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with boundary conditions ŒC.0/ D ŒC
in
and ŒC.l/ D ŒC
out
. Using these
boundary conditions the solution for the Nernst-Planck equation becomes
ŒC
out
e
u
z
2
FV
M
ˇ
l
ŒC
in
I D
(1.12)
e
l
z
V
M
F
RT
where D
. By defining the membrane's permeability as
ˇ
u
RT
lF
(1.13)
P D
one obtains that the current's equation through the membrane becomes
I D P
z
F
ŒC
out
e
ŒC
in
(1.14)
e
1
It is assumed now that there is flow of a large number of ions, e.g. K
C
,Na
C
, and
Cl
. At the equilibrium condition, the current through the membrane drops to zero,
i.e. it holds
(1.15)
I D I
K
C I
Na
C I
Cl
D 0
The potential of the membrane for which the zeroing of the current I takes place is
F
ln
n
P
K
ŒK
C
out
C
P
Na
ŒNa
C
out
C
P
Cl
ŒCl
in
o
(1.16)
RT
V
M
D
P
K
ŒK
C
in
CP
Na
ŒNa
C
in
CP
Cl
ŒCl
out
where variables P
K
, P
Na
, and P
Cl
denote the associated permeabilities. Equa-
tion (
1.16
) is the Goldman-Hodgkin-Katz and the associated voltage is measured
in mV.
1.2
Equivalent Circuits of the Cell's Membrane
1.2.1
The Electric Equivalent
The functioning of the cell's membrane can be represented as an electric circuit.
To this end: (1) the ion channels are represented as resistors, (2) the gradients of
the ions' concentration are represented as voltage sources, (3) the capability of the
membrane for charge storage is represented as a capacitor (Fig.
1.2
). This is depicted
in the following diagram:
For the charge that is stored in the membrane it holds
q D C
M
V
M
(1.17)
