Brain entropy, fractal dimensions and predictability: A review of complexity measures for EEG in healthy and neuropsychiatric populations

Abstract

There has been an increasing trend towards the use of complexity analysis in quantifying neural activity measured by electroencephalography (EEG) signals. On top of revealing complex neuronal processes of the brain that may not be possible with linear approaches, EEG complexity measures have also demonstrated their potential as biomarkers of psychopathology such as depression and schizophrenia. Unfortunately, the opacity of algorithms and descriptions originating from mathematical concepts have made it difficult to understand what complexity is and how to draw consistent conclusions when applied within psychology and neuropsychiatry research. In this review, we provide an overview and entry-level explanation of existing EEG complexity measures, which can be broadly categorized as measures of predictability and regularity. We then synthesize complexity findings across different areas of psychological science, namely, in consciousness research, mood and anxiety disorders, schizophrenia, neurodevelopmental and neurodegenerative disorders, as well as changes across the lifespan, while addressing some theoretical and methodological issues underlying the discrepancies in the data. Finally, we present important considerations when choosing and interpreting these metrics.

Keywords: EEG; complexity; entropy; fractal dimension; psychopathology.

FIGURE 1

Complexity measures can be structured broadly into indices of predictability and regularity. Measures of predictability capture either spatial or temporal dimensionality of the dynamical system, and measures of regularity reflect the complexity of the system's output either on single scales or multiscales.

FIGURE 2

Time‐delay embedding method to reconstruct an attractor in the phase space with delay (τ) = 2 and number of dimensions (m) = 3. The top left panel shows an example of a simulated signal. The middle left panel is the time series but delayed by 1 τ of 2 s. the bottom left panel is the time series delayed by 2 τ. the three time series are projected to a 3D space where each state vector (solid point) is plotted with the coordinates provided by the respective time series, forming an attractor (see plot on the right).

FIGURE 3

Illustrations of the difference between complexity, order and randomness, using various examples, namely, text, signal and pictorial examples. In the text example, words are arranged in alphabetical order in the ordered output, indiscriminately arranged in the random output and structured according to semantic and syntactic rules in the complex output. In the signal example, the ordered signal contains data points sorted according to amplitude, the random signal consists of data points sampled at random and the complex signal is a mixture of signals of multiple frequencies. In the pictorial example, pixels are vertically ordered by luminance in the ordered output, scrambled in the random output, and the complex arrangement of pixels creates a meaningful picture. The aim is to show that complex is not synonymous with random, an important conceptual distinction in complexity science.