Proceedings of the 2005 IEEE Engineering in Medicine and Biology 27th Annual Conference Shanghai, China, September 1-4, 2005

Multivariate Correlation Coefficient Decomposition and Its Application to Visual Evoked Potentials Zhang Junpeng*, Student Member, IEEE, Ma Dajun, Cui Yuan and Yong Liujun Abstract—Interactions between cortical areas are crucial for cognitive functioning. We develop a method, called Multivariate Correlation Coefficient Decomposition, to access such interactions. By decomposing multi-channel mutual correlation coefficient(CC ) matrix into individual CC, which is taken as mapping parameter, this method can map temporarily correlated sources activities. Both computer simulation and real Visual Evoked Potentials(VEP ) test show that, compared to traditional power mapping, the presented method is sensitive to the combination of the correlated brain sources with different energy levels. Thus, it is of theoretical significance and of practical value.

I. INTRODUCTION Brain electrical signal is one of the windows to understanding neural activities. It is of significance and impo1rtance to relate scalp recorded EEGs to its origins inside the brain. Brain sources can approximately be classified as correlated and uncorrelated sources. Correlations between signals of sensors covering different scalp areas are commonly taken as a measure of functional coupling. Functional connectivity between cortical areas appears as correlated time behavior of neural activity. Therefore, it would be of utmost interest to image interactions between different brain regions in the working human brain. Many methods aimed at localizing uncorrelated sources have been developed, such as multiple signals classification [1](MUSIC), simulated annealing [2], genetic algorithm [3], artificial neural network[4], and so on. Among these methods, MUSIC incorporates time-frequency characteristics of neural sources; however, MUSIC cannot separate correlated sources. Since proposed that coherent oscillatory brain activity might underlie functional connectivity [5], it is now a major concern in neurophysiological study. In recent years, the investigation of how to map correlated sources has received Manuscript received June 15, 2005.Zhang Junpeng is with Dept. of BiomedicalEngineering, Chengdu Medical Collge.(Phone:86-28-87676456; email:[email protected]). other authors work in the same affiliation as Zhang Junpeng.

0-7803-8740-6/05/$20.00 ©2005 IEEE.

much attention. For instance, J Gross proposed an algorithm, namely dynamic imaging of coherent sources [6], which can map correlated sources, but his method requires the selection of a reference region. The selection strongly depends on the users and it is difficult to be applied in practice. Motivated by multivariate phase synchronization clustering analysis [7], we develop a method to map the correlated sources. Different from MUSIC, This method, termed Multivariate Correlation Coefficient Decomposition (MVCCD), is designed for correlated sources. It avoids the procedures of selecting a reference region, which is necessary in Gross’s method. The main advantage of this method is that it can map multiple correlated sources with high spatial resolution and accuracy.

II. METHODS A.

Multivariate Correlation Coefficient Decomposition

(MVCCD) Suppose there are N signals Xik (i=1, …,N) with K samples (k=1,…,K). The correlation coefficient (C) between signal Xi and Xj is CCij, and all the CC consisted of a CC matrix. Suppose, beyond all the N signals, there is a reference signal Xr, which is the weighted sum of all the signals. The goal of MVCCD is to derive the strength of correlation between individual signal (one channel) and the reference signal from the CC matrices CCij. CC between individual signal and the reference is so called correlation index Pir, which measures the strength of correlation between the signal Xi and the weighted sum Xr of all the N signals. In general, CC between any two signals can be decomposed into two parts, CCij=pir pjr (1) Where CCij is the correlation coefficient between signal Xi and signal Xj. Pir is the correlation index between the signal Xi and the reference signal Xr. Equation (1) is based on the fact

Figure 1. Comparison between MVCCD and power mapping. marked by x-symbols, respectively.

Left: the 2D distribution of three coherent sources, whose locations are

Middle: the spatial distributions of the coherent sources by MVCCD; Right: the spatial distributions of

energy of scalp EEGs.

Figure 2. Visual evoked potential test. Left: the stimulus picture. The symbol “+” stands for the attentive point. Right: the sources imaging from real EEG data by MVCCD.

