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In-Depth Information
where
V.k;t/D
R
C1
1
e
ikx
V.x;t/
dx
V
0
D
R
C1
1
e
ikx
V
o
.x/
dx
(1.78)
I.k;t/D
R
C1
1
e
ikx
I.x;t/
dx
Using the definition of the inverse Fourier transform one has
2
R
C1
V.k;t/
dk
1
e
ikx
V.x;t/D
(1.79)
1
one finds that the solution V.x;t/is given by
V.x;t/D
R
C1
1
R
t
0
R
C1
r
M
G.x y;t/V
o
.y/
dy
C
G.x y;t s/I.y;s/
dyds
(1.80)
1
where the Green function G.x;t/ is defined as
e
t
e
x
2
=.
4
2
t
/
1
q
4
2
t
G.x;t/ D
(1.81)
If the membrane is in rest state at time instant 0, i.e. V
0
.x/ D 0 and for I.x;t/ D
I
0
ı.x/ı.t/, then
e
tx
2
4
2
e
.
t
/
r
M
I
0
p
4t
V.x;t/D
(1.82)
If I.x;t/ D I
o
u
.x/, i.e. a step input is applied at position x, then
q
t
q
t
i
(1.83)
h
e
x
erfc
x
p
e
x
erfc
x
p
r
M
I
o
4
V.x;t/D
2
p
t
2
p
t
C
p
R
x
e
y
2
dy, while in steady-state the solution becomes
2
where erfc D
4
e
j
x
j
r
M
I
0
(1.84)
V
ss
.x/ D
1.8
Ion Channels and Their Characteristics
1.8.1
Types of Ion Channels
In the set of equations that constitute the Hodgkin-Huxley model one can
distinguish the K
C
and Na
C
ion channels, whereas in the Morris-Lecar model
one can distinguish the Ca
C
model. However, there may be more ion channels
across the neuron's membrane which are described as follows [
16
,
60
,
65
].
