Information Technology Reference
In-Depth Information
or equivalently
I
cap
.x;t/ C I
ion
.x;t/ DI
L
.x C x;t/ C I
L
.x;t/
(1.37)
out of which one obtains [
16
,
65
]
@V
M
@t
C .2˛x/i
ion
.2˛x/c
M
˛
2
r
L
˛
2
r
L
@V
M
@x
.x C x;t/
@V
M
@x
.x;t/
D
(1.38)
After dividing both sides of the equation with 2˛x and taking x)0 cable's
equation is produced
@
2
V
M
@x
2
c
M
@V
M
@t
˛
2r
L
(1.39)
D
i
ion
V
M
.x;t/
r
M
Setting i
ion
D
one gets
@
2
V
M
@x
2
c
M
@V
M
@t
˛
2r
L
V
M
r
M
D
(1.40)
Cable's equation for the neural cell is formulated as follows [
16
,
65
]
D
2
@
2
V
M
@x
2
M
@V
M
@t
V
M
(1.41)
where D
q
ar
M
2r
L
and
M
D c
M
r
M
.
Next, it is assumed that the neural cell has the form of a semi-infinite cable
for x>0, in which a step current I
M
.0/ D I
0
is injected, while the associated
boundary condition is V
M
.0/. The solution of the partial differential equation given
in Eq. (
1.41
) provides the spatiotemporal variation of the voltage V.x;t/.
By examining the solution of the PDE in steady state, i.e. when
@V
M
@t
D 0,the
partial differential equation becomes an ordinary differential equation of the form
2
d
2
V
ss
dx
2
(1.42)
V
ss
D 0
˛
2
r
L
@V
M
@x
From Eq. (
1.35
), using that I
0
D
one obtains the boundary condition
dV
ss
dx
.0/ D
r
L
˛
2
I
0
(1.43)
while another boundary condition is that V tends to 0 as x!1. The solution of the
steady state (spatial component) of the cable equation is finally written as
˛
2
e
x
r
L
I
0
V
ss
.x/ D
(1.44)
Time constant affects the variation of voltage.
