# Measuring Partial Phase Locking Value to Detect Synchronization in Multivariate Gaussian Systems

Sergul Aydore (Signal and Image Processing Institute, University of Southern California), Dimitrios Pantazis (McGovern Institute for Brain Research, Massachusetts Institute of Technology), Richard Leahy (Signal and Image Processing Institute, University of Southern California)

The phase locking value (PLV) is a widely used measure of brain synchronization and is often used to detect cortical networks that play a major role in cognitive integration. A limitation of pairwise PLV analysis is that it does not differentiate between direct and indirect interactions in a multiple-node network. Schelter et al [1] have investigated inversion of the matrix of pairwise PLVs to compute a partial phase locking value. This is analogous to inversion of cross-correlation and cross-spectral matrices to compute partial correlations and partial coherence, respectively. An alternative approach was proposed by Cadieu et al [2] who develop a multiple-node phase coupling model based on multivariate extension of the Von Mises distribution. Conditional phase coupling measures can be determined in a straightforward manner from this model. Both of these approaches use phase only information and implicitly assume the statistical independence between amplitude and phase in the signals being analyzed.

We consider the multidimensional circular Gaussian model and show that phase and amplitude coupling are not independent. As a result, the associated phase-coupling distribution requires marginalization with respect to amplitude. Based on this analysis we derive an analytical expression for PLV and partial PLV. Not surprisingly, these expressions are able to accurately reveal phase coupling in simulated Gaussian data. However, we also applied these measures to data generated using coupled Roessler oscillators and compared the estimated coupling values with those determined using Cadieu's multivariate model [2] and Schelter's partial PLV method [1]. As illustrated in Fig. 1, even for this nonlinear system, the PLV based on a Gaussian model appears to outperform the alternatives in terms of either avoiding false positive interactions or through reduced estimator variance.

[1] B. Schelter et al. (2006), ‘Partial
phase synchronization for multivariate synchronizing system’, *Phys. Rev. Lett.*, 96, 208103.

[2] Cadieu, C. et al.
(2010), ‘Phase coupling estimation from multivariate phase statistics’, *Neural Computation*, vol 22, no. 12, pp.
1-20.