Biomedical Engineering Reference
In-Depth Information
orientations at the seed and target voxels by simultaneouslymaximizing the canonical
(magnitude/imaginary) coherence between these voxels.
7.7 Envelope Correlation and Related Connectivity Metrics
7.7.1 Envelope Correlation
Coherence is a metric that measures the phase relationship between two spectra, and
naturally it is sensitive to phase jitters. This property becomes problematic when we
try to estimate brain interactions at higher frequencies, such as gamma and high-
gamma activities. This is because, at high frequencies, a small time jitter could cause
a large phase jitter, and coherence may not be able to detect connectivity relation-
ships. That is, the coherence may not be appropriate as a connectivity metric at high
frequencies. The envelope-to-envelope correlation [
14
] is considered an appropriate
metric for such cases.
To compute the envelope correlation, we first convert the seed and target time
courses into their analytic signals, such that
u
t
)
(
i
ˀ
d
t
.
A
[
u
(
t
)
]=
u
(
t
)
+
(7.99)
−
t
t
On the left-hand side of the equation above,
indicates an operator that creates
an analytic signal of the real-valued time signal in the parentheses. Let us define
analytic signals from the seed- and target-voxel time courses
u
S
(
A
[·]
t
)
and
u
T
(
t
)
, such
that
e
i
ʸ
S
(
t
)
,
A
[
u
S
(
t
)
]=
A
S
(
t
)
(7.100)
e
i
ʸ
T
(
t
)
,
A
[
u
T
(
)
]=
A
T
(
)
t
t
(7.101)
where
A
S
(
t
)
and
A
T
(
t
)
are the amplitudes of the seed and target analytic signals,
and
are their instantaneous phases. The envelope correlation is the
correlation between the amplitudes,
A
S
(
ʸ
S
(
t
)
and
ʸ
T
(
t
)
t
)
and
A
T
(
t
)
, and is computed such that
j
=
1
A
T
(
t
j
)
A
S
(
t
j
)
ʘ
=
j
=
1
A
T
(
2
.
(7.102)
2
j
=
1
A
S
(
t
j
)
t
j
)
It is obvious in the above equation that common interferences such as the algorithm
leakage cause spurious correlation, and the seed blur should exist in an image of
envelope correlation.
