Biological Physics: Energy, Information, Life

458 Chapter 12. Nerve impulses[[Student version, January 17, 2003]]

seawater, 100%cNa

+

0

50%cNa

+

0

33%cNa 0 +

1 2

-80

-40

0

40

membrane potential,

mV

time, ms

Figure 12.7: (Experimental data.) The role of sodium in the conduction of an action potential. One of the top
traces was taken on a squid axon in normal seawater before exposure to low sodium. In the middle trace, external
sodium was reduced to one-half that in seawater, and in the bottom trace, to one third. (The other top trace was
taken after normal seawater was restored to the exterior bath.) The data show that the peak of the action potential
tracks the sodium Nernst potential across the membrane, supporting the idea that the action potential is a sudden
increase in the axon membrane’s sodium conductance. [After Hodgkin & Katz, 1949]

the membrane know to change them in just the right sequence to create a traveling, stereotyped
wave?Sections 12.2.4–12.3.2 will address these questions.

12.2.3 The time course of an action potential suggests the hypothesis of voltage gating Contents[[Student version, December 8, 2002]] vii

voltage gating

The previous sections have foreshadowed what is about to come. We must abandon the Ohmic
hypothesis, which states that all membrane conductances are fixed, in favor of something more in-
teresting: The temporary reversal of the sign of the membrane potential reflects a sudden increase
ingNa+(Equation 12.14 instead of Equation 11.9), so thatgtottemporarily becomes dominated by
the sodium contribution instead of by potassium. The chord conductance formula (Equation 12.3
on page 447) then implies that this change drives the membrane potential away from the potas-
sium Nernst potential and toward that of sodium, creating the temporary reversed polarization
characteristic of the action potential.
In fact, the cable equation shows quite directly that the Ohmic hypothesis breaks down during
anerve impulse. We know that the action potential is a travelingwave offixed shape, moving at
some speedθ.For such a travelingwave the entire historyV(x, t)iscompletely known once we
specify its speed and its time course atonepoint: We then haveV(x, t)=V ̃(t−(x/θ)), where
V ̃(t)=V(0,t)isthe curve shown in Figure 12.6b.^8 Then we also have thatddVx=−^1 θddVt ̃′

∣∣

∣t′=t−(x/θ),

bythe chain rule of calculus. Rearranging the cable equation (Equation 12.7) then gives us the
total membrane currentjq,rfrom the measured time courseV ̃(t)ofthe membrane potential at a
fixed position:

jq,r=

aκ

2 θ^2

d^2 V ̃

dt^2

−C

dV ̃

dt

. (12.15)

Applying Equation 12.15 to the measured time course of an action potential, sketched in
Figure 12.8a, gives us the corresponding time course for the membrane current (Figure 12.8b).

(^8) Recall the image of travelingwaves assnakes under the rug (Figure 4.12b on page 120).