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the difference K
r between the process dimensionality and the cointegration
rank; and
the alternative hypothesis.
As a result, the selected percentage points of the asymptotic distribution of the test
statistic
λ LR were tabulated by Johansen and Juselius in [13].
18.3 The Role of Phase Synchronization in Neural Dynamics
The word “synchrony” originates from a combination of two Greek words
συν
(syn, meaning common) and
(chronos, meaning time), and it can be trans-
lated as “happening at the same time.” A concept of synchronization can be de-
fined as a process of active adjustment between the rhythms of different oscillating
systems due to some kind of interaction or coupling between them [28]. Synchro-
nization phenomena were discovered in the late seventeenth century by C. Huygens
who first observed synchronization between two pendulum clocks hanging from a
common support [12]. Since then, the study of synchronization between dynamical
systems became an active field of research in many scientific and technical disci-
plines, including solid state physics [24], plasma physics [34], communication [3],
electronics [27, 22], laser dynamics [8, 36, 39], and control [30, 37].
Complex physiological systems, such as heart and brain, also display synchro-
nization. The presence of synchronization processes in physiological systems was
discovered by B. van der Pol in the beginning of the twentieth century. In particular,
he first applied oscillation theory to the human heart [29]. The role of synchro-
nization in neural dynamics is an important area of research in neuroscience. Much
effort is given to investigation of synchronization phenomena on all different levels
of organization of brain tissue, starting with pairs of individual neurons to larger
scales, such as within a given area of the brain or between distinct parts of the brain.
Recent findings indicate that long-range synchronization can be detected not only in
microelectrode studies [38,33], but also in the studies using surface recordings [32].
It has been shown that synchronization is a significant attribute of the signal
recorded from the patients affected by several neurological disorders. In particular,
researchers have found that epilepsy [20] and Parkinson's disease [40] manifest as
a pathological form of the synchronization process.
Several studies in neuroscience emphasize major difference between synchrony
as an appropriate estimate of phase relation, and the classical measures of coherence
or spectral covariance [2,1]. Le Van Quyen et al. discuss two important limitations of
coherence [31]. The first limitation arises because the standard approaches for mea-
suring coherence [4] based on Fourier analysis are known to be highly dependent
on the stationarity of the measured signal, whereas the signals recorded from the
brain, such as EEG, appear to be clearly nonstationary. The second limitation stems
from the fact that classical coherence is a measure of spectral covariance. Hence, it
is not able to separate the effects of amplitude and phase in the relations between
two signals. Thus, coherence gives only an indirect and approximate indication of
phase synchrony.
χρ
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