Structure of Membranes - Membrane Biophysics, Biological and Medical Physics, Biomedical Engineering - page 26

terms, e.g., a term representing inactivation would be necessary in the case of a third
power in the exponential power series expansion, making it much less elegant and
not convincing.
Following a detailed analysis [ 12 ], the generalized form of

g K can be con-

structed as

g

K0 e − t /τ n 4

1 / 4 1 / 4 1 / 4

g K = g

− − − − g

(2.21)

K K

where g K − is the asymptotic value of the conductance, and g K0 is the conductance at
t 0. Also, τ n is the inverse of the sum of the rate constants describing the timescale

of the resultant net inward flow of ions. The proposed equation is a best fit to the
experimental results, as presented in the original work [ 12 ].

=

The Sodium Conductance

The transient change in sodium conductance g Na was described by considering two
variables, both of which obey first-order equations. Following a few formal assump-
tions [ 12 ], g Na was found to fit very well to experimental observations by taking the
following form:

g Na = g Na [ e − t /τ m

3 e − t /τ h

1 − ] (2.22)

Here, g Na is the value which the sodium conductance would attain in the case when the
proportions of the inactivating molecules on the outside boundary of the membrane
τ m and τ h are the inverse values of the net transfer rate constants for inside and
outside directions, respectively.
A detailed analysis of the rate constants and other related aspects of membrane
conductance described in the Hodgkin-Huxley models is not only interesting but
also very important. However, due to space limitations and the scope of this topic,
we invite the reader to study the original material presented in the ground-breaking
papers published by this pair independently and with others in the early 1950s [ 8 -
12 , 19 ]. Below, we discuss a few more models which were subsequently built on the
basis of the Hodgkin-Huxley model in order to perform a better qualitative analysis,
and to better understand the various aspects of the Hodgkin-Huxley model.

2.3.5 The FitzHugh-Nagumo Model

Before discussing the FitzHugh-Nagumo model, we first further reduce the Hodgkin-
Huxley model to a more generalized form. Based on the experiments performed
by Hodgkin and Huxley on the squid giant axon between 1948 and 1952, they
constructed a model for patch clamp experiments. This provided a mathematical
description of the axon's excitable nature. Here, a key model assumption was that
the membrane contains channels for potassium and sodium ion flows. Following the