Biomedical Engineering Reference
In-Depth Information
Combining Eqs. (8.1), (8.2), and (8.3),
I
o
4
πr
2
a
r
=
=−
σ
e
∇
J
φ
e
.
(8.5)
We know, however, that components of
∇
φ
e
can only be in the radial direction, so
σ
e
dφ
e
I
o
4
πr
2
a
r
−
dr
=
(8.6)
integration with respect to
r
yields
I
o
4
πσ
e
r
φ
e
=
.
(8.7)
Note that in Eq. (8.7),
φ
e
is the same for any surface where
r
is constant (concentric spheres).
We have mentioned potentials before but it is worth giving a formal definition here. The electric
potential at a point is simply the work required to move a single charge (e.g.,
e
−
,
K
+
) from infinity to
that point against the electric field. Therefore, as
r
,
φ
e
must equal zero.
Equation (8.7) is useful when there is one current source. When more than one current source is
present, we can sum up the effects of all the currents. First, we can assume each current is generated within
some small volume,
dV
, and has some incremental effect on the overall
φ
e
. Differentiating Eq. (8.7)
→∞
I
o
dV
4
πσ
e
r
dφ
e
=
(8.8)
and to sum up the effects
I
m
(x,y,z)
r
1
4
πσ
e
φ
e
(x
,y
,z
)
=
dV
(8.9)
(x
x
)
2
y
)
2
z
)
2
r
=
−
+
(y
−
+
(z
−
(8.10)
where
r
is the distance from each current source,
(x,y,z)
, to the recording point,
(x
,y
,z
)
. Note that
we have assumed that the
I
o
of Eq. (8.8) is the membrane current,
I
m
, emanating from a small patch of
membrane. In this way, we can theoretically calculate potentials in a bath surrounding an active neuron,
given all of the
I
m
sources (see Fig. 8.1).
8.1.1 Forward and Inverse Problems
If all of the
I
m
and
r
terms are known,
φ
e
may be calculated using Eq. (8.9). This calculation is known as
the
forward problem
and has a unique solution. Equation (8.9) also demonstrates a fundamental problem
encountered in all extracellular recordings.
φ
e
is an
average
of the effect of many current sources, weighted
by the radii. As in any average, information is lost. Therefore, if you record
φ
e
there is no way to back
calculate all of the
I
m
sources. In fact, you may not even know how many current sources compose
φ
e
.
Computing
I
m
sources from
φ
e
is called the
inverse problem
.Despite the limitations of the inverse problem,
φ
e
is much easier to record and is the typical measurement made for both research and clinical purposes.
Although we will focus on the forward problem here, there exists a large body of research on gaining the
maximum amount of information from
φ
e
. We will explore some of these methods in Ch. 9.
