Biomedical Engineering Reference
In-Depth Information
According to bifurcation theory, when these parameters change slowly over time, or
the system is close to a bifurcation, dynamics slow down and conditions of
stationarity are better satisfied.
Theoretically, if the state space is of
d
dimensions, we can estimate up to
d
Lyapunov exponents. However, as expected, only
d
+ 1 of these will be real. The
rest are spurious [61]. The estimation of the largest Lyapunov exponent (
L
max
)ina
chaotic system has been shown to be more reliable and reproducible than the esti-
mation of the remaining exponents [78], especially when
d
is unknown and changes
over time, as in the case of high-dimensional and nonstationary data such as EEGs.
Before we apply the STL
max
to the epileptic EEG data, we need to determine the
dimension of the embedding of an EEG segment in the state space. In the ictal state,
temporally ordered and spatially synchronized oscillations in the EEG usually per-
sist for a relatively long period of time (in the range of minutes for seizures of focal
origin). Dividing the ictal EEG into short segments ranging from 10.24 to 50 sec-
onds in duration, estimation of
from ictal EEGs has given values between 2 and 3.
These values stayed relatively constant (invariant) with the shortest duration EEG
segments of 10.24 seconds [79, 80]. This implies the existence of a low-dimensional
manifold in the ictal state, which we have called an epileptic attractor. Therefore, an
embedding dimension
d
of at least 7 has been used to properly reconstruct this epi-
leptic attractor. Although
d
for interictal (between seizures) EEGs is expected to be
higher than that for ictal states, we have used a constant embedding dimension
d=
7
to reconstruct all relevant state spaces over the ictal and interictal periods at differ-
ent brain locations. The strengths in this approach are that: (1) the existence of irrel-
evant information in dimensions higher than 7 might not have much influence on
the estimated dynamic measures, and (2) reconstruction of the state space with a
low
d
suffers less from the short length of moving windows used to handle
nonstationary data. A possible drawback is that related information to the transi-
tion to seizures in higher dimensions will not be accurately captured.
The STL
max
algorithm is applied to sequential EEG epochs of 10.24 seconds in
duration recorded from electrodes in multiple brain sites. A set of STL
max
profiles
over time (one STL
max
profile per recording site) is thus created that characterizes
the spatiotemporal chaotic signature of the epileptic brain. A typical STL
max
profile,
obtained by analysis of continuous EEGs at a focal site, is shown in Figure 3.27(a).
This figure shows the evolution of STL
max
as the brain progresses from preictal
(before a seizure) to ictal (seizure) to postictal (after seizure) states. There is a grad-
ual drop in STL
max
values over tens of minutes preceding the seizure at this focal site.
The seizure is characterized by a sudden drop in STL
max
values with a subsequent
steep rise in STL
max
that starts soon after the seizure onset, continues to the end of
the seizure, and remains high thereafter until the preictal period of the next seizure.
This dynamic behavior of STL
max
indicates a gradual preictal reduction in
chaoticity at the focal site, reaching a minimum within the seizure state, and a
postictal rise in chaoticity that corresponds to the brain's recovery toward normal,
higher rates of information exchange. What is more consistent across seizures and
patients is an observed synchronization of STL
max
values between electrode sites
prior to a seizure. This is shown in Figure 3.27(b). We have called this phenomenon
preictal dynamic entrainment (dynamic synchronization), and it has constituted the
basis for the development of the first prospective epileptic seizure prediction algo-
ν
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