Biomedical Engineering Reference
In-Depth Information
6.4.5 Lyapunov Exponent
During the past decade, several studies have demonstrated experimental evidence
that temporal lobe epileptic seizures are preceded by changes in dynamic properties
of the EEG signal. A number of nonlinear time-series analysis tools have yielded
promising results in terms of their ability to reveal preictal dynamic changes essen-
tial for actual seizure anticipation.
It has been shown that patients go through a preictal transition approximately
0.5 to 1 hour before a seizure occurs, and this preictal state can be characterized by
the
Lyapunov exponent
[12, 22-29]. Stated informally, the Lyapunov exponent
measures how fast nearby trajectories in a dynamical system diverge. The noted
approach therefore treats the epileptic brain as a dynamical system [30-32]. It con-
siders a seizure as a transition from a chaotic state (where trajectories are sensitive to
initial conditions) to an ordered state (where trajectories are insensitive to initial
conditions) in the dynamical system. The Lyapunov exponent is a nonlinear mea-
sure of the average rate of divergence/convergence of two neighboring trajectories in
a dynamical system dependent on the sensitivity of initial conditions. It has been suc-
cessfully used to identify preictal changes in EEG data [22-24]. Generally,
Lyapunov exponents can be estimated from the equation of motion describing the
time evolution of a given dynamical system. However, in the absence of the equation
of motion describing the trajectory of the dynamical system, Lyapunov exponents
are determined from observed scalar time-series data,
x
(
t
n
)
x
(
n t
), where
t
is the
sampling rate for the data acquisition. In this situation, the goal is to generate a
higher dimensional vector embedding of the scalar data
x
(
t
) that defines the state
space of the multivariate brain dynamics from which the scalar EEG data is derived.
Heuristically, this is done by constructing a higher dimensional vector
x
i
from the
data segment
x
(
t
) of given duration
T
, as shown in (6.13) with
=
defining the embed-
ding delay used to construct a higher dimensional vector
x
from
x
(
t
) with
d
as the
selected dimension of the embedding space and
t
i
being the time instance within the
period [
T
τ
−
(
d
−
1)
τ
]:
[
]
(
)
()( )
( )
x
i
=
xt
,
xt
−
τ
,
,
xt
−
d
−
1
τ
(6.13)
i
i
i
>
d
min
,
x
i
provides a faithful representation of the phase space for the dynamical sys-
tems from which the scalar time series was derived. A suitable practical choice for
d
,
the embedding dimension, can be derived from the “false nearest neighbor” algo-
rithm. In addition, a suitable prescription for selecting the embedding delay,
The geometrical theorem of [33] tells us that for an appropriate choice of
d
,is
also given in Abarbanel [34]. From
x
i
a most stable short-term estimation of the larg-
est Lyapunov exponent can be performed that is referred to as the
short-term largest
Lyapunov exponent
(STL
max
) [24]. The estimation
L
of STL
max
is obtained using
(6.14) where
τ
x
ij
(0)
=
x
(
t
i
)
−
x
(
t
j
) is the displacement vector, defined at time points
t
i
t
, and where
N
is the total number of local STL
max
that will be estimated within the time period
T
of
the data segment, where
T
and
t
j
and
x
ij
(
Δ
t
)
=
x
(
t
i
Δ
t
)
−
x
(
t
j
Δ
t
) is the same vector after time
Δ
=
N
Δ
t
+ (
d
−
1)
τ
:
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