2010 Special IssueEstimation of genuine and random synchronization in multivariate neural series
Introduction
In neurological studies, synchronization is recognized as a key feature to indicate the information process in a normal or abnormal brain (e.g. Buzsáki, 2004, Buzsáki and Draguhn, 2004, Fell et al., 2001, Varela et al., 2001). The estimated synchronization of the experimental and clinical data, for instance multivariate neural signals, become the signatures of brain pathologies, or brain functions, or the early diagnosis and monitoring of brain disorder. (Aarabi et al., 2008, Carmeli et al., 2005, Darvas et al., 2009, Knyazeva et al., 2008, Rudrauf et al., 2005, Stam et al., 2007). In order to investigate synchronization in the brain, multiple electrodes are often used to record the neural signals in different areas of the brain simultaneously. Therefore, how to estimate the synchronization index of multivariate neural signals has become a crucial issue in neural signal processing.
Some methods have been developed to estimate the synchronization index (or correlation coefficient) between two neural series, for instance cross-correlation, spectrum-based coherence, synchronization likelihood, mutual information, nonlinear interdependence, phase synchronization, correntropy coefficient, event synchronization, and so on (Arnhold et al., 1999, Brown and Kocarev, 2000, Bruns, 2004, Carter, 1987, Lachaux et al., 1999, Le Van Quyen et al., 2005, Quian Quiroga, and Kraskov et al., 2002, Quian Quiroga, and Kreuz et al., 2002, Stam and van Dijk, 2002, Xu et al., 2008). To analyze multivariate neural series, we may repetitively use bivariate measure methods to obtain the synchronization index among neural series. However, how to obtain a golden synchronization index in multivariate neural signals is still a bottleneck problem (Allefeld and Kurths, 2004, Pereda et al., 2005). Recently, an S-estimator has been developed to estimate the synchronization in multi-channel EEG series (Carmeli et al., 2005). In this method, the quantified synchronization is inversely proportional to the embedding dimension of the dynamical system, and is independent of the total power and the time dimension of the neural signals. The disadvantage of the S-estimator is that the estimated synchronization index includes random and/or artifact components, because the synchronization is often estimated over finite-length data, so the estimated synchronization includes, to some extent, random and/or artifact information (Müller et al., 2008, Plerou et al., 2002). In a recent study, a concept of genuine synchronization was proposed for the first time by Müller et al. (2008) to remove the random (and/or artifact) components in multivariate neural signals. In this method, the genuine cross-correlation strength (TCS) was estimated by means of the significant deviation of the eigenvalues (or partial eigenvalues) of the linear zero-lag cross-correlation matrix. However, all of eigenvalues contain rich information (Kwapień, Drożdż, & Oświeçimka, 2006), so this information should be used.
In this paper, we propose a method to estimate the GSI and the RSI in multivariate neural signals. To test the performance of the method, a multi-channel neural mass model (MNMM) (Cui, Li, & Gu, 2009) is applied to generate multivariate neural signals. The effects of different coupling coefficients, signal to noise ratios (SNRs) and time-window widths on the method are investigated. Application of the method to the multivariate long-term EEG recording of a 35 year-old male suffering from mesial temporal lobe epilepsy is demonstrated as well.
Section snippets
Correlation matrix analysis
Equal-time correlation is a simple method to measure the synchronization between two series. Consider multivariate neuronal data , , , where is the channel number and is the number of data points in time window . To provide the same scale for all the neuronal population activities, the normalized data are first calculated by , where and are the mean and standard deviation of , respectively. Then the equal-time correlation
Simulated multivariate EEG series
Typical simulated EEG series of the 10 channels and corresponding normalized spectra of different coupling coefficients are shown in Fig. 2. The weight parameter is . With the increase of coupling strength, the spectra of simulated EEG series vary from multiple frequencies to a single and low frequency.
The effects of frequency band on the S-estimator, the GSI and the RSI
The uncoupled EEG series are simulated by the MNMM with three different weight parameters. Fig. 3(a) shows the simulated 10-channel EEG series over 1 s. Fig. 3(b) shows the
Conclusions
In this paper, the GSI and the RSI are proposed to quantify a genuine synchronization and a random synchronization in multivariate neural series by means of a correlation matrix analysis and the surrogate method. A multi-channel neural mass model is employed to generate multi-channel neural time series; then the performance of the proposed method is evaluated by the simulated EEG series, including the effect of the frequency of EEG signals, coupling coefficients, SNRs and time-window widths.
Acknowledgements
This work was supported by Program for New Century Excellent Talents in University (NECT-07-0735), National Natural Science Foundation of China (90820016), Natural Science Foundation of Hebei and “973” Program of China (2007CB512501).
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