Biomedical Engineering Reference
In-Depth Information
3.3.4) and empirically fitted curves to the experimental data. They modeled the
time dependency on the ionic conductance by a channel variable or activation coef-
ficients, which indicate the probability of a channel being open. In general, the con-
ductance for a time-dependent channel is written in terms of the channel variable
(
x
), ranging from zero to one, and the maximum conductance. They developed a
set of four differential equations to demonstrate action potential of the membrane.
There are sophisticated programs, which allow simulating various action poten-
tials. Whenever a short (millisecond range) inward current pulse (in the nano-am-
pere range) is applied to a patch of axonal membrane, the membrane capacitance is
charged and the membrane potential depolarizes. As a result
n
and
m
are increased.
If the current pulse is sufficient in strength then the generated
I
Na
will exceed
I
K
,
resulting in a positive feedback loop between activation
m
and
I
Na
. Since
t
m
is very
small at these potentials, the sodium current shifts the membrane potential beyond
0 mV. As sodium inactivation
h
and
I
K
increase, the membrane potential returns
to its resting potential and even undershoots a little due to the persistent potas-
sium current. If
I
inj
0, it can be proved that the rest state is linearly stable but it is
excitable if the perturbation from the steady state is sufficiently large. For
I
inj
=
0
there is a range where repetitive firing occurs. Both types of phenomena have been
observed experimentally in the giant axon of the squid. Hodgkin and Huxley per-
formed their voltage clamp experiments at 6.3°C. Higher temperatures affect the
reversal potentials since the Nernst equation is temperature dependent. Tempera-
ture affects the transition states of ion channels dramatically. Higher temperatures
lead to lower time constants and decreased amplitudes. A multiplication factor for
each time constant has been developed to account for temperature. Although the
Hodgkin-Huxley model is too complicated to analyze, it has been extended and
applied to a wide variety of excitable cells. There are experimental results in both
skeletal muscle and cardiac muscle, which indicate that a considerable fraction of
the membrane capacitance is not “pure,” but has a significant resistance in series
with it. This leads to modifications in the electrical equivalent circuit of the mem-
brane. Understanding the behavior of ion channels has also allowed simplification
of the model. One such is the Fitzhugh-Nagumo model, based on the time scales of
the Na
+
and K
+
ion channels. The Na
+
channel works on a much faster time scale
than the K
+
channel. This fact led Fitzhugh and Nagumo to assume that the Na
+
channel is always in equilibrium, which allowed them to reduce the four equations
of the Hodgkin and Huxley model to two equations.
=
3.3.5 Intracellular Recording of Bioelectricity
Measurements of membrane potential, both steady-state and dynamic changes,
have become key to understanding the activation and regulation of cellular re-
sponses. Developing mathematical models that depict physiological processes at
the cellular level depends on the ability to measure required data accurately. One
has to measure or record the voltage and/or current across the membrane of a cell,
which requires access to inside the cell membrane. Typically, the tip of a sharp
microelectrode (discussed in Section 3.5) is inserted inside the cell. Since such mea-
surements involve low-level voltages (in the millivolts range) and have high source
resistances, usage of amplifiers is an important part of bioinstrumentation signals
