Biomedical Engineering Reference
In-Depth Information
CHAPTER 3
Active Membranes
In the previous chapter we considered the response of a membrane to a small stimulus. In these situations
the resistance of the membrane was linear and
I
ion
was modeled as a simple resistor and battery in series.
In this chapter, we consider what happens when a stimulus causes
V
m
to reach threshold. The result is
that
R
m
no longer behaves linearly a property that may be represented in a circuit model as the
variable
resistance
in Fig. 3.1.
i
I
S
tim
R
m
C
m
I
ion
I
Cm
E
rest
e
Figure 3.1:
A nonlinear resistive membrane.
3.1 THEHODGKIN-HUXLEYMODEL
The first physiologically accurate nonlinear model of
I
ion
was published in 1952 by Hodgkin and Huxley.
To create the model, Hodgkin and Huxley combined a brilliant experiment a number of assumptions
were about the biophysics of ion channels. Despite its relative simplicity, it remains the gold-standard of
ionic membrane models.
3.1.1 The Parallel Conductance Model
The first assumption was that
I
ion
was composed of three currents that acted independently of one another.
These currents were Sodium (
I
Na
), Potassium (
I
K
), and a generic
leakage
current (
I
L
). Mathematically,
C
m
dV
m
dt
I
m
=
+
I
Na
+
I
K
+
I
L
.
(3.1)
Experimental data showed that
I
L
was a linear current, while
I
Na
and
I
K
were nonlinear. Therefore,
the circuit analog, taking into account the independence assumption and nonlinear ionic currents, can
