Wave Dynamics in the Transmission of Neural Signals - Advanced Models of Neural Networks - page 136

where A D A 0 is the charge in the area of the membrane as a function of

x and t and c D p 1=. s A / is the density-dependent velocity of the wave. Here,
0 is the density of the membrane at physiological conditions. To the extent that the
compressibility is independent of the density and the amplitude of the propagating
density wave A << 0 the wave's velocity is approximately constant (c D c 0 ).
Thus, the wave equation is simplified in the form

@ 2

D c 0 @ 2

@t 2 A @x 2 A

(6.21)

The compressibility s depends on temperature and on density of the lipid
membranes. The lipids of the membrane can change from the liquid to the gel
state. At densities near the phase transition, where the two phases co-exist, a small
increase in pressure can cause a significant increase in density by converting lipids
from liquid to gel. Near this phase transition, the compression modulus becomes
significantly smaller. In the latter case, the wave's velocity is approximated by

c 2

A s D c 0 C p A

1

C q. A / 2 D

(6.22)

with p<0and q>0.
A more complicated form of the wave's propagation in the nerve is obtained by
including the term h@ 4 A =@x 4 with h>0. This term shows that compressibility
decreases at higher frequencies and leads to a propagation velocity which increases
with increasing frequency. The term also shows a linear dependence on frequency
of the pulse propagation velocity. Changes in coefficient h affect the spatial size of
the solitary waves but not their functional form. Substituting Eqs. ( 6.22 )in( 6.21 )
and including also the abovementioned compressibility factor h@ 4 A =@x 4 ,the
equation describing the wave propagation becomes

@ 2

gh @ 4

@t 2 A @x fŒc 0 C p A

@

C q. A / 2 @x A

@x 4 A

D (6.23)

In the above equation A is the change in lateral density of the membrane A D

A 0 , A is the lateral density of the membrane, 0 is the equilibrium lateral

density of the membrane, c 0 is the velocity of small amplitude wave, p and q are
parameters defining the dependence of the wave velocity on the membrane's density.
Equation ( 6.23 ) is closely related to Boussinesq equation.

6.4.2 Comparison Between the Hodgkin-Huxley

and the Soliton Model

In the following, some major features of the two neuron models (the

Hodgkin-Huxley model and the soliton model) are summarized [ 10 , 80 ].