A Measure of Multivariate Phase Synchrony Using Hyperdimensional Geometry

Phase synchrony has been used to investigate the dynamics of subsystems that make up a complex system. Current measures of phase synchrony are mostly bivariate focusing on the synchrony between pairs of time series. Bivariate measures do not necessarily lead to a complete picture of the global interactions within a complex system. Current multivariate synchrony measures are based on either averaging all possible pairwise synchrony values or eigendecomposition of the pairwise bivariate synchrony matrix. These approaches are sensitive to the accuracy of the bivariate synchrony indices, computationally complex and indirect ways of quantifying the multivariate synchrony. Recently, we had proposed a method to compute the multivariate phase synchrony using a hyperdimensional coordinate system. This method, referred to as Hyperspherical Phase Synchrony (HPS), has been found to be dependent on the ordering of the phase differences. In this paper, we propose a more general hyper-spherical coordinate system along with a new higher-dimensional manifold representation to eliminate the dependency on the ordering of the signals' phases. This new framework, referred to as Hyper-Torus Synchrony (HTS), is shown to be equivalent to the root-mean-square of a sufficient set of squared phase-locking values whose phase differences contain information about all oscillators in the network. The statistical properties of HTS are given analytically and its performance is evaluated thoroughly for both synthetic and real signals.