Information Technology Reference
In-Depth Information
The current of the i-th electrode is
I
electrode
A
i
i
electrode
D
(1.65)
with A
i
D 2˛
i
L denoting the surface of segment i. Thus, one arrives at a set of
coupled differential equations
I
electrode
A
1
c
1
dV
1
dt
V
1
C
r
m
1
D g
1;2
.V
2
V
1
/ C
(1.66)
I
electrode
A
2
c
2
dV
2
dt
V
2
C
r
m
2
D g
2;1
.V
1
V
2
/ C
or equivalently
c
1
dV
1
dt
V
1
V
2
V
1
r
1
C
r
M
1
D
C i
1
(1.67)
c
2
dV
2
dt
V
2
V
1
V
2
r
2
C
r
M
2
D
C i
2
Next, it is assumed that the current is injected in only one out of the two cylinders,
that is i
2
D 0.Itisassumedthatr
1
D r
2
D r and r
M
1
D r
M
2
D r
M
.From
Eqs. (
1.66
) and (
1.67
) it holds
c
1
dV
1
dt
C c
2
dV
2
dt
V
1
C
V
2
r
M
C
D i
1
(1.68)
Thus, finally about the segment's resistance one has
V
1
i
1
r
M
.r
C
r
M
/
rC2r
M
D
(1.69)
while the variation of voltage V at the i-th segment is generalized as follows
C
P
k;j
V
k
V
j
C
j
dV
j
dt
V
j
R
j
D
C I
j
(1.70)
R
jk
where R
kj
denotes the resistance of the connection between the k-th and the j-th
segment.
1.7.2
Computing Cable's Equation in Neurons
It is assumed that the cable is defined in the interval .0;l/ and that its cross-section
is circular with diameter d.x/. The cable is segmented in n-parts and the distance
from the origin is defined as x
j
D jh, where h D
l
n
is the elementary length. Each
part has surface equal to A
j
D d
j
h where d
j
D d.x
j
/ is the diameter and the
associated surface of the cross-section is equal to
d
j
4
. It is assumed that the special
resistance and the capacitance of the membrane are c
M
and r
M
, respectively, while
