Examination of scale-invariant characteristics of epileptic electroencephalograms using wavelet-based analysis☆
Graphical abstract
Introduction
Physiological signals and systems have been shown to exhibit an extraordinary range of patterns and behaviors [1]. The introduction of the concept of fractals and corresponding quantitative measures has provided additional avenues of investigation and applications in biology and medicine including neuroscience [2], [3]. The mathematical concept of a fractal is commonly associated with irregular objects that exhibit a geometric property called scale-invariance or self-similarity [1], [4]. Fractal forms are composed of subunits resembling the structure of the macroscopic object [1] which in nature can emerge from statistical scaling behavior in the underlying physical phenomena [5].
The existence of fractal behaviors in the brain has been widely accepted at various levels including anatomic, functional, pathological, molecular, and epigenetic [2]. Although such characteristics and behaviors may be visually apparent, they can be difficult to capture using conventional computational analyses and measures [6]. Conventional methods such as Fourier analysis are simple and quick but may lead to erroneous interpretations [7]. Computational analyses and measures for quantifying and characterizing fractal behaviors and complexity such as the Hurst exponent, the scaling exponent, fractal dimension and various forms of entropy [8], [9], [10] have been applied in epilepsy research to characterize brain behaviors and for epileptic seizure detection. Recent studies (e.g., [11], [12]) documented fractal organization of the brain and nervous system.
An important class of statistical scale-invariant or self-similar random processes is the processes [5]. A traditional mathematical model and the empirical properties of processes have largely been inspired by the fractional Brownian motion framework [5], [13], [14]. In general, models of processes are represented using a frequency domain characterization and exhibit power law behavior [15] that can be characterized in the frequency domain by . The wavelet transform is a natural tool for characterizing self-similar or scale-invariant signals and plays a significant role in the study of self-similar signals and systems [5], in particular processes [5], [13]. The spectral exponent that specifies the distribution of power in processes from low to high frequencies can be characterized in terms of the slope of the log-variance of the wavelet coefficients versus scale graph.
The Hurst exponent H is the most common quantitative measure that is used to characterize the long-range correlation or self-similarity of signals. There are a number of methods for estimating the Hurst exponent H. The most commonly used method is the rescaled range (R/S) analysis [16]. Another method is detrended fluctuation analysis (DFA), originally introduced in [17], is a robust method for estimating the Hurst exponent H. The Hurst exponent is in the range . When , the process is considered to have short-range correlation [21], indicating that an increase in the process is more probably followed by a decrease and vice versa (anti-persistence). The process is considered to have long-range correlation when (persistence), indicating that an increase in the process is more probably followed by an increase and vice versa [18]. A value of indicates that the signal is uncorrelated.
In this study, we apply the wavelet-based representation for processes and use the spectral exponent to examine the scale-invariant or self-similar characteristics of the dynamics of the brain of subjects with epilepsy. A number of different wavelet bases are used to investigate their effects on the estimation of the spectral exponent. Furthermore, the scale-invariant or self-similar characteristics of intracranial EEG data obtained during different pathological states of the brain (ictal and interictal states) assessed from the spectral exponent are compared to the Hurst exponent H.
The remainder of this paper is organized as follows. Section 2 presents background information and methods. Data and the experimental setup are discussed in Section 3. Section 4 details and discusses the computational results. Finally, Section 5 summarizes the results and provides concluding remarks.
Section snippets
Processes
In general, models of processes are represented using a frequency domain characterization. The dynamics of processes exhibit power-law behavior [15] and can be characterized in the form of [13]over several decades of frequency , where and are, respectively, the Fourier transform and the variance of the signal , and denotes the spectral exponent. The spectral exponent specifies the distribution of spectral content from low to high frequencies, and an
Experimental data
The data used in this study consists of two sets of intracranial EEG data of epilepsy patients obtained from the Department of Epileptology, University of Bonn (available online at http://epileptologie-bonn.de/cms/front_content.php?idcat=193), that originated from the study presented in [24]. The intracranial EEG data sets, referred to as F and S, were recorded using intracranial electrodes from five epilepsy patients whom had achieved complete seizure control after resection of one of the
Spectral exponents of the intracranial EEG signals
The mean and the standard deviation of the of the wavelet coefficients of the intracranial EEG signals of data sets F and S obtained using Db1, Db2, Db6, and Db25 are summarized in Table 1. It is observed that the of the wavelet coefficients of the intracranial EEG signals of both data sets tends to increase from a lower level to a higher level. At any level, the of the wavelet coefficients of the intracranial EEG signals of data set S tends to be higher than that of
Conclusions
The computational results suggest that the intracranial EEG signals obtained during seizure activity tend to be less scale-invariant or self-similar than that during non-seizure periods. This structural characteristics of the intracranial EEG signals can be clearly observed as the intracranial EEG signals obtained during epileptic seizure activity tend to have dominant scales or frequencies. These scale-invariant or self-similar characteristics of the intracranial EEG data observed from the
Acknowledgments
This work is supported by a TRF-CHE Research Grant for New Scholar, jointly funded by the Thailand Research Fund (TRF) and the Commission on Higher Education (CHE), the Ministry of Education, Thailand, under Contract No. MRG5280189.
Suparerk Janjarasjitt received his M.S. degree in 2002 and Ph.D. degree in 2006 in systems and control engineering from Case Western Reserve University, Cleveland, Ohio, USA. He is currently an Assistant Professor at the Department of Electrical and Electronic Engineering, Ubon Ratchathani University, Ubon Ratchathani, Thailand.
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Suparerk Janjarasjitt received his M.S. degree in 2002 and Ph.D. degree in 2006 in systems and control engineering from Case Western Reserve University, Cleveland, Ohio, USA. He is currently an Assistant Professor at the Department of Electrical and Electronic Engineering, Ubon Ratchathani University, Ubon Ratchathani, Thailand.
Kenneth A. Loparo received his Ph.D. degree in systems and control engineering from Case Western Reserve University, Cleveland, Ohio, USA in 1977. He is the Nord Professor of Engineering and currently serves as the Chair of Department of Electrical Engineering and Computer Science, Case Western Reserve University.
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Reviews processed and approved for publication by Editor-in-Chief Dr. Manu Malek.