460 Chapter 12. Nerve impulses[[Student version, January 17, 2003]]
in the membrane’s permeabilities to various ions. Net current flows across a membrane whenever
the actual potential differenceV deviates from the “target” value given by the chord formula
(Equation 12.3 on page 447). But the target value itself depends on the membrane conductances.
If these suddenly change from their resting values, so will the target potential; if the target switches
from being more negative thanV to more positive, then the membrane current will change sign.
Since the target value is dominated by the Nernst potential of the most permeant ion species, we
can explain the current reversal by supposing that the membrane’s permeability to sodium increases
suddenly during the action potential.
So far we have done little more than restate Idea 12.13. To go farther, we must understand
whatcausesthe sodum conductance to increase. Since the increase does not begin until after the
membrane has depolarized significantly (Figure 12.8, stage “B”), Hodgkin and Huxley proposed
that
Membrane depolarization itself is the trigger that causes the sodium conduc-
tance to increase.(12.16)
That is, they suggested that some collection of unknown molecular devices in the membrane allow
the passage of sodium ions, with a conductance depending on the membrane potential. Idea 12.16
introduces an element ofpositive feedbackinto our picture: Depolarization begins to open the
sodium gates, which increases the degree of depolarization, which opens still more sodium gates,
and so on.
The simplest precise embodiment of Idea 12.16 is to retain the Ohmic hypothesis, but with the
modification that each of the membrane’s conductances may depend onV:
jq,r=∑
i(V−ViNernst)gi(V). simplevoltage-gating hypothesis (12.17)In this formula the conductancesgi(V)are unknown, but positive, functions of the membrane
potential. Equation 12.17 is our proposed replacement for the Ohmic hypothesis, Equation 11.8.^9
The proposal Equation 12.17 certainly has a lot of content, even though we don’t yet know
the precise form of the conductance functions appearing in it. For example, it implies that the
membrane’s ion currents are still Ohmic (linear in ln(c 1 /c 2 )), if we holdV fixed while changing the
concentrations. However, the membrane current is a nownonlinearfunction ofV,acrucial point
for the following analysis.
Before proceeding to incorporate Equation 12.17 into the cable equation, let’s place it in the
context of this book’s other concerns. We are accustomed to positive ions moving along the electric
field, which then does work on them; they dissipate this work as heat, as they drift against the
viscous drag of the surrounding water. This migration has the net effect of reducing the electric
field: Organized energy (stored in the field) has been degraded to disorganized (thermal) energy.
But stage “B” of Figure 12.8b shows ions moving inward,oppositelyto the potential drop. The
energy needed to drive them can only have come from the thermal energy of their surroundings.
Can thermal energy really turn back into organized (electrostatic) energy? Such unintuitive energy
transactions are possible, as long as they reduce thefreeenergy of the system. The axon started
out with a distributed excess of free energy, in the form of its nonequilibrium ion concentrations.
Chapter 11 showed how the source of this stored free energy is the cell’s metabolism, via the
membrane’s ion pumps.
(^9) The symbol ∆Vappearing in Equation 11.8 is abbreviated asVin this chapter (see Section 12.1.3a) on page
451.