Identifying phase synchronization clusters in spatially extended dynamical systems

arXiv:1003.2597v1 [physics.data-an] 12 Mar 2010 Published as Phys. Rev. E 74, 051909 (2006). Copyright 2006 by the American Physical Society. Identifying phase synchronization clusters in spatially extended dynamical systems Stephan Bialonski1, 2, ∗and Klaus Lehnertz1, 2, 3, † 1Department of Epileptology, Neurophysics Group, University of Bonn, Sigmund-Freud-Str. 25, D-53105 Bonn, Germany 2Helmholtz-Institute for Radiation and Nuclear Physics, University of Bonn, Nussallee 14-16, 53115 Bonn, Germany 3Interdisciplinary Center for Complex Systems, University of Bonn, R¨omerstr. 164, 53117 Bonn, Germany (Published 14 November 2006) We investigate two recently proposed multivariate time series analysis techniques that aim at detecting phase synchronization clusters in spatially extended, nonstationary systems with regard to field applications. The starting point of both techniques is a matrix whose entries are the mean phase coherence values measured between pairs of time series. The first method is a mean field approach which allows to define the strength of participation of a subsystem in a single synchronization cluster. The second method is based on an eigenvalue decomposition from which a participation index is derived that characterizes the degree of involvement of a subsystem within multiple synchronization clusters. Simulating multiple clusters within a lattice of coupled Lorenz oscillators we explore the limitations and pitfalls of both methods and demonstrate (a) that the mean field approach is relatively robust even in configurations where the single cluster assumption is not entirely fulfilled, and (b) that the eigenvalue decomposition approach correctly identifies the simulated clusters even for low coupling strengths. Using the eigenvalue decomposition approach we studied spatiotemporal synchronization clusters in long-lasting multichannel EEG recordings from epilepsy patients and obtained results that fully confirm findings from well established neurophysiological examination techniques. Multivariate time series analysis methods such as synchronization cluster analysis that account for nonlinearities in the data are expected to provide complementary information which allows to gain deeper insights into the collective dynamics of spatially extended complex systems. PACS numbers: 87.19.La, 05.45.Tp, 05.45.Xt, 05.10.-a I. INTRODUCTION Spatially extended complex dynamical systems may be thought of being composed of numerous constituents (dy- namically formed subsystems) each having its own dy- namics. Typically the relevant state variables of such sys- tems can not be observed directly but only through some observation function that projects the high-dimensional state space onto an observation space of much lower di- mension, resulting in a set of time series. Multivari- ate analyses of such time series might then help to gain deeper insights into the collective dynamics of spatially extended systems. Although a number of time series analysis methods have been developed over the past (see [1–5] for an overview) most techniques either allow to characterize single time series (univariate approaches) or to investigate relationships between two time series (bi- variate approaches). However, applying bivariate tech- niques to pairs of time series – taken from a multichan- nel recording – does not necessarily allow to identify the relevant information in the full data set. The latter is of particular interest for scientific fields investigating spa- tially extended dynamical systems, such as meteorology, economics, social science or neurosciences, where a com- plex but relatively sparse connectivity between subsys- ∗Electronic address: bialonski@gmx.net †Electronic address: klaus.lehnertz@ukb.uni-bonn.de tems prevails. Understanding brain function – both dur- ing physiological and pathophysiological conditions (as e.g. in the case of epilepsy) – requires a characterization and quantification of the collective behavior of neural net- works generating signals at different areas. In principle, multivariate time series analysis tech- niques can be used to investigate mutual relationships between arbitrary numbers of time series. A large va- riety of methods [6] aim at revealing additional infor- mation by classifying time series into different groups. In addition to the classical principal component analysis (also known as Karhunen-Loeve transform) [7] indepen- dent component analysis [8] provides a decomposition of data into independent source signals, and if the as- sumption of independence holds, it can be regarded as a suitable method. If independence can not be assumed, mutual information based methods might be more ap- propriate [9, 10]. The partial coherence [3] measures the fraction of coherence between two time series that is not shared with a third time series. Whereas the partial co- herence is based on the assumption of linearity and thus does not capture nonlinear interactions, the recently pro- posed concept

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