270 BIOPHYSICAL STUDIES ON NERVE AND MUSCLE 



given concentrations; (b) leaky membranes, through which the Na + and K + 

 permeate, or diffuse down their respective gradients; (c) electrical charges 

 permanently fixed within the hundred angstroms or so of effective thickness 

 of membrane; or (d) changes in shape of acetylcholinesterase (ACE), an 

 enzyme located on the surface of nerve (Figure 10-3 illustrates) and thought 

 by some to be the cover whose shape determines whether or not Na + , or K h , 

 or both, can enter the slip through the pores in the membrane. 



Two quantitative theories permeate the literature on nerve transmission: 

 the use of the electrical cable theory to describe the spread of a localized 

 electrical disturbance; and the description of ionic currents through the 

 membrane as a function of permeability. 



Early in the century electrical engineers had worked out the effect of a 

 break, or a series of breaks, in the insulation of an electric cable having a 

 metallic conductor inside and salt water outside. By 1938 Curtis and Cole 

 had used this application of Ohm's law to describe how a localized dis- 

 turbance in a nerve membrane can spread on down the nerve. The key 

 expression is: 



d V 



d 2 V = r x + r 2 

 dx 2 r 



E- V - rC m 



dt 



where /: is the concentration-cell voltage across the membrane in the ab- 

 sence of a disturbance, i.e., when the membrane is resting; V is the "action" 

 voltage at any time, /, at a distance, x, along the surface from the site of the 

 disturbance, 0; r, and r 2 are the electrolytic resistances (ohms), between 

 and x, in the outside conductor and the inside conductor, respectively; 

 r is the resistivity of the membrane (fixed, unknown thickness) in ohm cm 2 ; 

 and C m is the capacitance of the membrane, which is being depolarized (dis- 

 charged) at a rate dV / dt. The expression teaches that the depolarization oc- 

 curs at a rate which increases as the divergence (spread) of voltage along the 

 surface increases, and decreases as the resistances to ion flow (r„ r 2 , and 

 r) increase. 



By 1952 Hodgkin and Huxley had described measured changes in mem- 

 brane conductance of the giant axon of the squid in terms of change in the 

 permeabilities of the simple ions of the external and internal media. The 

 principle ideas of this theory will now be given. 



Currents through the membrane are considered to charge (or discharge) 

 the membrane capacitance and to leak Na + , K + , and other ions as well. 



Thus: 



/ = C m dV/dt + / Na+ + / K+ + / 



where /is total current, and the 7,'s are the currents due to the different ions. 

 Then each /, is expressed as being the product of the membrane conduct- 

 ance (g t ) and the driving voltage for that ion. Thus: I t = g t A V 



