Biomedical Engineering Reference
In-Depth Information
thalamus was reported in [
13
,
14
]. We also show, in Sect.
9.5
, that lPAC and xPAC
can be detected from a healthy subject using hand-motor MEG data.
9.4 Quantification of Phase-Amplitude Coupling
9.4.1 Instantaneous Amplitude and Phase
In the PAC analysis, it is essential to calculate the instantaneous amplitude of the
high-frequency signal and the instantaneous phase of the low-frequency signal. The
procedure to obtain these quantities in the source space analysis is summarized as
follows. Let us define the voxel time course as
x
. The voxel time course is band-
pass filtered into specific HF and LF bands of interest, resulting in the LF signal
x
L
(
(
t
)
.
The Hilbert transform is then applied to extract their instantaneous amplitudes
and phases, such that
t
)
and the HF signal
x
H
(
t
)
A
[
x
L
(
t
)
]
=
A
L
(
t
)
exp [
i
ʸ
L
(
t
)
]
,
(9.1)
A
[
x
H
(
t
)
]
=
A
H
(
t
)
exp [
i
ʸ
H
(
t
)
]
,
(9.2)
A
[·]
where
indicates an operator that converts a real-valued signal into its analytic
signal. In the equations above,
A
L
(
)
and
A
H
(
)
t
t
are the instantaneous amplitudes of
the LF and HF signals, and
are the instantaneous phases of the LF
and HF signals. Once amplitudes and phases of the LF and HF signals are obtained,
we proceed with the evaluation of the strength of the PAC using measures described
in the following subsections.
ʸ
L
(
t
)
and
ʸ
H
(
t
)
9.4.2 Amplitude-Phase Diagram
To evaluate the strength of the PAC, it is necessary to derive an empirical distribution
of the HF signal amplitude,
A
H
(
t
)
, with respect to the phase of the LF signal,
ʸ
L
(
t
)
.
To obtain such an empirical distribution, we observe
A
H
(
for a time window con-
taining several cycles of the LF signal, and the amplitude values
A
H
(
t
)
are classified
according to the phase of the LF signal. Let us divide the phase value into total
q
intervals such as
t
)
j
−
1
j
q
]
ʔ
1
,ʔ
2
,...,ʔ
q
, where
ʔ
j
=[
2
ˀ
,
2
ˀ
. There are multiple
q
(and usually many) values of
A
H
(
t
)
obtained when
ʸ
L
(
t
)
has a value within
ʔ
j
.The
mean of such values of
A
H
(
t
)
is computed, and it is denoted
ʨ
j
, i.e.,
ʨ
j
=
A
H
(
t
)
ʸ
L
(
t
)
∈
ʔ
j
,
(9.3)
