**3.3 Synchronization**

While signal complexity is a property of a single time series or EEG channel, transient synchronized activity is a measure of the interaction between different channels and an indication of communication and coordination between different brain regions. Synchronization may be used as a marker for diagnosing underlying mental disorders that involve aberrant long-range connectivity in the brain and may also reveal causal mechanisms (Whittington, 2008). The complexity of synchronization patterns appears to change during network development and reflects different neural wiring schemes and levels of cluster organization (Fuchs et al., 2007).

The synchronization patterns of complex networks have been shown to be closely related to the topology of the network (Arenas et al., 2006) and are related to brain connectivity (Sakkalis et al., 2008). Synchronization between sensors is an indicator of connectivity between brain regions on a scale commensurate with the sensor spacing. EEG signals are believed to derive from pyramidal cells aligned in parallel in the cerebral cortex and hippocampus (Sörnmo and Laguna, 2005), which act as many interacting nonlinear oscillators (Nunez and Srinivasan, 2006). Synchronization between gamma activity (typically defined to be 30-50 Hz range) is believed to be involved in long-range communication between brain regions. A possible link between gamma activity and the hemodynamic response measured by fMRI was found in a study of auditory response. Distinct activations in the gamma frequency range were found in subcortical structures, including the anterior cingulate cortex (ACC) and thalamus (Mulert et al., 2007).

Synchronized oscillations are transiently stable, thus form and decay rapidly. Synchrony can result from a common input oscillator, such as in cardiac synchronization. It can also be an emergent, self-organized phenomenon that is related to the network structure itself. The latter is of particular relevance to the search for psychiatric biomarkers that are associated with complex behaviors. The complexity of synchronization patterns appears to change during network development and reflects different neural wiring schemes and levels of cluster organization (Fuchs et al., 2007). It is thus reasonable to suppose that the developing brain will show different but characteristic synchronization patterns at different developmental stages. While the fundamental neurophysiological correlates of these patterns may be difficult to ascertain, they nevertheless may serve as a marker for normal and abnormal brain functional development. The emergence of a "social brain network" during early childhood was found in a study of evoked response potentials (ERPs) in 3-, 4-, and 12-month-old infants viewing faces of different orientation and direction of eye gaze (Johnson et al., 2005), suggesting a particular pattern of brain connectivity that develops in early childhood. Default mode networks (DMNs) found in adults and (negatively) associated with particular cognitive and sensorimotor activities were lacking in a study of (premature) 3 month old infants (Fransson et al., 2007). However, resting state networks have recently been shown to emerge in the first year of life, suggesting development of brain networks and their potential disruption in neurodevelopmental disorders (Supekar et al.; Uddin et al., 2010).

There is no consensus at this time on the best methods for determining nonlinear synchronization in neurological data and a number of different algorithms have been proposed (Kreuz et al., 2007; Sakkalis et al., 2009). Although strong signal synchronization would likely be detected by any of several nonlinear synchronization measures, two measures that are based on different algorithmic approaches are chosen here (see (Sakkalis et al., 2009) for more thorough discussion of some of the relative merits of each).

Two methods outlined here for **c**omputing bivariate synchronization matrix are: (1) the synchronization likelihood (SL) method (Montez et al., 2006); and (2) the instantaneous cross modulation from the circular phase Hilbert transform-based synchronization index (HI), which is robust to signal noise and short time windows (Gans et al., 2009).

Synchronization indices may be searched for correlation in each frequency bands using centered moving averages (Bashan et al., 2008). This approach will find weak or strong correlations with time lags. For each pair, the relative phase index can be computed and stored in a correlation matrix. At each time, some channels may be synchronized. A clustering algorithm can be applied to all channels at a single (averaged) time segment. The result is analogous to gene expression profile clustering (Ramoni et al., 2002). Statistical significance of clusters can be determined by assigning a numerical label to each channel involved in a cluster and the fractional overlap of clusters in different individuals computed. Synchronized clusters may also exhibit very low frequency oscillations, with frequencies of 1 to 0.01 hertz. These have been found in fMRI studies of default mode networks (Broyd et al., 2009; Greicius et al., 2008; Uddin et al., 2010).

Synchronization likelihood (SL) is a method based on the assumption that neurons are highly nonlinear devices, hence methods from chaotic dynamical systems may effectively capture the relevant dynamics of the system (Sakkalis et al., 2009). It is an unbiased generalized synchronization method that relies on detection of simultaneously occurring patterns that may differ in two time series. A method for automatically computing all but two user parameters for the SL algorithm has been developed and will be used here (Montez et al., 2006).

