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Table 18.6: Post-ictal cointegration rank: Results of the Johansen procedure for the
multiple series during 2 s after seizure for three consecutive seizures. Significance
level is 1%. Full rank is denoted by
†
Seizure #
Short lag
Long lag
pr
pr
Seizure 1
p
=
2
,
r
=
10
p
=
20
,
r
=
12
Seizure 2
p
=
2
,
r
=
13
p
=
20
,
r
=
10
Seizure 3
p
=
2
,
r
=
16
†
p
=
20
,
r
=
13
estimated under a long lag parameter may not adequately represent the underlying
processes in seizures 1 and 2. On the other hand, seizure 3 is estimated based on
almost 1,200 sample values, and therefore, the long lag model of a longer seizure 3
may be more realistic, than the long lag models for shorter seizures 1 and 2. Overall,
the models based on a short lag
p
for all three seizures provide an evidence of abso-
lute synchronization among the channels. Whereas, the preseizure and postseizure
models are more likely to be less restricted, and seem to exhibit a cointegration rank
between 9 and 16.
18.6 Conclusion
Recent success in application of phase synchronization to analysis of dynamic pro-
cesses in epileptic brain motivated us to develop a concept of
generalized synchro-
nization
. This new concept based on our original idea to extend the condition of
classical synchronization from the classical bivariate case to a more general multi-
variate case by studying a cointegrating relationship in the multiple time series. The
proposed approach allows one to analyze the synchrony among different parts of
the common interrelated system (such as a human brain), by modeling the phases
extracted from a finite number of signals in the systems by means of cointegrated
vector autoregression. Interestingly, the cointegration rank in the cointegrated VAR
model of the phase time series can be viewed as a measure of synchrony among
the phases of different components of the EEG signal. We applied our multivariate
approach to phase synchronization on the EEG data recorded from the patients with
absence epilepsy. The results of our experiments indicate that the new method is
capable of capturing phase synchronization in multivariate EEG during seizures.
Not only this new measure of multivariate phase synchrony can be tested on var-
ious biomedical data, such as multichannel EEG recorded from an epileptic brain,
but also the new multiple phase synchronization can be employed in different areas
of applied and theoretic research (including physics, communication, electronics,
laser dynamics, and control) for studying synchronization among several dynamical
systems or a system that consists of several parts.