(Xi Xi !)T (Xr  Xr !) (Xj Xj !)T (Xr  Xr !) . || Xi Xi !||.|| Xr  Xr !|| || Xj Xj !||.|| Xr  Xr !||

Pir.Pjr

T

T

(Xi Xi !) (Xj Xj !) (Xr  Xr !) (Xr  Xr !) . || Xi Xi !||.|| Xj Xj !|| || Xr  Xr !||.|| Xr  Xr !|| and

( X r   X r ! )T ( X r   X r !) 1 || X r   X r !|| . || X r   X r !|| Thus Pir.Pjr

( Xi   Xi !) T ( Xj   Xj !) || Xi   Xi !|| . || Xj   Xj !||

CC ij

This derivation shows that each CC can be decomposed into two parts, and all the cases of equation (1) with different i and j consist of a group of equations. In order to estimate the value of Pir from these equations, an optimization algorithm may be utilized to minimize the sum of square weighted errors [7]

¦E

2 ij

with E ij

(CC ij  pir p jr ) 2

(2)

i , j !i

The simplest method is to directly call the Matlab function lsqnonlin in optimazation toolbox which is designed for solving nonlinear least-squares problems, so it is suitable for this model. In this study we do the calculation by calling lsqnonlin.

B.

Imaging the correlated sources

As the correlated sources contribute a lot to the reference signal, the weighted sum of all the signals, the Pir values of all the correlated channels may be larger then the other channels, thus, by relate the values of Pir to color depth, we may get a high resolution mapping of correlated sources. For comparison, the power mapping [8-10] is also calculated in this study.

III. COMPUTER SIMULATION TEST The head is modeled as a concentric 4-sphere model. In this model, the radii are 7.9 cm, 8.1 cm, 8.5 cm and 8.8 cm, respectively, for the outer boundaries of the inner brain sphere, the cerebrospinal fluid layer (CSF), the skull layer and the scalp layer. And the conductivities are 0.461 A (V m)-1, 1.39 A (V m)-1, 0.0058 A (V m)-1 and 0.461 A (V m)-1, respectively. Suppose that there are three radial dipoles (Q =3) in the volume conductor model, the locations of which in Cartesian coordinates are respectively (x, y, z) =˄0.0688, -0. 2116, 0. 9749˅φ6.5cm, (-0.1341, 0.4126, 0.9010)φ6.5cm, ˄-0.4126ˈ-0.1341ˈ0.9010˅φ 6.5cm. The moments of three dipoles are 1, 0.12, and 0.5, respectively. The mutual correlation coefficients among the three sources are 0.8979, 0.7899 and 0.9443, respectively. Fig.1 shows the results. By projecting the three correlated sources onto the x-y plane, the left sub-figure of fig.1 is obtained. Middle shows the 2D source distribution obtained

by applying MVCCD to scalp EEGs. Right is the sources distribution by the traditional power mapping. We can conclude that, compared to power mapping, MVCCD has higher spatial resolution. Power mapping may miss the sources that are highly correlated but have low power, but MVCCD doesn’t.

VI. ACKNOWLEDGEMENT This work is supported by NSFC No.90208003 and the 973 project No.2003CB716106, and the real data was provided by Beijing Cognitive Lab.

(1)

References

IV. REAL EEG DATA TEST 119-channel ERP data are obtained. The stimulus in the right visual field is a checkboard shown in Fig.2. Both the ERP data and electrode coordinate data are group averages of ten subjects. The averaged electrode coordinate values are further normalized to the unit sphere surface. Fig.2 (the right) is the sources imaging. This result is consistent with the visual gateway theory when the stimulation appears in the right visual field, the main reaction is located on the left visual cortex.

V. CONCLUSION In this work we have implemented and tested a method for mapping correlated sources. It enables mapping correlated sources by simple algorithm. It overcomes the major drawback of power mapping, and the latter may miss the correlated sources with low power. The simulation test also proves that MVCCD is robust to noise and to the choice of reference electrodes. However, in the present, this method only can map the 2D locations of correlated sources. Future work is to further develop this method and enable mapping the 3D locations of correlated sources. In the same time, we will tempt to incorporate MVCCD with MUSIC, and realize the complete localization of both correlated sources and uncorrelated ones.

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