Instantaneous cross modulation (synchronization) of EEG channels can be computed using the Hilbert transform method (Gans et al., 2009). This method is robust to noise and detects synchronization across all frequency bands. The n:m cyclic relative phase index y1,2 between two signals, f1(t) and f2(t), at a specific time t is computed over a time interval using a sliding window as:

$$\Psi\_{1,2}^{n,m}(t) = \left| n\phi\_1(t) - m\phi\_2(t) \right|\_{t'} \mod 1 \tag{3}$$

where f(t)=arctan(H(y)/y) and H(y) is the Hilbert transform of the time series y. This approach is stable for nonstationary data (Gans et al., 2009), which is appropriate for our

and 12-month-old infants viewing faces of different orientation and direction of eye gaze (Johnson et al., 2005), suggesting a particular pattern of brain connectivity that develops in early childhood. Default mode networks (DMNs) found in adults and (negatively) associated with particular cognitive and sensorimotor activities were lacking in a study of (premature) 3 month old infants (Fransson et al., 2007). However, resting state networks have recently been shown to emerge in the first year of life, suggesting development of brain networks and their potential disruption in neurodevelopmental disorders (Supekar et

There is no consensus at this time on the best methods for determining nonlinear synchronization in neurological data and a number of different algorithms have been proposed (Kreuz et al., 2007; Sakkalis et al., 2009). Although strong signal synchronization would likely be detected by any of several nonlinear synchronization measures, two measures that are based on different algorithmic approaches are chosen here (see (Sakkalis

Two methods outlined here for **c**omputing bivariate synchronization matrix are: (1) the synchronization likelihood (SL) method (Montez et al., 2006); and (2) the instantaneous cross modulation from the circular phase Hilbert transform-based synchronization index (HI),

Synchronization indices may be searched for correlation in each frequency bands using centered moving averages (Bashan et al., 2008). This approach will find weak or strong correlations with time lags. For each pair, the relative phase index can be computed and stored in a correlation matrix. At each time, some channels may be synchronized. A clustering algorithm can be applied to all channels at a single (averaged) time segment. The result is analogous to gene expression profile clustering (Ramoni et al., 2002). Statistical significance of clusters can be determined by assigning a numerical label to each channel involved in a cluster and the fractional overlap of clusters in different individuals computed. Synchronized clusters may also exhibit very low frequency oscillations, with frequencies of 1 to 0.01 hertz. These have been found in fMRI studies of default mode networks (Broyd et

Synchronization likelihood (SL) is a method based on the assumption that neurons are highly nonlinear devices, hence methods from chaotic dynamical systems may effectively capture the relevant dynamics of the system (Sakkalis et al., 2009). It is an unbiased generalized synchronization method that relies on detection of simultaneously occurring patterns that may differ in two time series. A method for automatically computing all but two user parameters for the SL algorithm has been developed and will be used here

Instantaneous cross modulation (synchronization) of EEG channels can be computed using the Hilbert transform method (Gans et al., 2009). This method is robust to noise and detects synchronization across all frequency bands. The n:m cyclic relative phase index y1,2 between two signals, f1(t) and f2(t), at a specific time t is computed over a time interval using a sliding

> () () () , 1,2 1 2 , *n m <sup>Ψ</sup> t ntmt* = − φ

 φ

where f(t)=arctan(H(y)/y) and H(y) is the Hilbert transform of the time series y. This approach is stable for nonstationary data (Gans et al., 2009), which is appropriate for our

mod 1 (3)

et al., 2009) for more thorough discussion of some of the relative merits of each).

which is robust to signal noise and short time windows (Gans et al., 2009).

al., 2009; Greicius et al., 2008; Uddin et al., 2010).

al.; Uddin et al., 2010).

(Montez et al., 2006).

window as:

data. The mod 1 term ensures that significant phase differences will be detected even in the presence of noise-induced phase jumps. In most cases n=m=1 is commonly assumed, though cross correlation of signals with n not equal m will be used here. Two signals are defined to be synchronized when is less than a specified constant. This algorithm is stable for nonstationary data and will detect synchronization without the need to distinguish between noise and chaos (Gans et al., 2009). A sliding window will be used to compute sync over 5 minutes. Ψ1,2

A number of methods have been used for determining synchronization in neurological data and a number of different algorithms have been proposed (Kreuz et al., 2007; Sakkalis et al., 2009). Although useful, many of these methods have difficulties with nonstationary, nonlinear signals and either fail to find true synchrony or introduce spurious synchronization (Fine et al., 2010). Spurious synchronization due to volume conduction effects can be removed by applying a spatial algorithm to ICA decomposition (Hironaga and Ioannides, 2007). ICA eliminates volume conduction effects while maintaining the same time resolution, thus still allowing generalized synchronization to be computed.

As Fourier spectrum can only give meaningful interpretation to linear and stationary processes, its application to data from nonlinear and nonstationary processes is problematical. A relatively new method for extracting *instantaneous* phase and frequency information from both linear and nonlinear, chaotic signals is the Huang-Hilbert transform (Huang and Wu, 2005; Huang et al., 2009). Determination of instantaneous phase and frequency is usually accomplished using the Hilbert transform method (Kreuz et al., 2007). However, this is only appropriate for monofrequency analytic signals that have a single center of rotation in the complex plane. The Empirical Mode Decomposition (EMD) introduced by Huang makes no assumptions about linearity. The EMF decomposes a nonlinear, nonstationary time series into adaptively determined characteristic time scales of each of the components (Huang et al., 2009). These component functions are termed intrinsic mode functions (IMF) and are analogous to Fourier components in a traditional linear decomposition. The IMFs computed using the empirical mode transform (EMF) have the property of a single center of rotation in the complex plane, ideally satisfying the requirements for the application of the Hilbert transform to determine instantaneous phase and frequency (Fine et al., 2010; Huang et al., 2009).

