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E R D .g Cl E Cl C g K E K C g Na E Na /r M
(1.26)
1
g Cl C g K C g Na
r M D
(1.27)
it holds that
c M dV M
dt
.V M E R /
r M
I.t/
A
D
C
(1.28)
Considering that the conductance coefficients are constants and that the currents
have reached their steady state one obtains
g Cl E Cl C g K E K C g Na E Na C I=A
g Cl Cg K Cg Na
V ss D
(1.29)
Therefore, in case that there is no external current, the membrane's voltage V M in
the steady state is the weighted sum of the equilibrium potentials E Cl , E K , and E Na
(according to Nernst's equation) for the three ionic channels.
1.2.2
Membrane's Time Constant
It is assumed that the external current applied to the membrane is I.t/ D I 0 between
time instant t D 0 and t D T . Assuming that the cell's shape is approximated by a
sphere with surface A D 4 2 , the normalized (over surface and time unit) current
is
(
I 0
I.t/
4 2 D
4 2 if 0t<T
0 otherwise
I M .t/ D
(1.30)
Using Eq. ( 1.28 ) and considering E R D 0 for the membrane's potential, the
following partial differential equation holds
c M dV M
dt
V M
r M
D
C I M .t/
(1.31)
Solving the differential equation with respect to V M .t/ one obtains the change in
time of the membrane's potential which is described by Eq. ( 1.32 ) and is depicted
in Fig. 1.5
( r M I 0
t
m / for 0<t<T
V M .T/e t
4 2 .1 e
V M .t/ D
(1.32)
for t>T
M
 
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