Biomedical Engineering Reference
In-Depth Information
important if the resistivity of the internal medium is low compared to the specific
resistance of the bounding membrane. A small blood vessel can also be treated as a
cable. Resistivity of the internal medium is related to the resistance,
R
i
, by
Resistivity
×
L
(10.42)
R
=
i
2
π
a
where
L
is the length of the cable and
a
is the radius of the cable. For simplification
of the derivation
r
i
is defined as resistances per unit length of cylinder, that is,
R
Resistivity
(10.43)
r
==
i
i
2
L
π
a
Similarly
r
e
is the external resistance per unit length of the cylinder. Consider a
case where a membrane is divided into three compartments of equal length
z
[Fig-
ure 10.9(b)]. Each compartment is assumed to be an isopotential patch of mem-
brane. The membrane of each subcylinder is represented by a parallel combination
of membrane capacitance
c
m
Δ
Δ
z
and a circuit for the ionic conductances in the mem-
brane. The total current through a membrane patch is
I
m
(
z
)
z
and current flowing
in the direction of increasing
x
is assigned a positive sign. The membrane current
varies with distance
x
down the cylinder. If
I
m
and
c
m
are membrane current and
capacitance per unit length of the cylinder, then multiplying
Δ
z
gives the total cur-
rent and capacitance in a subcylinder.
r
m
is the membrane
resistance for the length
of the cylinder with the units of ohm.length.
The membrane potential inside the cell is represented as
Δ
ΔΦ
i
(
z
) and outside
the cell as
ΔΦ
e
(
z
). Assuming that the potentials also vary with distance down the
cylinder, they also become a functions of distance
x
. The membrane potential is
ΔΦ
i
(
z
)
ΔΦ
e
(
z
). As the potentials vary along the length of the cylinder, there will be
currents
I
i
(
z
) and
I
e
(
z
) flowing between the nodes;
I
i
(
z
) is the total current flowing
down the interior of the cylinder and
I
e
(
z
) is the total current flowing parallel to
the cylinder in the extracellular space. In the brain there are many cylinders from
different neurons packed together, so there are many extracellular currents.
I
e
(
z
)
is only the portion of the extracellular current associated with the cylinder under
study. The internal current
I
i
(
z
) flows through resistance
r
i
Δ
−
z
, which is the resist-
ance of the solutions inside the cylinder between the center of one subcylinder and
the center of the next.
r
e
Δ
z
is similarly defined as the resistance in the extracellular
space between the center of two subcylinders, that is, as the resistance to the flow
of current
I
e
(
z
). Ohm's law for current flow in the intracellular and gives:
ΔΦ
i
(
z
)
−
ΔΦ
i
(
z
z.
Rearranging and taking the limit as
+
Δ
z
)
=
I
i
(
z
)
r
i
Δ
Δ
x
goes to 0,
∂ΔΦ
(10.44)
i
=−
Izr
()
i
i
∂
z
