Biomedical Engineering Reference
In-Depth Information
z
z
(a)
(b)
second source
second source
first source
first source
y
y
10
10
third source
third source
8
8
6
6
sensor array
sensor array
-2
0
2
-2
0
2
y (cm)
y (cm)
Fig. 8.1
The coordinate system and source-sensor configuration used in the numerical experiments.
The plane at
x
=
0 cm is shown. The
small circles
show the locations of the three sources, and the
bold arrows
schematically show their causal relationships assumed in the experiments.
a
The first
experiment with bivariate causal time series.
b
The second experiments with trivariate causal time
series
s
1
(
0
s
1
(
t
)
.
90
t
−
1
)
=
s
2
(
t
)
0
.
16 0
.
8
s
2
(
t
−
1
)
−
s
1
(
0
.
50
t
−
2
)
+
+
e
(
t
).
(8.121)
−
0
.
2
−
0
.
5
s
2
(
t
−
2
)
The time series of the third source,
s
3
(
)
, was generated using Gaussian random
numbers. Thus, the third source activity was independent from either the first or
the second source activities. The causal relationship assumed in this experiment is
depicted in Fig.
8.1
a.
Then, simulated sensor recordings were computed by projecting the time series
of the three sources onto the sensor space by using the sensor lead field. A small
amount of simulated sensor noise was added. The Champagne source reconstruction
algorithm was applied to the simulated sensor recordings. Here, three-dimensional
reconstruction was performed on a region defined as
t
−
4
≤
x
≤
4,
−
4
≤
x
≤
4,
and 6
≤
z
≤
12 cm with a voxel interval equal to 0.5 cm. Reconstructed source time
series
s
1
(
t
)
,
s
2
(
t
)
, and
s
3
(
t
)
were obtained as the time series at voxels nearest to the
assumed source locations.
Once
were obtained, the MVAR coefficients between these time series
were estimated by using the least-squares method in Sect.
8.7.1
. Using the esti-
mated MVAR coefficients, we computed the spectral Geweke causality described in
Sect.
8.4
, as well as coherence. The results are shown in Fig.
8.2
a.
In these results, coherence (Eq. (
8.51
)) can detect an interaction between the first
and second source activities. The spectral Geweke causality in Eqs. (
8.60
) and (
8.61
)
detects the unidirectional information flow from the first source activity to the second
source activity. We also computed the partial directed coherence (PDC) and the
directed transfer function (DTF), with results shown in Fig.
8.2
b. The PDC and DTF
s
1
(
t
)
,
s
2
(
t
)
