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circular variance computed via Hilbert transform, and reported a decease in syn-
chronization preceding epileptic seizures [28, 29]. Whereas in [24], Kraskov et al.
introduced phase synchronization with the phase based on the wavelet transform for
localization of interictal focus in temporal lobe epilepsy.
20.3 Methods
20.3.1 Chaos Theory and Epilepsy
Several studies applied chaos theory to analysis of EEG data [10, 33, 3]. In chaotic
systems, trajectories originating from very close initial conditions diverge exponen-
tially. The system dynamics can be characterized by the rate of the divergence of the
trajectories, which is measured by Lyapunov exponents and dynamical phase.
First, using the method of delays [31], the embedding phase space is constructed
from a data segment
x
(
t
)
with
t
∈
[
0
,
T
]
so that the vector
X
i
of the phase space is
given by
X
i
=(
x
(
t
i
)
,
x
(
t
i
+
τ
)
, ··· ,
x
(
t
i
+(
p
−
1
)
τ
)
,
(20.1)
where
t
i
∈
[
1
,
T
−
(
p
−
1
)
τ
]
,
p
is a chosen dimension of the embedding phase space,
and
denotes the time delay between the components of each phase space vector.
Next, the estimate
L of the short-term largest Lyapunov exponent STLmax
is com-
puted as follows:
τ
N
a
1
N
α
i
=
1
log
2
X
(
t
i
+
Δ
t
)
−
X
(
t
j
+
Δ
t
)
L
=
,
(20.2)
X
(
t
i
)
−
X
(
t
j
)
where
N
a
is the total number of local maximum Lyapunov exponents that are esti-
mated during the time interval
[
0
,
T
]
;
Δ
t
is the evolution time for the displacement
vector
X
(
t
i
)
−
X
(
t
j
)
;
X
(
t
i
)
represents the point of the fiducial trajectory such that
t
is an appropriately
selected vector that is adjacent to in the embedding phase space. In [8], Iasemidis
et al. suggested a method of estimating STLmax in the EEG data based on the Wolf's
algorithm for time series [38].
The short term largest Lyapunov exponent STLmax is proved to be an espe-
cially useful EEG feature for studying the dynamics of the epileptic brain [10,33,3].
Figure 20.1 shows an example of the STLmax curve derived from an EEG channel
over a 140 min time window that includes a seizure. In this example, the STLmax
values gradually decreases before the seizure and drops to the lowest point dur-
ing the ictal state. It immediately reverses to the highest point after the seizure
stops, a phenomenon that we called “seizure resetting” [12]. In addition, transi-
tions among interictal, preictal, ictal, and postictal states can be characterized by the
spatiotemporal changes in STLmax profiles among EEG channels [34]. Figure 20.2
shows a typical spatiotemporal pattern of STLmax profiles.
=
t
i
,
X
(
t
0
)=(
x
(
t
0
)
,
x
(
t
0
+
τ
)
, ··· ,
x
(
t
0
+(
p
−
1
)
τ
))
, and
X
(
t
j
)
