Biology Reference
In-Depth Information
action potential. In the propagated action potential, the local circuit currents
have to be provided by the net membrane current. At this point in their 1952
paper, H and H appeal to a well-known partial differential equation from cable
theory (which is a variant of Laplace's well-known heat diffusion partial dif-
ferential equation) relating the current to the second partial derivative of the
potential difference (V ) with respect to distance (x). This equation is given by
the expression
1
r
1
+
2
V
x
2
i
=
(27)
r
2
There are some simplifications then invoked, e.g., since r
1
r
2
r
1
can be
dropped. The expression for the current density for the fiber with a radius of a
then allows the equation to be rewritten as
a
2R
2
2
V
x
2
I
=
(28)
This relation is then substituted into Eqn [26], which yields a partial differential
equation that is 'not practical to solve as it stands' (p. 522). But a similarity is
noted for the condition of steady propagation, one which permits the equation
to be converted into an ordinary differential equation that can be solved numer-
ically, if laboriously given the computational tools available in 1952. This is the
propagated action potential equation and was written as
d
2
V
a
2R
2
2
dt
2
=
C
M
dV
dt
+
g
K
n
4
V
−
V
K
+
g
Na
m
3
hV
−
V
Na
+
g
l
V
−
V
l
(30)
where is a parameter that has to be estimated numerically, based on the
behavior of the equation at extreme boundary conditions (see p. 522 of H and
H for details). A section on numerical methods of solution of such equations is
interpolated in the 1952 article, and after a minor (abbreviational) substitution,
Eqn [30] is rewritten as Eqn [31] (not shown here, but see p. 524 of the original
article). This equation (either Eqn [30], or the equivalent Eqn [31], is solved
numerically, and graphs of the membrane conductances during a propagated
action potential are depicted. H and H's graphical results were shown in their
Fig. 17 that were based on numerical solutions of Eqn [31] showing components
of membrane conductances g during propagated action potential
V. This
figure is widely reproduced in standard neuroscience texts (also see their original
article, p. 530). Readers will also recognize such graphs of the conductances
as THE classical action potential result, which is represented, based on largely
qualitative considerations, in typical neuroscience textbooks.
−
