Information Technology Reference
In-Depth Information
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
10
20
30
40
50
60
70
80
time
Fig. 1.5
Diagram of membrane's voltage variation
1.3
Describing Membrane's Voltage Dynamics with Cable's
Equation
The previous analysis has considered that the neurons have a spherical shape. This
assumption does not hold for neurons' axon and in the case of neurons' dendrites
where the neurons' shape is better approximated by a cylinder. In the latter case it
is considered that the neurons have the shape of a long cylinder or cable, or cable of
radius a as depicted in Fig.
1.6
.
The equation of the membrane's voltage according to the cable's model is a
partial differential equation that describes the voltage's spatiotemporal variation
V
M
.x;t/. Voltage V
M
.x;t/ depends also on the coaxial currents I
L
that pass
through the cell [
21
].
The aggregate resistance of the cable, for radius a and length x is given by
r
L
x
˛
2
R
L
D
(1.33)
where r
L
is a special resistance. According to Ohm's law, for the change of voltage
along the cables's axis holds [
16
,
65
]
V
M
.x C x;t/ V
M
.x;t/ DI
L
.x;t/R
L
DI
L
.x;t/
x
˛
2
r
L
(1.34)
