468 Chapter 12. Nerve impulses[[Student version, January 17, 2003]]
gK+holds steady indefinitely at this value.
Thus, the simple voltage-gating hypothesis describes reasonably well the initial events following
membrane depolarization (points 1–2 above), which is why it gave a reasonably adequate description
of the leading edge of the action potential. In the later stages, however, the simple gating hypothesis
breaks down (points 3–4 above), and indeed here our solution deviated from reality (compare the
mathematical solutions in Figure 12.10 to the experimental trace in Figure 12.6b). The results
in Figure 12.12 show us what changes we should expect in our solutions when we introduce more
realistic gating functions:
- After half a millisecond, the spontaneous drop in sodium conductance begins to driveVback
down to its resting value. - Indeed, the slow increase in potassium conductance after the main pulse implies that the
membrane potential will temporarilyovershootits resting value, instead arriving at a value
closer toVKNernst+ (see Equation 12.3 and Table 11.1 on page 416). This observation explains
the phenomenon of afterhyperpolarization, mentioned in Section 12.1.1. - Once the membrane has repolarized, an equally slow process resets the potassium conductance
to its original, lower value, and the membrane potential returns to its resting value.
Hodgkin and Huxley characterized the full time course of the potassium conductance by assum-
ing that for every value ofV,there is a corresponding saturation value of the potassium conductance,
g∞K+(V). The rate at whichgK+relaxes to its saturation value was also taken to be be a function
ofV. These two functions were taken as phenomenological membrane properties, and obtained
byrepeating experiments like Figure 12.12 with command voltage steps of various sizes. Thus the
actual conductance of a patch of membrane at any time depends not on the instantaneous value
of the potential at that time, as implied by the simple voltage-gating hypothesis, but rather on
theentire past historyof the potential, in this case the time sinceV wasstepped fromV^0 to its
command value. A similar, but slightly more elaborate, scheme successfully described the rise/fall
structure of the sodium conductance.
Substituting the conductance functions just described into the cable equation led Hodgkin and
Huxley to a cable equation more complicated than our Equation 12.23. Obtaining the solutions
wasaprodigious effort, originally taking weeks to compute on a hand-cranked, desktop calculator.
But the solution correctly reproduced all the relevant aspects of the action potential, including its
entire time course, speed of propagation, and dependence on changes of exterior ion concentrations.
There is an extraordinary postscript to this story. The analysis described in this chapter implies
that as far as the action potential is concerned the sole function of the cell’s interior machinery is
to supply the required nonequilibrium resting concentration jumps of sodium and potassium across
the membrane. P. Baker, Hodgkin, and T. Shaw confirmed this rather extreme conclusion by the
extreme measure of emptying the axon of all its axoplasm, replacing it by a simple solution with
potassium but no sodium. Although it was almost entirely gutted, the axon continued to transmit
action potentials indistinguishable from those in its natural state (Figure 12.13)!
12.3.2 The patch-clamp technique allows the study of single ion channel behavior
behavior
Hodgkin and Huxley’s theory of the action potential was phenomenological in character: They
measured the behavior of the membrane conductances under space- and voltage-clamped conditions,