In regard to an unknown system such as the brain, researchers have developed the approaches for probing the system characteristics by analyzing its output or response recordable (e.g., Galka, 2000). As in most cases multi-channel neurophysiological signals are simultaneously recorded, univariate analysis alone cannot accomplish the assessment of the interdependence among channels. Therefore, it is necessary to make use of the multivariate analysis giving more insights into the brain dynamical mechanisms. Despite their capability of approaching specific aims, univariate and multivariate time-series analysis are mostly based on the widely-used, conventional time-domain and frequency-domain approaches (see, e.g., Bendat and Piersol, 2000). Unfortunately, these methods based on linear assumption cannot give any information about the nonlinear features of the signal.
Neurons are highly nonlinear, moreover, have been demonstrated to exhibit chaotic behavior (Matsumoto and Tsuda, 1988). Due to the intrinsic nonlinearity of neuronal activities, their integrative activities constituting the brain functions are nonlinear based on a sound hypothesis. Thus, other techniques based on the nonlinear dynamic theory have been introduced and been proved useful in EEG analysis (van Gils et al., 1997). These methods have been used to quantify underlying brain dynamics and evaluate EEG spatio-temporal complexity since two decades ago (Babloyantz et al., 1985; Babloyantz and Destexhe, 1986; Lo and Principe, 1989; Rapp et al., 1989; Pijn et al., 1991; van Putten, 2001). First encouraging results claimed that EEG signals showed chaotic structure (Babloyantz et al., 1985), but further studies did not find any strong evidence of chaos in EEG (Pijn, 1990, Theiler et al., 1992 and Theiler and Rapp, 1996). In the resent years, it is accepted that EEG signals are, at least in a general sense, not (low-dimensionally) chaotic (Lehnertz et al., 2000). In spite of that, nonlinear chaotic measures are still used for a more
practical goal even if there is no sign of chaos. Invariant quantities from the representation of the signals in the phase space may reveal nonlinear structures which are inaccessible by standard linear approaches (Stam, 2005).
Methods for univariate nonlinear time series analysis were originally applied to neurophysiological data about two decades ago (Babloyantz et al., 1985). Most popular tools from nonlinear dynamical theory used for EEG analysis are dimensional computation.
Complexity measure reflecting the dimensionality of underlying CNS dynamics provides a macroscopic view and thus the first measurand for quantifying an unknown system.
Thereby, this parameter denotes the number of state variables required to describe the temporal dynamics of the EEG signal (Grassberger & Procaccia, 1983) and provides an index that has been roughly interpreted as a measure of the irregularity or complexity of EEG dynamics. A number of studies have reported the fruitful results of characterizing the dynamic behavior of the CNS (central nervous system) under various physiological or mental states (for recent surveys: Elbert et al., 1994; Korn & Faure, 2003; Segundo, 2001;
Stam, 2005). It is well known that the dimensional complexity of the human EEG increases during various types of stimulation such as imagery (Schupp et al., 1994) and mental activity (Rapp et al., 1989; Mölle et al., 1999). Many studies also investigated the relationship between “brain complexity” and different states of consciousness. For example, correlation dimension has been useful in sleep-wake research (Pereda et al., 1998; Pradhan et al., 1995) and in studies of the depth of anesthesia (van den Broek et al., 2005; Widman et al., 2000).
Although univariate nonlinear method, such as correlation dimension, furnishes important information about the CNS characteristics, they mostly suffer from the problems of indirect estimation, computational inefficiency and bias from implementing parameters (Lo and Principe, 1989; Yaylali, 1996; Lo and Chung, 2000). To deal with the problems, we introduced the method “complexity index (δ)” (Lo and Chung, 2000; Lo and Chung, 2001)
with an efficient algorithm into long-term EEG analysis. Details of the algorithm are illustrated in Chapter IV.
In the past few years, researchers in engineering and medicine began employing several nonlinear multivariate techniques in neurophysiology. A number of studies have proposed the viewpoint of considering brain dynamics as a large ensemble of coupled nonlinear dynamical subsystems. Accordingly, significant nonlinear synchronization has been detected on the macroscopic scales of EEG channels in healthy subjects (Breakspear
& Terry, 2000a, 2002b; Stam et al., 1996, 2003; Feldmann & Bhattacharya, 2004). Various types of synchronization based on nonlinear dynamical theory have been demonstrated to be the more powerful mechanism than narrow-band frequency synchronization (e.g.
coherence function). Two relevant concepts are: generalized synchronization (Rulkov et al., 1995), a state in which a functional dependence between the systems exist, and phase synchronization (Rosenblum et al., 1996), a state in which the phases of the systems are correlated whereas their amplitudes may not be. In brief, these measures quantify, for a short time, the grade of predicting the state-space evolution in one system by that in the other, simultaneous system.
By analyzing the reconstructed phase space, such invariant quantities are theoretically useful in neurophysiology due to their ability to detect nonlinear interactions hidden to standard linear approaches. Nevertheless, the application of these methods to neurophysiological signals is not a plain subject. On the one hand, these signals are often noisy, non-stationary and of finite length. On the other hand, theoretical studies indicated that these indexes are not able to reveal any causal relationship among signals (Quian Quiroga et al., 2000). In this work, we go through these questions by using the modified similarity index, a robust set of interdependence measures (Arnhold et al., 1999; Quian Quiroga et al., 2000, 2002), for the analysis of meditation EEG. Instead of estimating predictions, similarity index quantifies how neighborhoods (i.e., recurrences) in one
attractor map into the other. This method has the advantages of sensibility to nonlinear interdependence and potential of detecting asymmetric relationships (Arnhold et al., 1999;
Le van Quyen, 1998). We thus investigated the capability of this multivariate method in characterizing the nonlinear interdependence behaviors of brain dynamics under Chan meditation. Details of the algorithm are illustrated in Chapter IV.