Biology Reference
In-Depth Information
where V is the potential difference (voltage), R and F the universal Boltzmann
and Faraday constants , T the temperature, z the valence of the ion, and X
o
and
X
i
are the concentrations of the ion outside and inside the cell. (Such laws are
constraints and foundations, but are not the complete derivational source, for the
later H and H equations I introduce further below (also see Bogen (2005) and
Craver (in press) on this point.))
Part II of the Hodgkin-Huxley paper is a 'mathematical description of mem-
brane current during a voltage clamp'. Equations for the sodium and potassium
currents as conductances are developed. The equations do not come from 'first
principles' but rather are empirical equations fitted from the voltage clamp data.
They are typically chosen based on simplicity, with a first order equation being
preferred over a second order, etc. A first order equation is satisfactory to rep-
resent a portion of the time course of nerve depolarization (a rapid change of
voltage across the membrane), but a fourth order equation is needed to repre-
sent the beginning of the potassium depolarization process. The equation for
potassium conductance, in the form that it could be compared with the empirical
results, was chosen as
g
K
1/4
g
K0
1/4
exp
4
−
g
K
1/4
t
n
g
K
=
−
−
It is a theoretical equation, to use H and H's language, based on the equivalent
circuit and the general empirically found form of the rise and fall of ion con-
duction during depolarization and repolarization. H and H doubt gives a 'correct
picture' of the membrane, though they do provide a possible physical basis
for the equation (see pp. 506-507 of the 1952 article). The equation contains
a constant
n
that can then be specified to be the best fit to experimentally
determined depolarizations of different potential membrane differences. Hodgkin
and Huxley found that there was reasonable agreement between theoretical and
experimental curves. H and H then go on to develop the somewhat more com-
plex reasoning leading to the equation for sodium conductance, which I shall not
discuss, but which can be found on pp. 512-515 of their 1952 paper. They also
develop equations for rate constants and , and the dimensionless proportions
nm, and h, of ions inside and outside the membrane, in Part II as well.
At the beginning of Part III of their 1952 paper, titled 'Reconstruction of
Nerve Behavior,' H and H summarize the equations they have developed in
Part II of that paper. The summary is from the H and H (1952) article and
the numbering of the equations in brackets comes from their original equation
numbers. The summary looks like this:
C
M
dV
g
K
n
4
V
g
Na
m
3
hV
I
=
dt
+¯
−
V
K
+¯
−
V
Na
+¯
g
l
V
−
V
l
(26)
