Individual Neurons - Computational Explorations in Cognitive Neuroscience - page 32

2.4.3 Electric Potential versus Diffusion: The

Equilibrium Potential

To compute the current that a given ion will produce,
we need some way of summarizing the results of both
the electric and diffusion forces acting on the ion. This
summarization can be accomplished by figuring out the
special equilibrium point where the electric and diffu-
sion forces balance each other out and the concentration
of ions will stay exactly as it is (even though individ-
ual ions will be moving about randomly). At this point,
there will be zero current with respect to this type of ion,
because current is a function of the net motion of these
ions. Of course, in a simple system with only electri-
cal forces, the equilibrium point is where the electrical
potential is actually zero. However, with different con-
centrations of ions inside and outside the neuron and
the resulting diffusion forces, the equilibrium point is
not typically at zero electrical potential.
Because the absolute levels of current involved are
generally relatively small, we can safely assume that the
relative concentrations of a given ion inside and outside
the cell, which are typically quite different from each
other, remain relatively constant over time. In addition,
we will see that the neuron has a special mechanism for
maintaining a relatively fixed set of relative concentra-
tions. Thus, the equilibrium point can be expressed as
the amount of electrical potential necessary to counter-
act an effectively constant diffusion force. This poten-
tial is called the equilibrium potential ,orthe reversal
potential (because the current changes sign [ reverses ]
on either side of this zero point), or the driving poten-
tial (because the flow of ions will drive the membrane
potential toward this value).
The equilibrium potential ( E ) is particularly conve-
nient because it can be used as a correction factor in
Ohm's law, by simply subtracting it away from the ac-
tual potential V , resulting in the net potential ( V ￿E ):

Figure 2.7: Sketch of diffusion in action, where both types of
particles (ions) move in the same direction, each independent
of the other, due to the accumulated effects of random motion.

same charge (e.g., K + ) somewhere else (figure 2.7). In
contrast, electricity doesn't care about different types of
ions (any positive charge is the same as any other) — it
would be perfectly happy to attract all Na + ions and
leave a large concentration of K + somewhere else.
Because it is just as reliable an effect as the electri-
cal force, and it is convenient to write similar force-
like equations for diffusion, we will treat diffusion as
though it were a direct force even though technically it
is not. Thus, we can use essentially the same terminol-
ogy as we did with electricity to describe what happens
to ions as a result of concentration differences (instead
of charge differences).
Imagine that we have a box with two compartments
of liquid separated by a removable barrier. With the
door closed, we insert a large number of a particular
type of ion (imagine blue food coloring in water) in one
compartment. Thus, there is a concentration difference
or gradient between the two compartments, which re-
sults in something like a concentration potential that
will cause these ions to move into the other compart-
ment. Thus, when the barrier is removed, a diffusion
current is generated as these ions move to the other side,
which, as is the case with electricity, reduces the con-
centration potential and eventually everything is well
mixed. The diffusion coefficient acts much like the elec-
trical conductance (inverse of resistance), and a rela-
tionship analogous to Ohm's law holds:

(2.4)

(2.3)

We will call this the diffusion corrected version of
Ohm's law, which can be applied on an ion-by-ion basis
to get the current contributed by each type of ion, as we
will see.

This diffusion version of Ohm's law is called Fick's
first law ,where I is the movement of ions (diffusion
current), D is the diffusion coefficient, and C is the con-
centration potential.