Biomedical Engineering Reference
In-Depth Information
f
12
, which
are asynchronous and synchronous, respectively, with the larger region. In this case
the small region may be considered part of the large region for dynamics in band
Δ
band
Δ
f
11
, while simultaneously producing dynamics in bands
Δ
f
11
and
Δ
f
11
.
The scalp potential and Laplacian are generated by the mesosource functions
P
1
(
r
,
t
) and
P
2
(
r
,
t
) integrated over the surfaces of their respective regions as given by
(1.5). Local scalp potential and Laplacian measures depend on the sum of contribu-
tions from each of the two mesosource regions. However, the relative contributions
of individual regions (dipole layers) can differ substantially. For example, if the
small and large regions have diameters in the 2- and 10-cm ranges, respectively, we
expect the following relation between surface potentials
f
12
, but separate for dynamics in
Δ
Φ
S
and Laplacians
L
S
:
Φ
Φ
L
L
S
1
S
1
<<
(1.11)
S
2
S
2
Relation (1.11) indicates that smaller cortical layers tend to make larger relative
contributions to the Laplacian, whereas larger regions contribute more to potential.
If the two regions generate dynamics with different dominant frequencies, scalp
potential spectra will differ from scalp Laplacian spectra, a prediction consistent
with experimental observations of spontaneous EEGs [14]. These data indicate that
large and small dipole layers can contribute to different frequencies within the alpha
band, and may or may not have overlapping frequencies.
My outline of the surface Laplacian in this chapter has been mostly qualitative,
but a number of quantitative studies generally support these ideas [2]. For example
a
four-sphere head model
(consisting of an inner brain sphere surrounded by three
spherical shells: CSF, skull, and scalp) may be used with Poisson's equation (1.3) to
estimate the relative sensitivity of the potential and surface Laplacian measures to
dipole layer source regions of different sizes. Figure 1.6 shows scalp potential
directly above the centers of dipole layers of varying angular extent, forming super-
ficial spherical caps in the four-sphere head model. The four curves shown in each
figure correspond to four different ratios of brain-to-skull conductivity. Each curve
in the upper part of Figure 1.6 shows scalp potential as a percentage of transcortical
potential
V
C
, which is roughly related to the local normal component of cortical
mesosource function
P
through Ohm's law; that is,
σ
V
d
CC
P
~
(1.12)
C
σ
C
and
d
C
are the local conductivity and thickness of cortex, respectively.
Transcortical potential has been estimated in experiments with mammals, typically
V
C
Here
V for spontaneous EEGs [4, 15]. Given this intracranial data, Fig-
ure 1.6 suggests maximum scalp potentials of roughly 30 to 150
100
−
300
μ
V for dipole lay-
ers with spherical cap radii of about 8 cm or gyral surface areas of several hundred
square centimeters.
In the lower part of Figure 1.6, the relative scalp Laplacian is plotted due to the
same dipole layers. While potentials are shown to be primarily sensitive to broad
dipole layers, Laplacians are sensitive to smaller layers, as implied by Figure 1.5. In
μ
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