Biomedical Engineering Reference
In-Depth Information
12.5 EQU
IVALENT CIRCUIT MODEL FOR THE CELL ME
MBRANE
In this section, an equivalent circuit model is developed using the tools previously
developed. Creating a circuit model is helpful when discussing the Hodgkin-Huxley model
of an action potential in the next section, a model that introduces voltage- and time-
dependent ion channels. As previously described, the nerve has three types of passive electri-
cal characteristics: electromotive force, resistance, and capacitance. The nerve membrane is a
lipid bilayer that is pierced by a variety of different types of ion channels, where each channel
is characterized as being passive (always open) or active (gates that can be opened). Each ion
channel is also characterized by its selectivity. In addition, there is the active
Na-K
pump that
V
m
across the cell membrane at steady state.
maintains
12.5.1 Electromotive, Resistive, and Capacitive Properties
Electromotive Force Properties
The three major ions,
Cl
, are differentially distributed across the cell mem-
brane at rest using passive ion channels, as illustrated in Figure 12.5. This separation of
charge exists across the membrane and results in a voltage potential
K
þ
,
Na
þ
, and
V
m
, as described by
Eq. (12.33) (the Goldman equation).
Across each ion-specific channel, a concentration gradient exists for each ion that creates
an electromotive force, a force that drives that ion through the channel at a constant rate.
The Nernst potential for that ion is the electrical potential difference across the channel
and is easily modeled as a battery, as shown in Figure 12.11 for
K
þ
. The same model is
Na
þ
and
Cl
, with values equal to the Nernst potentials for each.
applied for
Resistive Properties
In addition to the electromotive force, each channel also has resistance—that is, it resists
the movement of ions through the channel. This is mainly due to collisions with the channel
wall, where energy is given up as heat. The term conductance,
, measured in siemens (S),
which is the ease with which the ions move through the membrane, is typically used to
G
Cl
−
K
+
Na
+
Outside
E
K
Inside
Na
+
K
+
A
−
Cl
−
FIGURE 12.11
K
þ
Nernst potential. The polarity of the battery is given with the ground on the outside of the membrane in agreement
with convention. From Table 12.1, note that the Nernst potential for
K
þ
channel with a value equal to the
A battery is used to model the electromotive force for a
K
þ
is negative, which reverses the polarity of
K
þ
out of the cell.
the battery, driving
