Biomedical Engineering Reference
In-Depth Information
which is simplified to
2
1
i
1
i
2
ˁ(
r
)
=
ˉ
˃
∇
ʦ
(
r
)
−
μ˃(
i
ˉ)
+
μ˃
ʦ
(
r
)
−
ˉ
∇·
J
S
(
r
).
(A.30)
By substituting Eq. (
A.30
)into(
A.29
), we obtain
2
∇
ʦ
(
r
)
−
i
ˉμ(˃
+
i
ˉ)
ʦ
(
r
)
2
1
ˉ)
+
μ˃
1
2
=−
ˉ
˃
∇
ʦ
(
r
)
+
μ˃ (
i
ʦ
(
r
)
+
ˉ
∇·
J
S
(
r
).
i
i
The equation above results in
)
=
∇·
J
S
(
r
)
2
k
2
∇
ʦ
(
r
)
+
ʦ
(
r
ˉ
.
(A.31)
˃
+
i
In the absence of boundary conditions, Eqs. (
A.28
) and (
A.31
) have the following
well-known solutions.
r
)
4
J
S
(
e
−
ik
|
r
−
r
|
d
3
r
,
A
(
r
)
=
(A.32)
r
|
ˀ
|
r
−
V
and
∇
·
r
)
1
J
S
(
e
−
ik
|
r
−
r
|
d
3
r
.
ʦ
(
r
)
=−
(A.33)
4
ˀ(˃
+
i
ˉ)
|
r
−
r
|
V
Note that the derivation of these solutions has not involved any assumptions about the
frequency
. The typical values of these parameters,
together with assumptions about the frequency
ˉ
or the tissue parameters
˃, μ,
, allow simplifications of the above
equations suitable for EEG and MEG. The phase shifts of the electric potentials due
to capacitive effects are accounted by the denominator of Eq. (
A.33
):
ˉ
i
ˉ
˃
(˃
+
i
ˉ)
=
˃(
1
+
).
(A.34)
Thus capacitive effects may be ignored if
1. Although the very highest
frequencies generated by the neural activity corresponding to action potentials may
reach 1000 Hz, MEG and EEG signals are expected to be dominated by synaptic
and dendritic activity that involves slower time scales less than 100Hz (
ˉ/˃
ˉ<
200
ˀ
rad/s).
We make use of measured biological parameters to evaluate the dimensionless
quantity
ˉ/˃
. To a best approximation, biological tissues are not magnetizable,
10
−
7
H/m, the value of vacuum. The dielectric permittivity
μ
=
ˀ
×
thus
4
is tissue
ˉ/˃
.
and frequency dependent such that for brain and scalp,
0
01 at 100 Hz, and
ˉ/˃
0
.
03 at 1000 Hz. Thus, capacitive effects may be ignored to within 1%
errors.
