448 Chapter 12. Nerve impulses[[Student version, January 17, 2003]]
last Example.
Summarizing, Equation 12.3 is an approximation to the resting potential difference (Equa-
tion 11.12 on page 423).^4 Instead of describing a true steady state, Equation 12.3 describes the
quasi-steady (slowly varying) state obtained immediately after shutting off the cell’s ion pumps.
Wefound that both approaches give roughly the same membrane potential. More generally, Equa-
tion 12.3 reproduces a key feature of the full steady-state formula: The ion species with the greatest
conductance per area pulls ∆Vclose to its Nernst potential (compare Idea 11.13 on page 424). More-
over, a nerve cell can transmit hundreds of action potentials after its ion pumps have been shut
down. Both of these observations suggest that for the purposes of studying the action potential it’s
reasonable to simplify our membrane model by ignoring the pumps altogether and exploring fast
disturbances to the slowly varying quasi-steady state.
Capacitors Figure 12.3b contains a circuit element not mentioned yet: a capacitor. This symbol
acknowledges that some charge can flow toward a membrane without actually crossing it. To
understand this effect physically, go back to Figure 11.2a on page 412. This time, imagine that the
membrane is impermeable to both species but that external electrodes set up a potential difference
∆V.The figure shows how a net charge density,e(c+−c−), piles up on one side of a membrane,
and a corresponding deficit on the other side, whenever the potential difference ∆Vis nonzero.^5 As
∆V increases, this pileup amounts to a net flow of charge into the cell’s cytosol, and another flow
out of the exterior fluid, even though no charges actually traversed the membrane. The constant of
proportionality between the total chargeqseparated in this way and ∆Vis called thecapacitance,
C:
q=C(∆V). (12.4)
Gilbert says: Wait a minute. Doesn’t charge neutrality (the first two points listed on page 445)
say that charge can’t pile up anywhere?
Sullivan replies: Yes, but look again at Figure 11.2a: The region just outside the membrane has
acquired a net charge, but the region justinsidehas beendepletedof charge, by an exactly equal
amount. So the net charge in between the dashed lines hasn’t changed, as required by charge
neutrality. As far as the inside and outside worlds are concerned, it looks as though current passed
through the membrane!
Gilbert: I’m still not satisfied. The region between the dashed lines may be neutral overall, but
the region from the dashed line on the left to the center of the membrane is not, nor is the region
from the center to the dashed line on the right.
Sullivan: That’s true. Indeed, Section 11.1.2 showed it is this charge separation that creates any
potential difference across a membrane.
Gilbert: So is charge neutrality wrong or right?
Gilbert needs to remember a key point in our discussion of Kirchoff’s law. A charge imbalance over a
micrometer-size region will have an enormous electrostatic energy cost, and is essentially forbidden.
But the Example on page 229 showed that over a nanometer-size region, like the thickness of a
cell membrane, such costs can be modest. We just need to acknowledge the energy cost of such an
imbalance, which we do using the notion of capacitance. Our assertion that the currents into the
(^4) T 2 Actually, both of these equations are rather rough approximations, because each relies upon the Ohmic
hypothesis, Equation 11.8.
(^5) Actually, a larger contribution to a membrane’s capacitance is the polarization of the interior insulator (the
hydrocarbon tails of the constituent lipid molecules); see Problem 12.3.