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According to Takens, in order to properly embed a signal in the state space, the
embedding dimension
d
should at least be equal to
. Of the many different
methods used to estimate
D
of an object in the state space, each has its own prac-
tical problems [13]. The measure most often used to estimate
D
is the state space
correlation dimension
(
2
D
+
1
)
from experimental data have
been described in [1] and were employed in our work to approximate
D
in the ictal
state. The brain, being nonstationary, is not expected to be in a steady state in the
strict dynamical sense at any location. Arguably, activity at brain sites is constantly
moving through steady states, which are functions of certain parameter values at
a given time. According to bifurcation theory [5], when these parameters change
slowly over time (e.g., when the system is close to a bifurcation), dynamics slow
down and conditions of stationarity are better satisfied. In the ictal state, temporally
ordered and spatially synchronized oscillations in the EEG usually persist for a rela-
tively long period of time (in the range of minutes). Dividing the ictal EEG into short
segments ranging from 10.24 to 50 s in duration, estimation of
ν
. Methods for calculating
ν
from ictal EEG has
produced values between 2 and 3 [7], implying the existence of a low-dimensional
manifold in the ictal state, which we have called “epileptic attractor.” Therefore,
an embedding dimension
d
of at least 7 has to be used to properly reconstruct this
epileptic attractor.
Although
d
of interictal (between seizures) EEG data is expected to be higher
than that of the ictal state, a constant embedding dimension
d
ν
7 has been used
to reconstruct all relevant state spaces over the ictal and interictal periods at differ-
ent brain locations. The advantages of this approach are: (a) existence of irrelevant
information in dimensions higher than 7 might not influence much the estimated dy-
namical measures, and (b) reconstruction with high
d
requires longer data segments,
which may interfere with the nonstationary nature of the EEG. The disadvantage is
that possibly existing relevant information about the transition to seizures in higher
than
d
=
7 dimensions may not be captured.
The Lyapunov exponents measure the information flow (bits/s) along local eigen-
vectors as the system moves through such attractors. Theoretically, if the state space
is of
d
dimensions, we can estimate up to
d
Lyapunov exponents. However, as ex-
pected, only
D
=
1 of these will be real. The others are spurious [15]. Methods for
calculating these dynamical measures from experimental data have been published
in [8]. The estimation of the largest Lyapunov exponent (
L
max) in a chaotic sys-
tem has been shown to be more reliable and reproducible than the estimation of the
remaining exponents [20], especially when
D
is unknown and changes over time,
as in the case of high-dimensional and nonstationary data (e.g., interictal EEG).
A method to estimate an approximation of
L
max from nonstationary data, called
STL (short-term Lyapunov) [8, 7], has been developed via a modification of the
Wolf's algorithm used to estimate
L
max from stationary data [21]. The STL
max
algorithm is applied to sequential EEG segments recorded from electrodes in mul-
tiple brain sites to create a set of STL
max
profiles over time (one STL
max
profile per
recording site) that characterize the spatiotemporal chaotic signature of the epileptic
brain. The consistent observation across seizures and patients is the convergence of
STL
max
values between electrode sites prior to seizures. We have called this phe-
+
