Structure of Membranes - Membrane Biophysics, Biological and Medical Physics, Biomedical Engineering - page 24

In the Hodgkin-Huxley theory, besides the main currents, sodium and potassium
ion currents across the membrane, all other small currents are combined to form a
leakage current I L .
In the squid giant axon the I - V curves of open Na + and K + channels are approx-
imated by linear equations. Therefore, the membrane capacitive current equation
becomes:

C m d V

d t =− g Na (

V − V Na ) − g K ( V − V K ) − g L ( V − V L ) + I ext (2.18)

Here, I ext is the externally applied current, and g Na , g K and g L represent conductances
(reciprocal of Ohmic resistance) for Na + and K + ions and other ions responsible
for leakage currents, respectively. Voltages V Na , V K , and V L are membrane resting
potentials, corresponding to Na + ,K + ions and other leakage ions across the mem-
brane. The previous first-order ordinary differential equation can be rewritten as a
more general form of equation, representing a capacitive current of the membrane
according to:

d V
d t =− g eff (

C m V − V eq ) + I ext (2.19)

Here, g eff = g Na + g K + g L is the effective conductance across the membrane, and
V eq = (g Na V Na + g K V K + g L V L )/g eff is the membrane resting potential.
Specifically, in voltage-gated ion channels, the channel conductance g i is a

function of both time and voltage ( g n ( t , V ) ), while in leak channels, g L is a

constant ( g L ).
The above description can be summarized in a way similar to that in the origi-
nal paper of Hodgkin-Huxley [ 12 ]. The electrical behavior of a membrane may be
represented by a network, as shown in Fig. 2.5 . Here, current can be carried through
the membrane either by charging the membrane capacitor or by the movement of
ions Na + ,K + , etc. through the corresponding equivalent resistors in parallel. The
ionic current corresponding to a specific ion is proportional to the difference between
the membrane potential and the equilibrium potential for a specific ion. Here, the
proportionality constant is the Ohmic conductance for the corresponding ion.

Voltage- and Time-Dependent Conductance in the Hodgkin-Huxley Model

As explained earlier, the total membrane current I m can be subdivided into two main
categories, which are capacitive currents and ionic currents. Thus, under normal
conditions the following equation is valid:

C m d V I m =

d t + g Na (

V − V Na ) + g K ( V − V K ) + g L ( V − V L ) (2.20)

This equation gives the values of the membrane capacitance that are independent of
the magnitude and sign of V , and are little affected by the time course of V (see Table 1