A combined measure to differentiate EEG signals using fractal dimension and MFDFA-Hurst

Highlights

• The dynamics of the electroencephalogram signals of normal and epileptic patients is analyzed.

• Multifractal Detrended Fluctuation Analysis (MFDFA), Hurst exponent (H) and fractal dimension (D) are adopted.

• A new combined measure, namely the Combined Index (CI), is proposed.

• This procedure can avoid bias when a single index is chosen individually.

• The results indicate that the new CI is useful for investigating of the EEG dynamics and improve the efficiency of classification.

Abstract

The analysis of the brain electrical activity measured by means of electroencephalographic (EEG) records is a fundamental technique for the understanding and diagnosis of neurological diseases. This paper adopts the Multifractal Detrended Fluctuation Analysis, Hurst exponent (H) and fractal dimension (D) to analyze the dynamics of the EEG signals of normal and epileptic patients using a set of physiological time series. Furthermore, a new measure, namely the Combined Index, is proposed. The results indicate that the indices H and D are useful for building a combined measure of the EEG both for healthy and epileptic cerebral activity.

Introduction

Normal physiology of biological systems requires an intricate network to implement an efficient control function. These networks incorporate a mix of integration, differentiation, feedback loops, and other regulatory mechanisms that enable an organism to perform multiple activities, typical of complex systems with features adapting in time. The modeling of these phenomena is a challenging task and the non-stationary behavior of such mechanisms is an additional problem. In order to describe and quantify the dynamic characteristics of biological systems, different techniques related to complexity theory have been employed [1], [2], [3], [4], [5], [6], [7], [8].

The analysis of complex systems and the assessment of complexity emerged as important facets of the mathematical and physical sciences [9], [10], [11]. Complexity can be loosely defined as the difficulties faced when describing a signal or predicting its future behavior [12].

The historical development of the concepts of complexity has centered on measuring regularity using various metrics based on nonlinear time series (TS) analysis. The correlation dimension, entropy and Lyapunov exponents are indices often employed to measure the randomness and predictability of a given TS [13], [14], [15], [16], [17], [18].

Another alternative approach is to compute the fractal complexity of the TS. The Hurst exponent (H) and the fractal dimension (D) are measures of the fractal complexity or persistence of fractal processes such as fractional Gaussian noise or Brownian motion [19].

With age conditions and health disorders, there is a loss of complexity in the dynamics of many integrated physiological processes of an organism  [20], [21]. The estimates of complexity can be used to probe different aspects of complex signals, eventually affected by aging and disease, and can be applied to a range of physiological measures, including in respiratory signals [22], [23], [24], cardiac measures [25], [26], anesthesia dosage monitoring [27], [28] and electrophysiological signals from the brain.

The brain is recognized as one of the most complex dynamic systems due to the intricate structure of the neural network. The synapses allow the information transmission among neurons. Through the axon, electrical signs may be (or not) propagated to other cells. The so-called brain activity is formed by the superposition of all those signals and the correlation study of the electrical activity provides significant information about the system dynamics [29].

The recording of cerebral electrical activity can be obtained by means of electroencephalography. In practice, the electroencephalogram (EEG) is a non-invasive technique since it is performed through disc-shaped electrodes arranged in the scalp of the patients. The EEG is a bioelectrical manifestation of the nervous system activity. Often, the EEG seeks to pinpoint some problem as, for example, the location of epileptic foci detected from abnormal electrical activity in certain regions of the brain [30].

Epilepsy [31] is a serious neurological disease that affects people regardless of age, gender, or ethnicity. According to the World Health Organization, epilepsy is a serious brain disorder that affects approximately 50 million people (about 1% of the world’s population), 80% of whom live in underdeveloped, or developing countries. The problem is revealed by the occurrence of epileptic seizures, directly affecting the quality of life of patients, in the most varied aspects. Epilepsy is a disorder of the central nervous system characterized predominantly by spontaneous, recurrent and unpredictable interruptions of the normal functioning of the brain, called epileptic seizures. An epileptic seizure is defined as a transient occurrence of signs and/or symptoms caused by synchronous or excessive abnormal cerebral activity [32], [33].

The so-called epilepsy does not refer to a single disease. In fact, it encompasses a variety of disorders and conditions, that have in common an abnormally predisposition for seizures, reflecting the underlying brain dysfunction resulting from different causes. Often epilepsy does not have an identifiable cause, but in some cases it may be related to issues, such as, genetic factors, congenital abnormalities, birth trauma, metabolic or chemical imbalance in the body, head trauma, brain infection, brain tumors, stroke, cortical dysplasia, among others [31]. Since epilepsy is caused by abnormal brain activity, epileptic seizures can affect any process coordinated by the brain. Thus, the signs and symptoms of an epileptic seizure vary on a case-by-case basis and may involve changes in behavior, mood, sensations (visual, olfactory, auditory) and other cognitive functions, as well as loss of consciousness and movement disorders.

Several studies focused on the detection of epilepsy from EEG signals using non-linear methods that can detect and quantify both linear and nonlinear mechanisms and, thereby, reflect somehow the characteristics of the EEG [34], [35], [36], [37], [38], [39], [40], [41]. Non-linear features may be able to extract the hidden complexities embedded in the EEG TS. One of the earliest studies on this topic was developed by Babloyantz et al. [42], that used non-linear parameters such as the correlation dimension and the largest Lyapunov exponent to study the sleep wave signal. Such studies extended the potential application areas of EEG nonlinear analysis methodologies, but the identification of epilepsy still remains one relevant field of research [39].

The dynamic TS analysis of EEG signals may reveal complex phenomena associated with long-range correlation and distinct classes of non-linear interactions. This type of analysis is capable of providing useful diagnostic and prognostic information. Nonetheless, most of the papers found in the literature about EEG consider a single index. Moreover, when several indices are adopted, the fusion of information is not considered and merely an individual analysis of each index is followed. The analysis based on a single index is not sufficient to capture the EEG properties and, consequently, such aproaches lead to limited results [43].

Bearing these facts in mind, three mathematical tools [44], [45], [46], namely the Multifractal Detrended Fluctuation Analysis (MFDFA), Hurst exponent (H) and fractal dimension (D), are adopted in this paper to investigate the EEG of normal and epileptic humans. A new measure embedding the H and D indices is also designed. This approach eliminates the limitations that can occur when adopting a single index for EEG analysis.

The main goal of the paper is to have a deeper insight into the relation between loss of information processing in the brain and changes in the randomness pattern of the EEG when using some indices of assessment. A direct comparison between the indices for each set is drawn in order to evaluate how they perform when identifying whether there is, or not, a dependence between sets, or, by other words, to determine if the signal can indicate if the patient is healthy or not.

The paper is structured as follows. Section 2 describes the data characteristics and the methods of EEG used in the follow-up. Section 3 presents the results of the proposed techniques and Section 4 their discussion. Finally, Section 5 summarizes the main conclusions.