Biological Physics: Energy, Information, Life

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

5. 2.5 0 2.5 5. 7.5 10.

0.2

0.6

1.

x

=0

wrongθ

x

=6

λaxon

x

=3

λaxon

tθ/λaxon

v/v

2

Figure 12.10: (Mathematical functions.) Traveling-wave solution to the nonlinear cable equation (see Prob-
lem 12.6). The membrane potential relative to rest,v(x, t), is shown as a function of time at three different fixed
locations (three solid curves). Points at largerxseethewave go by atlater times, so this wave is traveling in the +x
direction. The parameters≡v 2 /v 1 has been taken equal to 3 for illustration. Comparing to Figure 12.2b on page
444 shows that this simplified model qualitatively reproduces the leading edge of the action potential. The dashed
line shows a solution to Equation 12.22 with a value of the front velocityθdifferent from that in Equation 12.24;
this solution is singular. Time is measured in units ofλaxon/θ.The potential relative to resting is measured in units
ofv 2 (see text).

diffusive spreading (electrotonus), not an action potential. Thus

a. Voltage gating leads to a graded, diffusive response for stimuli below

some threshold, but above-threshold, depolarizing stimuli yield a large, fixed-

amplitude response.^11

b. The above-threshold response takes the form of a traveling wave of fixed

shape and speed.

(12.25)

Our model, a mathematical embodiment of Idea 12.18, has captured many of the key features
of real nerve impulses, listed at the end of Section 12.1.1. We didn’t prove that thewaverapidly
forgets the precise nature of its initial stimulus, remembering only whether it was above threshold
or not, but this should seem reasonable in the light of the mechanical analogy (see Section 12.2.2).
Wealso get a quantitative prediction. The velocityθis proportional toλaxon/τ=

√

aκgtot/(2C^2 ),
times a factor independent of the axon’s radiusa.Thusthe model predicts that if we examine a
family of unmyelinated axons of the same general type, with the same ion concentrations, we should
find that the pulse speed varies with axon radius asθ∝

√

a.This prediction is in fact roughly borne
out in experimental data. Moreover, the overall magnitude of the pulse speed is≈λaxon/τ.Forthe
squid giant axon our estimates give this quantity as about 12mm/(2ms)=6ms−^1 ,within an order
of magnitude of the measured action potential speed of about 20ms−^1. This is an encouraging
preliminary result.

(^11) T 2 Our simple model does not treat hyperpolarizing initial conditions correctly (compare the dashed and solid
lines in Figure 12.9a atv<0).