Information Technology Reference
In-Depth Information
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