Wave Dynamics in the Transmission of Neural Signals - Advanced Models of Neural Networks - page 134

Fig. 6.1 Myelinated axis

The capacitance due to myelination is defined from the following relation

ln.a 2 =a 1 /
2C m d m L

1

c m D

(6.8)

where L is the length of the myelinated area, C m is the capacitance of the material,
and d m is the thickness of the cellular wall. The potential on the myelinated
membrane satisfies a diffusion PDE of the form

4a 1
R L

@ 2 V
@t 2

c m
L

@V

(6.9)

@t D

where R L is the transmembrane resistivity. The above diffusion equation can be also
written as

@t D D @ 2 V

@V

(6.10)

@t 2

4a 1 L

where D D

c m R 1 . It holds that the transmembrane conductance and the capacitance
of the myelinated areas is about 1,000 times smaller comparing to unmyelinated
areas. The membrane's potential varies linearly between the nodes

.V n V n 1 /x
L

(6.11)

V.x/D V n1 C

The current at the node is proportional to the gradient of the voltage at the
myelinated segments. Thus, at node n voltage satisfies the relation

AC m dV n
dt

DAI ionic .V n / C I n I n1 (6.12)

where A is the area of the membrane which corresponds to the node and the
longitudinal current is

4a 1
R l @x D 4a 1 .V n V n 1 /

@V

(6.13)

I n D

R 1 L

Area A is the surface of the membrane emerging at the node and is the length of
the node. By dividing with surface A Eq. ( 6.12 ) becomes