Biomedical Engineering Reference
In-Depth Information
38 CHAPTER 3. ACTIVEMEMBRANES
3.4.5 Changes in Nernst Potentials
Earlier it was stated that a bursting neuron may begin to change the intra and/or extracellular concentra-
tions. Even with pumps and exchangers, the cell may build up concentrations of some ions.Accompanying
these concentration changes, will be a change in the Nernst potentials, and thus the driving forces behind
the action potential. It is therefore important for some models to incorporate a way for concentrations
to change. The most common technique is to write differential equations for the intra and extracellular
ionic concentrations. For example, consider that there may be several
I
K
outward currents that could
potentially deplete the intracellular Potassium concentration,
K
+
]
i
.This change in
K
+
]
i
would impact
[
[
E
K
as well as
V
rest
through the Goldman-Hodgkin-Katz equation. A common formulation would be:
m
K
+
]
i
dt
d
[
I
K
1
+
I
K
2
+
I
K
3
=−
(3.34)
FV
i
K
+
]
e
dt
d
[
I
K
1
+
I
K
2
+
I
K
3
=
(3.35)
FV
s
where
F
is Faraday's constant,
V
i
is the volume of the intracellular space, and
V
s
is the small
shell
of
extracellular space surrounding the outside of the cell. In this formulation, any
K
+
ions leaving the
intracellular space must enter the extracellular space.
A situation where ionic concentrations may drastically change is during a disease. For example,
during ischemia (e.g., lack of O
2
) the concentration of ATP drops and the pumps that restore concen-
tration gradients become less efficient and may even fail. The result is a buildup of ions. If this buildup
continues, cells may die and lyse (e.g., pop) spilling their ions into the extracellular space and lead to a
drastic change in concentrations.
3.4.6 Intracellular Compartments and Buffers
The intracellular space in a cell is populated by many smaller organelles. Some of these organelles have
their own membranes, with their own ion channels, and are capable of transporting ions to and from the
intracellular space to the intra-organelle space.When this transport occurs, the concentration of that ion
will change in the intracellular space. Functionally, these organelles behave as a
buffer
. As in the previous
section, these changes in concentration can impact the Nernst potential of that ion. Figure 3.13 is a
schematic of a
Ca
2
+
buffer.
3.5
PHENOMENOLOGICALMODELS
Although the Hodgkin-Huxley mathematical model is simple to solve using todays computers, Hodgkin
and Huxley performed all of their calculations using calculators. In particular, computing exponentials
was difficult, so a number of simplified models were developed that captured the basic features of neuronal
action potentials.
3.5.1 Fitzhugh-NaghumoModel
In 1961, Fitzhugh and Nagumo independently developed a model based upon simple polynomials. In the
Fitzhugh-Naghumo model, the
m
,
h
,
n
, and
V
m
variables were reduced to only two differential equations.
