12.2. Simplified mechanism of the action potential[[Student version, January 17, 2003]] 463
In short, our model displays threshold behavior: Small disturbances get driven back tov=0,
but above-threshold disturbances drive to the other “fixed point”v 2 .Our program is now to repeat
the steps in Section 12.1.3, starting from step (b) on page 451 (step (a) is unchanged).
b′.Equation
Wefirst substitute Equation 12.20 into the cable equation (Equation 12.7). Some algebra shows
thatv 1 v 2 =gtot^0 /B,sothe cable equation becomes
(λaxon)^2
d^2 v
dx^2
−τ
dv
dt
=v(v−v 1 )(v−v 2 )/(v 1 v 2 ). nonlinear cable equation (12.21)Unlike the linear cable equation, Equation 12.21 is not equivalent to a diffusion equation. In general
it’s very difficult to solve nonlinear, many-variable differential equations like this one. But we can
simplify things, since our main interest is in finding whether there are any traveling-wave solutions
to Equation 12.21 at all. Following the discussion leading to Equation 12.15, awave traveling
at speedθcan be represented by a function ̃v(t)ofonevariable, viav(x, t)= ̃v(t−(x/θ)) (see
Figure 4.12b on page 120). Substituting into Equation 12.21 leads to anordinary(one-variable)
differential equation: (
λaxon
θ
) 2
d^2 ̃v
dt^2
−τ
d ̃v
dt=
̃v( ̃v−v 1 )( ̃v−v 2 )
v 1 v 2. (12.22)
Wecan tidy up the equation by defining the dimensionless quantities ̄v≡v/v ̃ 2 ,y≡−θt/λaxon,
s=v 2 /v 1 ,andQ=τθ/λaxon,finding
d^2 v ̄
dy^2
=−Qd ̄v
dy
+s ̄v^3 −(1 +s) ̄v^2 + ̄v. (12.23)c′.Solution
Youcould enter Equation 12.23 into a computer-math package, substitute some reasonable values for
the parametersQands,and look at its solutions. But it’s tricky: The solutions are badly behaved
(they blow up) unless you takeQto have one particular value (see Figure 12.10). This behavior
is actually not surprising in the light of Section 12.2.2, which pointed out that mechanical analog
system selects one definite value for the pulse speed (and henceQ). You’ll find in Problem 12.6
that choosing
θ=±
λaxon
τ√
2
s(s
2− 1
)
(12.24)
yields a traveling-wave solution (the solid curves in Figure 12.10).
d′.Interpretation
Indeed the hypothesis of voltage gating, embodied in the nonlinear cable equation, has led to the
appearance of traveling waves of definite speed and shape. In particular, the amplitude of the
traveling wave is fixed: It smoothly connects two of the values ofvfor which the membrane current
is zero, namely 0 andv 2 (Figure 12.9). We cannot excite such a wave with a very small disturbance.
Indeed clearly for small enoughvthe nonlinear cable equation is essentially the same as the linear
one (Equation 12.9 on page 452), whose solution we have already seen corresponds to passive,