Structure of Membranes - Membrane Biophysics, Biological and Medical Physics, Biomedical Engineering - page 23

has achieved a very high level of understanding following a few ground-breaking
discoveries, such as the seminal work of Alan Hodgkin and Andrew Huxley, who in
1952 developed the first quantitative model of the propagation of an electrical signal
along a squid giant axon. The Hodgkin-Huxley theory is applicable, not only to elec-
trophysiology, but also to applied mathematics through appropriate modifications.
The creation of a new field of mathematics called 'the study of excitable systems'
has been made possible, thanks to the remarkable simplification and extensions of
the Hodgkin-Huxley theory. We provide more details below.

The Hodgkin-Huxley Model of the Action Potential: A Quantitative Model

The cell membrane has well-defined biochemical and biophysical characteris-
tics. We have briefly described these two aspects earlier in this chapter. The
biophysical characteristics of a cell membrane are represented by the generally
accepted Hodgkin-Huxley model. The lipid bilayer is represented as a capacitor—
the low-dielectric ( ε ∼ 2) region inside the membrane relative to the outside region

with high dielectric values ( ε ∼

80) [ 20 ] makes the cell membrane an almost per-
fect capacitor. Voltage-gated and leak ion channels are represented by nonlinear ( g n )
and linear ( g L ) conductances, respectively. The electrochemical gradients driving ion
flow are represented by batteries ( E n and E L ), and ion pumps and exchangers are rep-
resented by current sources ( I p ). The voltage values for the batteries are determined
from the Nernst potentials of specific ionic species.
In an idealized cell, with a small portion of the membrane represented as equiv-
alent electrical circuit, we can apply the Hodgkin-Huxley model for calculating the
membrane current ( I m ) by the following current equation:

d V
d t +

I m = C m I K + I Na + I L (2.16)

Here, V is the membrane voltage, I K and I Na are the potassium and sodium currents,
respectively, and I L is the sum of all leakage currents due to the flow of other ions
moving passively through the membrane.
The charge stored in the capacitive membrane is q m

C m V where V is the
voltage across the capacitor, which is comparable to the transmembrane potential.
Earlier in this chapter, we have explained how currents across the membrane are
conserved quantities, which is possible due to the fact that the cell membrane is
modeled as a capacitor in parallel with ionic currents. Now, if ionic currents are
considered to depend on both transmembrane voltage V and time t , the membrane
capacitive current follows the formula

=

d V
d t +

C m I ion ( V , t ) = 0 (2.17)