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In particular, the importance of synchronization of neuronal discharges has been
shown by a variety of animal studies using microelectrode recordings of brain
activity [38, 33], and even at coarser levels of resolution by other studies in ani-
mals and humans [9]. The phase synchronization in the brain extracted from EEG
data using Hilbert or wavelet transforms has recently been shown to be an especially
promising tool in analysis of EEG data recorded from patients with various types of
epilepsy [35].
In this chapter, we introduce a novel concept of generalized phase synchroniza-
tion, which is based on vector autoregressive modeling. This new notion of phase
synchronization is constructed as an extension of the classical definition of phase
synchronization between two systems. Indeed, the phase synchronization is usu-
ally defined by imposing the condition that some integer combination of the instan-
taneous phases of two signals is constant. Often this condition is further relaxed
by allowing for a bounded linear combination of two phases, in order to account
for noise in the measurements. This classical approach to phase synchronization is
clearly bivariate. Since we are interested in investigating synchrony among several
areas in the brain, we would like to generalize the bivariate phase synchronization
to a multivariate case.
To construct a more general multivariate notion of phase synchronization, we
extend the classical definition by considering such a linear combination of phases
for a finite number of signals that represent a stationary process. All the individual
signals together form a common system described by some multivariate process. We
note that a vector process, such that a linear combination of its individual compo-
nents is a stationary process, can be modeled as a cointegrated vector autoregressive
time series.
Furthermore, we show that the cointegrated rank of the regression determines
how restricted the behavior of such system is. This means that the rank r of cointe-
grated autoregressive model, estimated from the multiple time series of the instanta-
neous phases, measures how large the vector subspace, which generates the changes
in the phase values, is.
This new measure of cointegration is also applied to absence epilepsy EEG data.
The data sets collected from the patients with other types of epilepsy are currently
being investigated.
This chapter is organized as follows. Section 18.2 introduces cointegrated vector
autoregressive processes, and various related testing procedures. In Section 18.3,
we discuss role of synchronization in brain dynamics, and give a definition of clas-
sical phase synchronization. Section 18.4 presents the Hilbert transform method
for extracting instantaneous phases from time series. To develop our multivariate
approach to studying phase synchrony in a complex system, such as brain, we ex-
tend the classical bivariate concept of phase synchronization based on cointegrated
vector autoregression in Section 18.5, and test our method on absence epilepsy
data.
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