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where D m;h;. The solution should satisfy the following boundary conditions
(6.5)
.V;U;m;h;n/.˙1/ D .V
rest
;0;m
1
.V
rest
/;h
1
.V
rest
/;n
1
.V
rest
//
6.3
Dynamics of the Solution of the Wave PDE
Neurons communicate over long distances and this is succeeded by electric
signals (action potentials) which are transmitted along the neuron's axis. Diffusion
phenomena describing nonlinear flow of ions across the membrane generate changes
in the potential of the membrane which are transmitted on its longitudinal axis. The
transmitted potential takes the form of a traveling wave. In several cases the wave
equation is computed from the solution of a reaction-diffusion PDE. The model
of propagating action potential in the form of a wave equation holds both for the
neuron's membrane and for dendrites.
For example, the neuron described by FitzHugh-Nagumo equation can be
considered to have dynamics of the Shilnikov type that is
@
2
V
@x
2
@V
@t
D
C f.V/
w
C I
(6.6)
@
w
@t
D .
v
kw
/
where f.
w
/ D
v
.1
v
/.
v
a/ with 0<a<1, >0and k0.If>0is
sufficiently small, then the wave differential equation has a solution (homoclinic
orbit). In particular, if f.
v
;
w
/ D I
v
C H.
v
a/
w
where H is a Heaviside
function, then the so-called McKean model is obtained.
The dynamics of the neuron described by Eq. (
6.6
) is called type-II neuron
dynamics. There is also the type-I neuron dynamics described by the diffusion PDE
@
2
V
@V
(6.7)
@t
D
@x
2
C f.V/
where f.
v
/ is a periodic function of period 2.
6.4
Myelinated Axons and Discrete Diffusion
It is assumed that the membrane's axis that is covered by myelin (a fatty substant
which one the one side serves as insulator of the axis and on the other side reduces
the capacitance of the membrane). It is also assumed that in regular intervals, known
as Ranvier nodes, the axis contacts the extracellular mean, and additionally that there
is high density of Na
C
channels (Fig.
6.1
).
The diameter of the axis is denoted as a
1
, while the diameter of the myelinated
axis is denoted as a
2
.
