Eckehard Olbrich

Dimensional complexity and spectral properties of the human sleep EEG

OBJECTIVE The relevance of the dimensional complexity (DC) for the analysis of sleep EEG data is investigated and compared to linear measures. METHODS We calculated DC of artifact-free 1 min segments of all-night sleep EEG recordings of 4 healthy young subjects. Non-linearity was tested by comparing with DC values of surrogate data. Linear properties of the segments were characterized by estimating the self-similarity exponent alpha based on the detrended fluctuation analysis which quantifies the persistence of the signal and by calculating spectral power in the delta, theta, alpha and sigma bands, respectively. RESULTS We found weak nonlinear signatures in all sleep stages, but most pronounced in sleep stage 2. Strong correlations between DC and linear measures were established for the self-similarity exponent alpha and delta power, respectively. CONCLUSIONS The dimensional complexity of the sleep EEG is influenced by both linear and nonlinear features. It cannot be directly interpreted as a nonlinear synchronization measure of brain activity, but yields valuable information when combined with the analysis of linear measures.

Chaos or noise: difficulties of a distinction

In experiments, the dynamical behavior of systems is reflected in time series. Due to the finiteness of the observational data set, it is not possible to reconstruct the invariant measure up to an arbitrarily fine resolution and an arbitrarily high embedding dimension. These restrictions limit our ability to distinguish between signals generated by different systems, such as regular, chaotic, or stochastic ones, when analyzed from a time series point of view. We propose to classify the signal behavior, without referring to any specific model, as stochastic or deterministic on a certain scale of the resolution epsilon, according to the dependence of the (epsilon,tau) entropy, h(epsilon, tau), and the finite size Lyapunov exponent lambda(epsilon) on epsilon.

Quantifying unique information

We propose new measures of shared information, unique information and synergistic information that can be used to decompose the mutual information of a pair of random variables (Y, Z) with a third random variable X. Our measures are motivated by an operational idea of unique information, which suggests that shared information and unique information should depend only on the marginal distributions of the pairs (X, Y) and (X,Z). Although this invariance property has not been studied before, it is satisfied by other proposed measures of shared information. The invariance property does not uniquely determine our new measures, but it implies that the functions that we define are bounds to any other measures satisfying the same invariance property. We study properties of our measures and compare them to other candidate measures.

Fading Signatures of Critical Brain Dynamics during Sustained Wakefulness in Humans

Sleep encompasses approximately a third of our lifetime, yet its purpose and biological function are not well understood. Without sleep optimal brain functioning such as responsiveness to stimuli, information processing, or learning may be impaired. Such observations suggest that sleep plays a crucial role in organizing or reorganizing neuronal networks of the brain toward states where information processing is optimized. Increasing evidence suggests that cortical neuronal networks operate near a critical state characterized by balanced activity patterns, which supports optimal information processing. However, it remains unknown whether critical dynamics is affected in the course of wake and sleep, which would also impact information processing. Here, we show that signatures of criticality are progressively disturbed during wake and restored by sleep. We demonstrate that the precise power-laws governing the cascading activity of neuronal avalanches and the distribution of phase-lock intervals in human electroencephalographic recordings are increasingly disarranged during sustained wakefulness. These changes are accompanied by a decrease in variability of synchronization. Interpreted in the context of a critical branching process, these seemingly different findings indicate a decline of balanced activity and progressive distance from criticality toward states characterized by an imbalance toward excitation where larger events prevail dynamics. Conversely, sleep restores the critical state resulting in recovered power-law characteristics in activity and variability of synchronization. These findings support the intriguing hypothesis that sleep may be important to reorganize cortical network dynamics to a critical state thereby assuring optimal computational capabilities for the following time awake.

A Geometric Approach to Complexity

We develop a geometric approach to complexity based on the principle that complexity requires interactions at different scales of description. Complex systems are more than the sum of their parts of any size and not just more than the sum of their elements. Using information geometry, we therefore analyze the decomposition of a system in terms of an interaction hierarchy. In mathematical terms, we present a theory of complexity measures for finite random fields using the geometric framework of hierarchies of exponential families. Within our framework, previously proposed complexity measures find their natural place and gain a new interpretation.

Shared Information—New Insights and Problems in Decomposing Information in Complex Systems

How can the information that a set {X 1,…,X n } of random variables contains about another random variable S be decomposed? To what extent do different subgroups provide the same, i.e. shared or redundant, information, carry unique information or interact for the emergence of synergistic information?

Complexity measures from interaction structures.

We evaluate information-theoretic quantities that quantify complexity in terms of kth-order statistical dependences that cannot be reduced to interactions among k-1 random variables. Using symbolic dynamics of coupled maps and cellular automata as model systems, we demonstrate that these measures are able to identify complex dynamical regimes.

INFERRING CHAOTIC DYNAMICS FROM TIME-SERIES : ON WHICH LENGTH SCALE DETERMINISM BECOMES VISIBLE

Abstract On large scales, a chaotic deterministic signal is indistinguishable from a random process. Determinism becomes visible only below a critical length scale. We analyse the dependence of this scale on the entropy of the signal and the minimal embedding dimension for state space reconstruction. The problem of the optimal choice of the delay time for flow data can also be discussed in this framework.

Analysis of oscillatory patterns in the human sleep EEG using a novel detection algorithm

The different brain states during sleep are characterized by the occurrence of distinct oscillatory patterns such as spindles or delta waves. Using a new algorithm to detect oscillatory events in the electroencephalogram (EEG), we studied their properties and changes throughout the night. The present approach was based on the idea that the EEG may be described as a superposition of stochastically driven harmonic oscillators with damping and frequency varying in time. This idea was implemented by fitting autoregressive models to the EEG data. Oscillatory events were detected, whenever the damping of one or more frequencies was below a predefined threshold. Sleep EEG data of eight healthy young males were analyzed (four nights per subject). Oscillatory events occurred mainly in three frequency ranges, which correspond roughly to the classically defined delta (0–4.5 Hz), alpha (8–11.5 Hz) and sigma (11.5–16 Hz) bands. Their incidence showed small intra‐ but large inter‐individual differences, in particular with respect to alpha events. The incidence and frequency of the events was characteristic for sleep stages and non‐rapid eye movement (REM)–REM sleep cycles. The mean event frequency of delta and sigma (spindle) events decreased with the deepening of sleep. It was higher in the second half of the night compared with the first one for delta, alpha and sigma oscillations. The algorithm provides a general framework to detect and characterize oscillatory patterns in the EEG and similar signals.

Dynamics of human sleep EEG

Abstract Several investigators of EEG time series reported a rejection of the null hypothesis of linear stochastic dynamics for epochs longer than 10 s . We examine whether this rejection is related to nonlinearity or to nonstationarity. Our approach is a combination of autoregressive (AR-) modeling and surrogate data testing. It is shown that the fraction of subsegments, for which the null hypothesis has to be rejected, increases with the length of the subsegments and can be related to fluctuations of the AR-coefficients on time scales in the range from 2 to 30 s .