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Thus, in the framework of generalized phase synchronization introduced above,
the cointegration rank represents a fundamental measure of synchrony in the multi-
channel signal, such as EEG. In particular, we say that the signal is
completely asyn-
chronous
, if the cointegration rank
r
is 0. On the other hand, when the multivariate
process is stable (i.e., the rank coincides with the dimension of the process,
r
=
K
),
the signal is said to be
perfectly synchronous
.
18.5.2 Absence Seizures
Absence seizures (or petit mal seizures) are known to occur in several forms of
epilepsy, whereas absence epilepsy refers to a type of epilepsy in which only the
absence seizures occur. Absence epilepsy is usually characterized by age of onset,
and often affects teenage population. Absence seizures usually begin in childhood
or adolescence, and often run in families, which may suggest a genetic predisposi-
tion. Absence seizures are marked by momentary lapses of consciousness. Absence
seizures often have no visible symptoms, although some patients may have purpose-
less movements during a seizure, such as rapidly blinking eyes. Absence seizures
often have a brief duration, and a person may resume the previous activity imme-
diately after the seizure [23]. These brief seizures can happen several times during
a day, but in some patients, the frequency of absence seizures can be as high as
hundred of times a day, which interferes with the daily activities of a child such as
school. In some cases of childhood absence epilepsy, the seizures stop when a child
reaches puberty. Absence seizures exhibit a characteristic spike-and-wave EEG pat-
tern at a 3 Hz frequency [23].
Figure 18.1 displays a multichannel EEG recording that includes an absence
seizure. The duration of the seizure is approximately 4 s. The figure vividly illus-
trates a characteristic spike-and-wave activity during the seizure.
18.5.3 Numerical Study of Synchrony in Multichannel EEG
Recordings from Patients with Absence Epilepsy
The proposed approach to studying synchronization among multiple channels was
applied to analysis of EEG data recorded from the patience with absence epilepsy.
First, the multiple time series of the instantaneous phases were extracted from the
raw EEG data using the Hilbert transform approach as described in Section 18.4.
In particular, we took advantage of the functions
hilbert
and
angle
readily
available in the MATLAB R 2006a environment.
The VAR modeling and testing were implemented using the R 2.6.1 statisti-
cal software. In our analysis of the instantaneous phases, we incorporated
ar
,
adf.test
,
po.test
,
cajolst
and other functions found in packages tseries
and urca.
