s1s2s3F1E1E2E3E4E5E6s3E7E8E9F2F3E10s3s4E11F4s5s5As5BF5E12E13F6F7s6s6As6Bs6CF8F9E14F10F11F12F13s7B1B2B3B4B5B6B7B8B9B10B11B12B13B14B15B16B17B18B19B20B21B22B23B24B25B26B27B28B29B30B31B32A phase-synchronization and random-matrix based approach tomultichannel time-series analysis with application to epilepsyIvan Osorio1and Ying-Cheng Lai21Department of Neurology, University of Kansas Medical Center, 3901 Rainbow Blvd., Kansas City,Kansas 66160, USA2School of Electrical, Computer and Energy Engineering, Department of Physics,Arizona State University, Tempe, Arizona 85287, USA(Received 22 Mar ch 2011; accepted 6 July 2011; published online 1 August 2011)We present a general method to analyze multichannel time series that are becoming increasinglycommon in many areas of science and engineering. Of particular interest is the degree of synchronyamong various channels, motivated by the recognition that characterization of synchrony in a systemconsisting of many interacting components can provide insights into its fundamental dynamics. Oftensuch a system is complex, high-dimensional, nonlinear, nonstationary, and noisy, rendering unlikelycomplete synchronization in which the dynamical variables from individual components approacheach other asymptotically. Nonetheless, a weaker type of synchrony that lasts for a finite amount oftime, namely, phase synchronization, can be expected. Our idea is to calculate the average phase-synchronization times from all available pairs of channels and then to construct a matrix. Due tononlinearity and stochasticity, the matrix is effectively random. Moreover, since the diagonalelements of the matrix can be arbitrarily large, the matrix can be singular. To overcome thisdifficulty, we develop a random-matrix based criterion for proper choosing of the diagonal matrixelements. Monitoring of the eigenvalues and the determinant provides a powerful way to assesschanges in synchrony. The method is tested using a prototype nonstationary noisy dynamical system,electroencephalogram (scalp) data from absence seizures for which enhanced cortico-thalamicsynchrony is presumed, and electrocorticogram (intracranial) data from subjects having partialseizures with secondary generalization for which enhanced local synchrony is similarly presumed.VC2011 American Ins titute of Physics. [doi:10.1063/1.3615642]An increasingly common practice in many fields of scienceand engineering is to record a large amount of data simul-taneously from an array of sensors (or channels) and thento analyze the data to probe the dynamics of the underlyingsystem. In realistic situations, the system contains multipleinteracting components, is nonlinear, nonstationary, andnoisy. Because of these characteristics, traditional methodssuch as those based on the Fourier power spectrum are of-ten ineffective. To develop methods to analyze multichan-nel data thus becomes an issue of paramount importanceand extremely broad interest. Here we present a methodbased on the ideas of stochastic phase synchronization andrandom matrices to extract information about the dynami-cal evolution of the underlying system. Generally, for areal system in a noisy environment, complete synchroniza-tion among the multiple signal generators from differentchannels is unlikely. Instead, in typical situations where thegenerators oscillate in time, a weaker type of synchronythat lasts for a finite amount of time, namely temporalphase synchronization, can occur. Our idea is then to cal-culate the average phase-synchronization times (APSTs)among all available pairs of channels and then to constructa matrix. Monitoring of the eigenvalues and the determi-nant of the synchronization-time matrix provides an effec-tive way to assess the degree of spatiotemporal synchrony.Due to the nonlinear and stochastic nature of the underly-ing system and environment, the synchronization-timematrices are effectively random matrices. For example,consider a set of multi-channel electrocorticogram (ECoG)recordings. During any time window of observation, theAPSTs obtained from all distinct pair of channels are ran-dom. Thus, for a given time window, the matrix elementsare uncorrelated or weakly correlated and can be effec-tively regarded as random with respect to each other. Wefind that the spectral properties of the synchronization-timematrixexhibitagreatdealofsimilaritytotheseofrandom matrices whose elements are drawn, for instance,from a Gaussian orthogonal ensemble. Moreover, any ma-trix element as a function of time also appears to be highlyrandom. What we face is thus random evolution of a ran-dom matrix. Looking for characteristic changes in the vari-ous properties of the random matrix in time may thereforeprovide an avenue to probing the change in the synchronyof the underlying system with high sensitivity. A technicalissue is the choice of the diagonal elements, which are inprinciple, infinite and, for a moving-window application,they are the size of the window. Consequently, a difficultyis that the window size is often much larger than theAPST, rendering singular the synchronization-time matrixand diminishing the matrix’s ability to discern systemchanges. We shall demonstrate that the spectral theory ofrandom matrices can be used to establish a criterion forchoosing the diagonal elements. Using coupled chaoticoscillators with time-varying coupling, we demonstrate the1054-1500/2011/21(3)/033108/11/$30.00VC2011 American Institute of Physics21, 033108-1CHAOS 21, 033108 (2011)Author complimentary copy. Redistribution subject to AIP license or copyright, see http://cha.aip.org/cha/copyright.jsppower of our method to detect characteristic changes inthe system. We then apply our method to multichannelECoG recordings from epileptic subjects to quantify theevolution of synchrony before, during, and after seizures,with the finding that epileptic seizures can be associatedwith either enhanced or reduced neuronal synchrony.I. INTRODUCTIONMultichannel data are becoming quite common in manyfields of science and engineering. Electroencephalogram(EEG) and electrocorticogram (ECoG) signals in medicineare one example. Recent years have witnessed an increasinguse of large-scale sensor network s in various civil anddefense applications, which typically generate a largeamount of multivariate data. Examples include monitoringand collection of information on objects ranging from plank-ton colonies,1endangered species,2soil and air contami-nants3to traffic