Biomedical Engineering Reference
In-Depth Information
·
ʸ
L
(
t
)
∈
ʔ
j
where
indicates taking the mean of amplitude values obtained when
ʸ
L
(
)
ʔ
j
. The mean amplitude
ʨ
j
is computed for all the phase bins
t
belongs to
ʔ
1
,...,ʔ
q
, and the resultant set of
ʨ
1
,...,ʨ
q
represents the distribution of the
mean HF signal amplitude with respect to the phase of the LF signal.
The plot of these mean amplitudes
ʨ
j
with respect to the phase bins is called the
amplitude-phase diagram, which directly expresses the dependence of the HF signal
amplitude on the phase of the LF signal. Therefore, if no PAC occurs,
A
H
(
t
)
does not
depend on
ʨ
j
is independent from the index
j
, resulting in
a uniform amplitude-phase diagram. On the contrary, if a PAC occurs, the amplitude
of the HF signal becomes stronger (or weaker) at specific phase values of the LF
signal, and the amplitude-phase diagram deviates from the uniform distribution. We
show several examples of the amplitude-phase diagram in Sect.
9.5
.
ʸ
L
(
t
)
. Thus, the value of
9.4.3 Modulation Index (MI)
The modulation index [
15
] is a measure that quantizes the deviation of the amplitude-
phase diagram from a uniform distribution. To compute the modulation index, we
first normalize the mean amplitude values,
ʨ
1
,...,ʨ
q
, by their total sum. That is,
the normalized mean amplitude value for the
j
th phase bin, expressed as
p
(
j
)
,is
obtained as
ʨ
j
q
i
=
p
(
j
)
=
1
ʨ
i
.
(9.4)
This
p
is the normalized version of the amplitude-phase diagram, which can be
interpreted as the empirically derived probability distribution of the occurrence of
A
H
(
(
j
)
t
)
obtained when
ʸ
L
(
t
)
belongs to
ʔ
j
. If no PAC occurs, there is no specific
relationship between
A
H
(
t
)
and
ʸ
L
(
t
)
, resulting in
p
(
j
)
having a uniform value.
Thus, the difference of the empirical distribution
p
from the uniform distribution
can be a measure of the strength of PAC. The modulation index,
(
j
)
M
I
, employs the
Kullback-Leibler distance to assess this difference, and is defined as
log
p
q
(
j
)
M
I
=
K
[
p
(
j
)
u
(
j
)
]
=
p
(
j
)
,
(9.5)
u
(
j
)
j
=
1
where
u
.
The time variation of the modulation index expresses temporal dynamics of the
PAC, which represents the changes in the functional states of a brain. To compute the
modulation index time variation, the source time course is divided into multiple time
windows, and a value of the modulation index is obtained from each time window.
Assuming that a total of
K
windows are used, we obtain a series of modulation index
values,
(
j
)
is the uniform distribution, which is
u
(
j
)
=
1
/
q
(
j
=
1
,...,
q
)
M
I
(
t
1
),M
I
(
t
2
), . . . ,M
I
(
t
K
)
, which expresses the temporal change of the
PAC strength.
