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Ionic channels appear throughout membranes in neurons and are
responsible for most of the interesting dynamic behavior. They play a
fundamental role in the properties and connections of neurons and
neural nets. So far, we have viewed the membrane as a passive cable
with inputs and inhomogeneities. However, as we have seen, cables
can transmit information only in a very analog fashion and for long
distances would require an enormous diameter. (Homework: Given
and
what diameter would
you need to get a length
constant of a meter? - This is less than the distance
traveled to the spinal cord from nerves in the foot. )
This problem can be resolved if there is a means to keep the potential
of the cable localized and high. This is exactly the problem that
action potentials solve. In a real nerve cell there are many species
of ions, calcium, potassium, sodium, chloride, magnesium, to name a
few of the most common. The basis for the resting membrane potential
is the balance of these ions in the cell and outside of the cell.
There are basically two forces at work in absence of ionically
selective channels: (i) passive diffusion in which case high
concentrations tend to move toward low concentrations and (ii)
electric forces which attempt to balance the charges on either side of
the membrane.
Hodgkin and Huxley won the Nobel prize for their elegant experimental
and theoretical work on the nature of the voltage gated channels in
the squid axon. This theory is the basis for all subsequent models of
ionic channels in nerve and other membranes. Some of the details may
differ, but the basic ideas are the same.
Channels facilitate the passive flow of ions across the
membrane. When they are gated by other forces such as calcium or
voltage, they can also provide great computational properties to the
neuron. Non-gated channels are responsible for the membrane potential.
Recall that the equilibrium potential of an ion is given by:
![\begin{displaymath}
E = 2.303\frac{RT}{ZF} \log\frac{\left[C\right]_o}{\left[C\right]_i}\end{displaymath}](img3.gif)
where C is the concentration, and at
we have
when Z=+1. Thus, since there are 20 mMoles of
potassium inside and 400 outside EK=-78mV. ENa=55mV and
ECl=-60mV.
Recall that the membrane potential is found from the Goldman equation:
![\begin{displaymath}
V_m = \frac{RT}{F}\ln \frac{\sum P_j \left[C_j\right]_o}
{\sum P_j \left[C_j\right]_i}\end{displaymath}](img6.gif)
where Pj are the permeabilities of the ions. At rest
PK:PNa:PCl=1:.04:.45
during the peak of the action potential
PK:PNa:PCl=1:20:.45
Figure 1:
Equivalent circuit for Squid Axon
 |
This is really the correct way to discern the membrane potential,
however, in modeling, we will make a much simpler equivalent
circuit. We will treat each ion channel as a conductor and a battery.
Note that the permeabilities act like conductances and the equilibrium
potentials act as batteries. Consider Figure 1
which ignores the pumps for sodium and potassium. Then the equations
for the membrane are:
|  |
(1) |
where I is the applied current. The membrane potential is defined as
a steady state of (1) that is the right-hand side must vanish.
This enables us to solve forV:

If
then
Vm=-69mV.
Homework:
- 1.
- Compute the sodium and potassium currents at rest (Hint: The
current of an ionic species is I=g(E-V) where E is the reversal
potential, g the conductance, and V the resting potential.)
- 2.
- What is the effect of gCl on the resting
potential. That is if gCl is small and positive, will this raise
or lower the potential.
- 3.
- Given that ECa=150mV suppose that
What is Vm?
- 4.
- Suppose that gNa increases 500 fold as it does during the
action potential. What is Vm in this case?
- 5.
- Again ignoring chloride and using the values in the example for
the conductances of sodium and potassium, how much current must you
inject to increase the potential by 10mV?
- 6.
- Rewrite (1) as

where
is the effective conductance. Using the given values
for the potassium and sodium conductances and noting that
where A is the area of the membrane and using
what
is the area of the membrane if the time constant is 1msec. (Hint: The
time constant is
)
Next: Voltage gated channels
Up: Channeling with Bard
Previous: Channeling with Bard
G. Bard Ermentrout
1/29/1998