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A phase-synchronization and random-matrix based approach to multichannel time-series analysis with application to epilepsy

### Abstract

We present a general method to analyze multichannel time series that are becoming increasingly common in many areas of science and engineering. Of particular interest is the degree of synchrony among various channels, motivated by the recognition that characterization of synchrony in a system consisting of many interacting components can provide insights into its fundamental dynamics. Often such a system is complex, high-dimensional, nonlinear, nonstationary, and noisy, rendering unlikely complete synchronization in which the dynamical variables from individual components approach each other asymptotically. Nonetheless, a weaker type of synchrony that lasts for a finite amount of time, 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 to nonlinearity and stochasticity, the matrix is effectively random. Moreover, since the diagonal elements of the matrix can be arbitrarily large, the matrix can be singular. To overcome this difficulty, we develop a random-matrix based criterion for proper choosing of the diagonal matrix elements. Monitoring of the eigenvalues and the determinant provides a powerful way to assess changes in synchrony. The method is tested using a prototype nonstationary noisy dynamical system, electroencephalogram (scalp) data from absence seizures for which enhanced cortico-thalamic synchrony is presumed, and electrocorticogram (intracranial) data from subjects having partial seizures with secondary generalization for which enhanced local synchrony is similarly presumed.

© 2011 American Institute of Physics

Received 22 March 2011
Accepted 06 July 2011
Published online 01 August 2011

Lead Paragraph:
An increasingly common practice in many fields of science and engineering is to record a large amount of data simultaneously from an array of sensors (or channels) and then to analyze the data to probe the dynamics of the underlying system. In realistic situations, the system contains multiple interacting components, is nonlinear, nonstationary, and noisy. Because of these characteristics, traditional methods such as those based on the Fourier power spectrum are often ineffective. To develop methods to analyze multichannel data thus becomes an issue of paramount importance and extremely broad interest. Here we present a method based on the ideas of stochastic phase synchronization and random matrices to extract information about the dynamical evolution of the underlying system. Generally, for a real system in a noisy environment, complete synchronization among the multiple signal generators from different channels is unlikely. Instead, in typical situations where the generators oscillate in time, a weaker type of synchrony that lasts for a finite amount of time, namely temporal phase synchronization, can occur. Our idea is then to calculate the average phase-synchronization times (APSTs) among all available pairs of channels and then to construct a matrix. Monitoring of the eigenvalues and the determinant of the synchronization-time matrix provides an effective way to assess the degree of spatiotemporal synchrony. Due to the nonlinear and stochastic nature of the underlying system and environment, the synchronization-time matrices are effectively random matrices. For example, consider a set of multi-channel electrocorticogram (ECoG) recordings. During any time window of observation, the APSTs obtained from all distinct pair of channels are random. Thus, for a given time window, the matrix elements are uncorrelated or weakly correlated and can be effectively regarded as random with respect to each other. We find that the spectral properties of the synchronization-time matrix exhibit a great deal of similarity to these of random matrices whose elements are drawn, for instance, from a Gaussian orthogonal ensemble. Moreover, any matrix element as a function of time also appears to be highly random. What we face is thus random evolution of a random matrix. Looking for characteristic changes in the various properties of the random matrix in time may therefore provide an avenue to probing the change in the synchrony of the underlying system with high sensitivity. A technical issue is the choice of the diagonal elements, which are in principle, infinite and, for a moving-window application, they are the size of the window. Consequently, a difficulty is that the window size is often much larger than the APST, rendering singular the synchronization-time matrix and diminishing the matrix’s ability to discern system changes. We shall demonstrate that the spectral theory of random matrices can be used to establish a criterion for choosing the diagonal elements. Using coupled chaotic oscillators with time-varying coupling, we demonstrate the power of our method to detect characteristic changes in the system. We then apply our method to multichannel ECoG recordings from epileptic subjects to quantify the evolution of synchrony before, during, and after seizures, with the finding that epileptic seizures can be associated with either enhanced or reduced neuronal synchrony.

Acknowledgments:
We thank Drs. Mark Frei and Liang Huang for valuable discussions. This work was supported by NIH. YCL was also supported by ONR under Grant No. N00014-08-1-0627.

Article outline:

I. INTRODUCTION
II. MATRIX OF AVERAGE PHASE-SYNCHRONIZATION TIME
III. USE OF RANDOM-MATRIX THEORY TO CHOOSE DIAGONAL ELEMENTS OF SYNCHRONIZATION-TIME MATRIX
IV. CONTROL STUDY: NETWORK OF COUPLED CHAOTIC OSCILLATORS
V. APPLICATION TO EPILEPTIC EEG AND ECOG BRAIN SIGNALS
A. Background
B. Results
VI. ISSUES
A. Random-matrix characteristics
B. Heuristic understanding of the evolution of determinant
C. Method of phase-coherence matrix
VII. CONCLUSION

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