Biomedical Engineering Reference
In-Depth Information
r
e
)
c
m
∂
2
V
m
∂x
2
∂V
m
∂t
+
V
m
r
m
=
(r
i
−
(4.11)
where we have replaced the
d/dt
terms by
∂/∂t
to indicate that this is a partial differential equation. A
bit of algebra yields the following:
∂
2
V
m
∂x
2
1
r
i
−
=
c
m
∂V
m
V
m
r
m
∂t
+
(4.12)
r
e
∂
2
V
m
∂x
2
r
m
r
i
−
∂V
m
∂t
+
=
r
m
c
m
V
m
(4.13)
r
e
or in the typical core-conductor form
λ
2
∂
2
V
m
∂x
2
∂V
m
∂t
+
=
τ
m
V
m
(4.14)
where
τ
m
=
r
m
c
m
is the
membrane time constant
in
msec
, as defined in Sec. 2.2 and
r
m
r
i
+
λ
=
(4.15)
r
e
is the
cable space constant
in units of
cm
.
4.1.2 A Simplification
One very common simplification to the cable equation is achieved by assuming that the extracellular
bath is much more conductive than the intracellular space. As a result, we can assume that
r
e
≈
0.Asa
result, all of
th
e
φ
e
potentials are equal and the extracellular bath acts as a ground. Therefore,
V
m
=
φ
i
and
λ
r
r
i
=
.
4.1.3 Units and Relationships
The units of the variables in Eqs. (4.1)-(4.15) can be confusing because they are different than the units
used in Ch. 2. The tables show the units used in parameters of the core conductor model and their
relationship to the membrane parameters.
Given these relationships we can rewrite Eq. (4.14) as:
β
C
m
∂
2
V
m
∂x
2
1
(R
i
+
∂V
m
∂t
+
V
m
R
m
=
(4.16)
R
e
)
where
β
=
2
/a
.
