Biomedical Engineering Reference
In-Depth Information
where
erfc
is the complimentary error function defined by
erfc(y)
=
1
−
erf(y)
and
0
2
π
e
−
z
2
dz
erf(y)
=
(4.25)
y
is defined as the error function (see Fig. 4.4). It is important to note that Eq. (4.24) is for an infinite cable
with a negative
i
stim
. The leading terms in Eqs. (4.23) and (4.24) will change if the nature of the stimulus
is changed or the cable is not infinitely long.
0.7
1
0.5
0.25
−0.2
0
−0.5
−0.75
−1
−2 −1.75−1.5−1.25 −1 −0.75−0.5−0.25
0
0.25 0.5 0.75
1
1.25 1.5 1.75
2
x
1.7
2
1.5
1.25
0.7
1
0.5
0.25
0
−2 −1.75−1.5−1.25 −1 −0.75−0.5−0.25
0
0.25 0.5 0.75
1
1.25 1.5 1.75
2
x
Figure 4.4:
Error and complimentary error functions.
Figure 4.5 shows the time and space solutions to Eq. (4.24) for different times (left panel) and
different locations in space (right panel). Because the specific numbers may vary for any given cable, the
plot is shown in terms of the general passive properties of the cable, i.e.,
λ
and
τ
m
). It is important to
note that the fall off in space is exponential only for the steady-state (
t
→∞
). Likewise, the rise in time
is exponential (as in an RC circuit) only at
x
=
λ
.
4.2 ACTIVE PROPAGATION INAXONS
In Sec. 4, we considered passive propagation down a cable as a model of dendritic propagation. The
membrane of the axon, however, has many nonlinear ion channels that are capable of generating an action
potential. The only modification needed to create an
active cable
is to replace the passive
I
ion
=
V
m
/r
m
with a more complex model as outlined in Ch. 3. Whereas a stimulus applied to a passive cable will be
attenuated
, in an active cable the action potential will propagate unattenuated. In this way, a signal that
reaches the axon hillock will be propagated unattenuated to the end of the axon. In all but the simplest
cases, it is not possible to derive an analytic solution when propagation is active.
To demonstrate the concept of active propagation, Fig. 4.6 shows propagation down an active
3
cm
long axon. The superthreshold stimulus was delivered to the left end of the axon at
t
=
0
msec
.In
