Biological Physics: Energy, Information, Life

12.1. The problem of nerve impulses[[Student version, January 17, 2003]] 445

R=1/(gA)

VNernst

I=jqA

∆V=V 2 −V 1

(in)

1

2

(out)

2

a

RNa+ RK+ RCl− C

2 (in)

1 (out)

VNaNernst+ VNernstK+ VClNernst−

b

Figure 12.3:(Circuit diagrams.) (a)Duplicate of Figure 11.4, for reference. (b)Discrete-element model of a small
patch of cell membrane of areaA. The orientations of the battery symbols reflect the sign convention in the text:
Apositive value ofViNernstmeans that the upper wire entering the corresponding battery is at higher potential than
the lower wire. Three representative ion species can flow between the interior and exterior of the cell, corresponding
toi=Na+,K+,and Cl−.Each species has its own resistanceRi=1/(giA)and entropic driving forceVNernsti .The
capacitanceC=CAwill be discussed later in this subsection. The dashed arrow depicts the circulating current flow
expected from the data in Table 11.1. The effect of the sodium–potassium pumps described in Chapter 11 is not
shown.

suffices to send an action potential all the way to the end of even the longest axon.

• Indeed, the entire time course of the action potential is the same at all distant points
  (Figure 12.2b). That is, the action potential preserves its shape as it travels, and
  that shape is “stereotyped” (independent of the stimulus).^1


• After the passage of an action potential, the membrane potential actually overshoots
  slightly, becoming a few millivolts more negative than the resting potential, and then
  slowly recovers. This behavior is calledafterhyperpolarization.


• Foracertain period after transmitting an action potential, the neuron is harder to
  stimulate than at rest; we say it is “refractory” to further stimulation.

Our job in Sections 12.2–12.3 will be to explain all of these remarkable qualitative features of the
action potential from a simple physical model.

12.1.2 The cell membrane can be viewed as an electrical network

Iconography Section 11.2.2 on page 418 described the electrical properties of a small patch
of membrane using circuit diagram symbols from first-year physics (see Figure 12.3a). Before
elaborating on this figure, we should pause to recall the meanings of the graphical elements of a
schematic circuit diagram like this one, and why they are applicable to our problem.
The figure shown consists of “wires,” a resistor symbol, and a battery symbol. Schematic circuit
diagrams like this one convey various implicit claims:

1.No significant net charge can pile up inside the individual circuit elements: The charge into

one end of a symbol must always equal the charge flowing out the other end. Similarly,

2.Ajunction of three wires implies that the total current into the junction is zero.

(^1) T 2 This statement requires a slight qualification. Close to the stimulating point, stronger stimuli indeed lead
to a faster initial depolarization, as the membrane gets to threshold faster. These differences die out as the action
potential travels down the axon, just as theentireresponse to a hyperpolarizing response dies out (see the three lines
showing responses at different distances in Figure 12.1b).