Information Technology Reference
In-Depth Information
C
m
dV
n
dt
DI
ionic
.V
n
/ C D.V
nC1
2V
n
C V
n1
/
(6.14)
4a
1
where D D
R
l
L
. Thus the PDE of the voltage along the myelinated axis is turned
into a system of equivalent ordinary differential equations.
C
m
d
dt
D DŒV.t C / 2V.t/ C V.t / I
ionic
.V;
w
; /
dw
dt
(6.15)
D g.V;
w
/
where parameter
w
represents gating variables, calcium, etc.
It can be set f.V/ DI
ionic
.V/ and it can be assumed that f.V/ has three
roots, V
rest
, V
thr
, and V
ex
, which mean resting state, threshold, and excited state,
respectively. As boundary conditions, one can assume V.1/ D V
ex
and V
1
D
V
ex
. Using the approximation about the second derivative with respect to time
2
V
00
D V.t C / 2V.t/ C V.t /
(6.16)
one arrives at the ordinary differential equation
2
L
C
m
V
0
D f.V/C
4a
1
R
1
V
00
(6.17)
p
L
where is the discretization step. By performing the change of variables D
the wave equation is written as
C
m
p
L
V
D f.V/C
4a
1
R
l
V
(6.18)
Assume that c is
th
e velocity of the wave traveling along the myelinated axis. It
holds that c D
p
L
, and consequently for the myelinated part it holds
q
L
c
L
c
myelin
D
'
(6.19)
For the values D 1m, L D 100m one obtains c
myelin
' 10c.
6.4.1
Solitons in Neuron's Dynamical Model
In the following approach of modelling for the neuron's membrane dynamics, the
neuron's axon is considered to be a one-dimensional cylinder with lateral excitations
moving along the coordinate x. The dynamics of this cylinder is described by a
wave-type equation [
10
,
80
,
86
,
103
]
@x
c
2
@x
A
@
2
A
@t
2
@
(6.20)
D
