12.2. Simplified mechanism of the action potential[[Student version, January 17, 2003]] 461
Note that Equation 12.17 states that the conductances track changes in potential instanta-
neously. Section 12.2.5 will show how this simplified conductance hypothesis already accounts for
muchof the phenomenology of the action potential. Section 12.3.1 will then describe how Hodgkin
and Huxley managed to measure the conductance functions, and how they were forced to modify
the simplified voltage-gating hypothesis somewhat.
12.2.4 Voltage gating leads to a nonlinear cable equation with traveling-wave solutions
wavesolutions
Wecan now return to the apparent impasse reached in our discussion of the linear cable equation
(Section 12.2.1): There seemed to be no way for the action potential to gain access to the free energy
stored along the axon membrane by the ion pumps. The previous subsection motivated a proposal
for how to get the required coupling, namely the simple voltage-gating hypothesis. However, it
left unanswered the question posed at the end of Section 12.2.3: Who orchestrates the orderly,
sequential increases in sodium conductance as the action potential travels along the axon? The full
mechanism discovered by Hodgkin and Huxley is mathematically rather complex. Before describing
it qualitatively in Section 12.3, this subsection will implement a simplified version, in which we can
actually solve an equation and see the outline of the full answer to this question.
Let’s first return to our mechanical analogy, a chain that progressively shifts from a higher to a
lower groove (Figure 12.5a). Section 12.2.2 argued that this system can support a traveling wave
of fixed, speed and definite waveform. Now we must translate our ideas into the context of axons,
and do the math.
Idea 12.12 said that the force needed to pull each successive segment of chain over its potential
barrier came from theprevioussegment of chain. Translating into the language of our axon, this
idea suggests that while the resting state is a stable steady state of the membrane,
- Once one segment depolarizes, its depolarization spreads passively to the
neighboring segment; - Once the neighboring segment depolarizes by more than 10mV,the positive
feedback phenomenon described in the previous subsection sets in, triggering a
massive depolarization; and - The process repeats, spreading the depolarized region.
(12.18)
Let’s begin by focussing only on the initial sodium influx. Thus we imagine only one voltage-gated
ion species, which we call “Na+.” We also suppose that the membrane’s conductance for this ion,
gNa+(v), depends only on the momentary value of the potential disturbancev≡V−V^0. (As
mentioned earlier, these assumptions are not fully realistic; thus our simple model will not capture
all the features of real action potentials. Section 12.3.1 will discuss an improved model.)
Adetailed model would use an experimentally measured form of the conductance per area
gNa+(v), as imagined in the dashed line of Figure 12.9a. We will instead use a mathematically
simpler form (solid curve in the figure), namely the function
gNa+(v)=g^0 Na++Bv^2. (12.19)Hereg^0 Na+represents the resting conductance per area; as usual we lump this in with the other
conductances, givingg^0 tot. Bis a positive constant. Equation 12.19 retains the key feature of
increasing upon depolarization (positivev); moreover it is always positive, as a conductance must
be.