Biomedical Engineering Reference
In-Depth Information
where
θ
is the
propagation velocity
and is a measure of the speed at which the action potential moves down
the cable. The reason Eq. (4.26) works is because the shape of the action potential is the same at every
point on the cable. The only parameter that changes is
when
the action potential occurs in time. Using
the chain rule twice
∂V
m
∂x
=
1
θ
∂V
m
∂t
(4.27)
∂
2
V
m
∂x
∂
2
V
m
∂t
1
θ
2
=
.
(4.28)
Substitution into Eq. (4.16) yields
d
2
V
m
dt
a
2
R
i
θ
2
I
m
=
.
(4.29)
Upon careful inspection, Eq. (4.29) has a deeper meaning. As long as
I
ion
does not change throughout
the cable, a family of solution exists with the only requirement being:
a
2
R
i
θ
2
=
K
(4.30)
a
2
R
i
K
θ
=
(4.31)
where
K
is a constant.The usefulness of Eqs. (4.30) and (4.31) are that only one set of
a
and
θ
are needed
to find
R
i
K
. Once
R
i
K
is known, the effect of any change in
a
on
θ
can be predicted.
4.2.1 Saltatory Conduction
A problem with using active propagation to transmit an impulse over long distances is that the signal will
take time to traverse the axon. In fact, in some regions of the body an axon can be up to one meter long.
To compensate, the nervous system has developed a clever solution to speed up propagation. Schwann
cells (a type of glial cell) create a
myelin sheath
around the axon. The presence of this sheath makes it
nearly impossible for current to cross the cell membrane.The impact is that membrane directly under the
sheath has a much higher
R
m
and the thicker membrane decreases
C
m
. Therefore, membrane covered in
myelin is effectively
passive
.
As shown in Fig. 4.7, the sheath does leave small regions of the neuron cell membrane exposed,
called
Nodes of Ranvier
, which may fire an action potential. If an active patch of membrane fires at a
Node of Ranvier, current will flow to the right but will not be able to easily cross the cross the membrane
because of the high
R
m
. Instead, most current will follow the path of least resistance and jump farther
down the cable. So, in effect, the high resistance of the myelin creates a short circuit that skips quickly
from one Node of Ranvier to the next. Multiple Sclerosis (MS) is a disease that causes inflammation and
scaring of the myelin sheath leading to degradation of neural impulse propagation. The symptoms are
changes in sensations, muscle spasms, a lack of coordination and balance, pain, and eventually cognitive
and emotional impairment.
