evolution.