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negative if close pairs in
Y
would correspond mainly to distant pairs in
X
.
H
(
k
)
(
)
is a linear measure thus it is more sensitive to weak dependencies compared to
mutual information. Arnhold et al. [2] also showed
H
was more robust against noise
and easier to interpret than
S
. Since
H
is not normalized Quian Quiroga et al. [26]
introduced another
N
X
|
Y
(
X
|
Y
)
:
R
(
k
)
N
n
=
1
1
N
R
n
(
X
)
−
(
X
|
Y
)
n
N
(
k
)
(
X
|
Y
)=
,
(19.19)
R
n
(
X
)
which is normalized between 0 and 1. The opposite interdependencies
S
(
Y
|
X
)
,
H
(
Y
|
X
)
, and
N
(
Y
|
X
)
are defined in complete analogy and they are in general not
equal to
S
, respectively. Using nonlinear interdepen-
dencies on several chaotic models (Lorenz, Roessler, and Henon models) Quian
Quiroga et al. [24] showed the measure
H
is more robust than
S
.
The asymmetry of above nonlinear interdependencies is the main advantage over
other synchronization measures. This asymmetry property can give directionality of
nonlinear interdependence between different cortical regions, and reflects different
properties of brain functions when it is important to detect causal relationships. It
should be clear that the above nonlinear interdependencies measures were bivari-
ate measures. Finally, although directional measures quantify the “driver-response”
relationship for a given input, the system under study might be driven by other un-
observed sources.
(
X
|
Y
)
,
H
(
X
|
Y
)
, and
N
(
X
|
Y
)
19.4 Statistical Tests and Data Analysis
All mutual informations between all pairs of electrodes were computed (exclud-
ing reference channels and channels with themselves). In Fig. 19.3, we present the
amount of mutual information for every patient before and after treatment. For ev-
ery heatmap figure, every axis corresponds to the channels and the intensity of each
pixel correspond to the amount of mutual information (in nats). Qualitatively, we
can see that the first column plots are darker than the second column plots, implying
that a mutual information decoupling occurs. In order to statistically validate this
assumption we performed a paired
t
-test with replacement.
For this we used bootstrap resampling technique [8] to investigate the variability
of the strength of interdependence among different brain cortical areas. In boot-
strap resampling, we randomly sample, with replacement, 10.24 s continuous EEG
recordings. We emphasize that resample should be performed on the parts of EEG
where no SWD is presented.
The reference A1 and A2 channels (inactive regions) were excluded from the
analysis. Two sample
t
-test (
N
=
α
=
.
05) was used to test the statistical
differences on mutual information and nonlinear interdependence during, before,
and after treatment. Low mutual information between different cortex regions were
observed in our subjects with less severity of ULD. Furthermore, for each patient
both mutual information between different brain cortical regions decreased after 2
30,
0