After computation of IMFs for each EEG channel, the IMF components with the highest power will be used to compute an instantaneous phase coherence matrix, R, using a sliding window. Following (Bialonski and Lehnertz, 2006), R is computed:

$$R\_{ij} = \left(\frac{1}{w} \sum\_{t=0}^{w-1} e^{i(\phi\_t - \phi)t} \right) \text{ / mod } 1 \tag{4}$$

where w is the number of time samples in the time series segment or window, i and j designate the channel number (or the IMF component of the channel) and f(t)=arctan(H(y)/y) and H(y) is the Hilbert transform of the IMF component. The mod 1 term ensures that significant phase differences will be detected even in the presence of noise-induced phase jumps. The Hilbert transform obtains the best fit of a sinusoid to each IMF at every point in time, identifying an *instantaneous frequency* (IF), along with its associated *instantaneous amplitude* (IA). The IF and IA provide a time-frequency decomposition of the data that is highly effective at resolving non-linear and transient features. This algorithm is stable for nonstationary data and will detect synchronization without the need to distinguish between noise and chaos (Gans et al., 2009). An example of bivariate synchronization between two EEG sensors in the right medical parietal region is shown in figure 2. The synchronization likelihood in this example was computed using the only the first three IMFs from each sensor, without searching for cross band synchronization.

To identify synchronized clusters of EEG channels, a method based on an eigenvector space method, using eigenvalues of R, can be used, following the algorithm developed and applied in (Allefeld and Bialonski, 2007; Bialonski and Lehnertz, 2006; Fine et al., 2010). The outcome of this algorithm will be synchronized clusters of EEG channels. These may be mapped onto scalp plots and the identified clusters compared to default mode networks that have been identified in young children (Sauseng and Klimesch, 2008; Supekar et al., 2009; Supekar et al., 2010). It will be of particular interest to determine if synchronization clusters are significantly correlated with functional networks in the brain and are biomarkers of abnormalities in brain network function (Assaf et al., 2010; Kennedy and Courchesne, 2008). To date, most research on functional brain networks, including the default mode network, has relied on functional MRI. Networks determined by fMRI reflect only the hemodynamic or metabolic response of neurons (Power et al., 2010). This can be considered a kind of amplitude correlation but not true synchronization of brain regions.

If synchronization of electrical activity can be shown to be an alternative measure of brain network activity it would open up much more exploration of the role of brain networks in cognitive activity, brain computer interfaces and neuropsychiatric disorders. Aberrations to default mode networks have been implicated in a number of brain disorders (Broyd et al., 2009) including post traumatic stress syndrome (Daniels et al., 2010), social phobias (Gentili et al., 2009), depression (Sheline et al., 2009), ADHD (Uddin et al., 2008), autism (Di Martino et al., 2009), and schizophrenia (Lagioia et al., 2010). fMRI is far too expensive to be used routinely as a clinical screening and monitoring tool. Yet the apparent widespread role of synchronized brain networks in many neuropsychiatric disorders suggests that a less expensive and easy to administer technology for analyzing brain networks would be widely useful in clinical practice.

## **3.3 Recurrence quantitative analysis**

Several univariate measures of time series complexity and a number of approaches for computing the degree of synchronization between signals have been used to analyze EEG data. Applications of these methods to psychiatric care and mental health continue to show promise. A more general framework for characterizing the dynamics of complex systems may be to construct recurrence plots (Marwan et al., 2007) and compute quantitative properties. The idea to use recurrence plots as a representation of complex system dynamics was first proposed by (Eckmann et al., 1987) in the late 1980s. The original tool presented a graphical means for visualizing differences in system dynamics. Methods for quantifying the small scale structures in recurrence plots were devised and shown to be capable of revealing system parameters and transitions that are not easily obtained by other methods (Marwan et al., 2002; Zbilut et al., 2002). Some dynamical parameters, such as K2 entropy and mutual information can also be derived from recurrence plots without RQA methods by computing the distribution of line lengths in the recurrence plot (Marwan et al., 2007). Readers are referred to the references for reviews of this unifying approach to nonlinear systems analysis.

(Schinkel et al., 2007) demonstrated that a single measure from RQA analysis could detect a change in the N400 response in single trials when subjects were presented with an oddball task, suggesting that RQA may be a sensitive measure of transient brain states. Few studies have been done using RQA to determine more stable brain functional characteristics. This may be a promising new field for research on EEG biomarkers of psychiatric disorders.
