A new measure to characterize multifractality of sleep electroencephalogram

Abstract

Traditional methods for nonlinear dynamic analysis, such as correlation dimension, Lyapunov exponent, approximate entropy, detrended fluctuation analysis, using a single parameter, cannot fully describe the extremely sophisticated behavior of electroencephalogram (EEG). The multifractal formalism reveals more "hidden" information of EEG by using singularity spectrum to characterize its nonlinear dynamics. In this paper, the zero-crossing time intervals of sleep EEG were studied using multifractal analysis. A new multifractal measure Δ as α was proposed to describe the asymmetry of singularity spectrum, and compared with the singularity strength range Δα that was normally used as a degree indicator of multifractality. One-way analysis of variance and multiple comparison tests showed that the new measure we proposed gave better discrimination of sleep stages, especially in the discrimination between sleep and awake, and between sleep stages 3 and 4.The research of nonlinear dynamics in electroencephalogram (EEG) has made much headway in recent years. Nonlinear analysis methods have been successfully applied to the studies of brain functions and pathological changes in EEG [1][2][3] . These studies also proved that EEG exhibited at least partly chaotic characteristics. The detrended fluctuation analysis (DFA) [4] , which has been widely used recently, revealed the longrange power-law correlation in EEG, indicating time scale invariant and fractal structure [5,6] . However, EEG is also rather noisy, displaying short-term decorrelation like white noise, and consequently, the EEG has been traditionally considered as a linear stationary random process. The paradoxical combination of short-term decorrelation and long-range correlation, stochastic and deterministic suggests that a single nonlinear parameter, such as largest Lyapunov exponent, correlation dimension, fractal dimension, scaling exponent, etc., may not be able to fully characterize the "stochastic chaos" (as named by Freeman [7] ) nature of EEG.The long time behavior of chaotic, nonlinear dynamic systems can often be characterized by (mono) fractal or multifractal measures. Monofractals are homogeneous in the sense that they have the same scaling properties, characterized by a single singularity exponent throughout the entire signal. In contrast, multifractals can be decomposed into many (possibly infinite) sub-sets characterized by different exponents. Multifractal signals are intrinsically more complex and inhomogeneous than monofractals. Multifractal models have been used to account for scale invariance properties of various objects in very different domains ranging from the energy dissipation or the velocity field in turbulent flows [8] to underlying hierarchical structure in proteins [9] . Physiologic signals generated by complex self-regulating systems, such as heartbeat interval, electrocardiogram, gait etc. have been proven to be multifractal, and the degree of multifractality often relates to pathological state or natural aging process [10][11][12] ....

Spatio-temporal analysis of monofractal and multifractal properties of the human sleep EEG

“…Fractal properties of the human sleep EEG have already been investigated by other research groups (Pereda et al, 1998;Shen et al, 2003;Lee et al, 2004;Acharya et al, 2005;Ma et al, 2006;Leistedt et al, 2007a,b), however, these studies were limited by low sampling rate (100 Hz in Acharya et al, 2005;Ma et al, 2006;Leistedt et al, 2007a,b;128 Hz in Shen et al, 2003) or low number of EEG channels (Fpz-Cz and Pz-Cz in Acharya et al, 2005;Ma et al, 2006;Cz in Leistedt et al, 2007a,b;C3 in Pereda et al, 1998;Shen et al, 2003;C4 in Lee et al, 2004). To assess MOF properties Pereda et al (1998) estimated the fractal exponentˇusing coarse grain spectral analysis.…”

“…Pereda et al (1998) also found higherˇvalues for deeper sleep stages. Ma et al (2006) assessed MUF properties of the sleep EEG estimating the range of the singularity strength and the asymmetry of the singularity spectrum. These measures also correlated with the sleep depth, exhibiting higher values for deeper sleep stages.…”

Multifractal mass exponent spectrum of complex physiological time series

“…Results of one sample sc4002e0 are presented in Figure 4 as an example, where Figure 4(a) gives the artificially judged hypnogram provided by the database according to Rechtschaffen and Kales (R&K) criterion [30], among which Wake denotes clear state, S1-S4 express gradually deepening sleep stages from degree 1 to 4, and REM represents rapid eye movement period. Arrows in this figure point out locations of sudden awakening during the course of sleep [15]. Figures 4(b) and (c) are the exhibitions calculated from K τ(q) and Δα respectively, which suggest that these indicators can all make good judgments on the above sleep states.…”

“…Since the conception of multifractal structure was first reported in 1986 by Halsey et al [5], it had been uncovered and greatly developed in a number of classical realms of science. In biology, important conclusion has been drawn, such as to the studies of DNA sequences [6,7], heartbeat information [8][9][10][11][12][13], cerebral electrical signals [14,15] and so on.…”

Complexity and characteristic frequency studies in ECG signals of mice based on multiple scale factors

“…Nevertheless, existing data also suggest that a single parameter or fractal dimension may be insufficient to characterize such complicated behaviors as human bioelectrical activities. Because of their multiple dimensions, multifractal theories [5][6][7][8][9][10][11][12][13][14][15][16] are more effective for examining the diverse levels of the fractal system.…”