Biomedical Engineering Reference
In-Depth Information
becomes less dense, and so the buoyant force begins to decrease. At some elevation, the buoyant force and
gravitational forces will exactly cancel out and the balloon will no long move up or down. The height of
the balloon is the point at which two forces balance and is similar to the resting membrane voltage. As the
balloon slowly leaks Helium, however, the buoyant force will decrease. The partially deflated balloon will
sink to reestablish the balance of buoyant gravitational forces. Eventually, the balloon will have leaked so
much Helium that it can not overcome the force of gravity.
2.4.3 Definition of Resting Membrane Voltage
Although we have an intuitive feel for
V
res
m
, we can now derive a more formal mathematical definition.
At rest the membrane voltage is not changing so
dV
m
dt
0. If no stimulus is being applied, then Eq. (2.8)
reveals that
I
ion
must equal zero. The meaning is that no
net
current is crossing the membrane. We can
only say net current because in general,
I
ion
may be composed of many currents which may balance one
another.
As
F
φ
and
F
C
are the driving forces behind ion movement, if
F
C
=
=
F
φ
then charged particles will
cross the membrane, i.e.,
I
ion
=
0 this means that
F
C
=
F
φ
, or alternatively, the current
due to the potential gradient,
I
φ
, is equal to the current due to the concentration gradient,
I
c
. In the next
sections we will use Fick's Law and Ohm's law to define
I
φ
and
I
c
.
0). For
I
ion
=
2.4.4 Fick's Law and Chemical Gradients
Fick's Law describes the
flux
(
mol
m
2
s
) of ions through an area of membrane with a thickness of
dx
due to
·
a concentration gradient.
D
dC
dx
I
c
=−
(2.24)
where the
diffusion coefficient
,
D
(
m
2
s
), is a material property of the membrane and is a measure of how
easily ions can pass.
C
is a concentration in
mol
m
3
.
2.4.5 Ohm's Law and Electrical Gradients
As stated earlier, Ohm's Law for a passive membrane is
V
m
=
I
φ
R
m
(2.25)
φ
i
−
V
m
R
m
=
φ
e
R
m
I
φ
=
.
(2.26)
The specific membrane resistivity,
R
m
, may be thought of as the ability of some ion to pass across the
thickness (
dx
) of the membrane
μ
p
C
·
|
Z
|
dx
R
m
=
(2.27)
Z
where
μ
p
is the ion mobility.
Z
is the ion valence, so
Z/
|
Z
|
is the charge sign (+ or -).
C
is the ionic
concentration.
