Biomedical Engineering Reference
In-Depth Information
Taking
d
/
dx
both sides,
∂
I
∂
⎛
r
∂ΔΦ
⎞
∂
⎛
r
⎞
−=
i
e
−
e
I
⎜
⎟
⎜
⎟
T
∂∂ +∂
z
zrr z
⎝
⎠
∂ +
zrr
⎝
⎠
i
e
i
e
The second term on the right side is zero due to the conservation of current
along the axial direction. Substituting (10.46a),
∂
⎛
r
∂ΔΦ
⎞
Iz
()
=
e
(10.47)
⎜
⎟
m
∂
zr r
⎝
+∂
z
⎠
i
e
I
m
is written as the sum of the ionic current
I
ion
(
x,
ΔΦ
, t
) through the
r
m
and the
current through the membrane capacitance as
∂ΔΦ
(10.48)
Izx I z
()
Δ=
(,
ΔΦ Δ+
,)
tzCmz
t
Δ
m
ion
∂
The ionic current is in general a complex and nonlinear function of membrane
potential modeled, for example, by Hodgkin-Huxley type equations. As for
I
m
,
I
ion
is ionic current per unit length of membrane cylinder. Also,
ΔΦ
I
=−=
I
I
−
I
.
ion
m
e
e
m
Substituting (10.47) into (10.48),
ΔΦ
∂ΔΦ
∂
⎛
r
∂ΔΦ
⎞
−+
IC
=
e
⎜
⎟
e
m
r
∂
t
∂
z
⎝
r
+∂
r
z
⎠
m
i
e
Typically the external current is neglected. Rearranging and substituting
τ
m
=
r
rr
λ =
m
r
m
c
m
and
,
+
c
i
∂ΔΦ
∂ ΔΦ
2
2
(10.49)
τ
=
λ
− ΔΦ
m
2
∂
t
∂
z
Equation (10.49) is referred to as the cable equation and often external current
is used.
is known
as the length constant (or space constant) of the cable and depends on the resistance
of cable membrane, the resistivity of the internal medium, and the geometry of the
cable cross section. Hence, differences in physical properties (e.g., diameter and
membrane properties) and differences in potential occur between compartments
rather than within them. These are not always measurable individually. However,
the space constant
τ
m
is the time constant defined in a single-compartmental model.
λ
is a useful parameter commonly used to quantify the extent to
which voltage changes spread;
λ
λ
is the length of a cable with the same diameter (as
