256 HANDBOOK OF PHVSIOLOGV ^ CIRCULATION I 



Na 



I' 



2 



TIME (mSEC) 



FIG. 13. Sodium conductance (gua) ''"d potassium con- 

 ductance (gi^) as functions of time in a voltage-clamped squid 

 axon. Membrane voltage was held at the resting level until 

 time zero when the membrane was suddenly depolarized by a 

 fixed millivoltage denoted by the number attached to each 

 curve. Open circles are experimental points and the solid 

 curves were calculated from equations 9 and 10 in the text for 

 values of the parameters which gave the best fit. The corre- 

 spondence between measured and calculated values is good 

 except at early times; the observed delay in the rise 

 of gjj is greater than the delay in the theoretical curves. This 

 difference can be seen for a depolarization of 1 09 mv. Tempera- 

 ture, 6°C. [From Hodgkin (53).] 



As for K+, the a and /3 terms are rate constants which 

 depend on vohage, temperature and [Ca++]o but 

 not on time. Depolarization increases a,n and 0^^, 

 and decreases 0m and ah, since m starts to increase 

 and h to decrease following depolarization. 



The equations describing gK and gN,, can be easily 

 solved for voltage-clamp conditions. Since S is a 

 constant, the a and /3 terms are also constants and 

 the solutions for n, m, h are exponential. Figure 13 

 shows the close relations between the appropriate 

 solutions of these equations and the experimental 

 voltage-clamp data. The circles are experimental 

 measurements and the solid curses are solutions of 

 equations 9 and 10 with appropriate values of the 

 parameters. The numbers beside the curves give 

 the depolarization from the resting level in millivolts. 

 The fit is quite satisfactory, the principal difference 

 being that the theoretical curves for gx rise with some- 

 what less delay than the experimental curves. This 

 difference is not obvious in the curves of figure 1 3 but 

 can be seen clearly in figure 3 of (60) [cf. also (20)]. 



Fitting of equations a and 10 to the experimental 



data for different, fixed depolarizations provides 

 values of the a and j3 terms at the different clamping 

 voltages used. The fitted values of these parameters 

 are reasonably well described by continuous functions 

 of S, e.g., 



in = o.oi(-AS -f 10) / exp( ) ^ ' ' 



(m) 



/3„ = 0.125 ^^P ( — i£/8o) 



where AS = 8 — Sr. Similar expressions describe Om, 

 /3m, "h, and j3h. Equations 9, 10, and 11 describe the 

 variations in time and voltage of R^a and Rk in 

 figure 35. 



PREDICTION OF THE .\CTION POTENTL-VL (60). With ex- 

 plicit expressions for the a and jS terms, gNa and gK, 

 the behavior of the membrane under space-clamp con- 

 ditions can be predicted by solving equations 7b, 8, 

 9, I o, and 1 1 for S as a function of time. For the space 

 clamp, Im is a constant and the set of equations de- 

 scribing S is: 



Cs -f- li = Im = constant, I, = Ka -f- Ik -|- Ii 



iNa = gNam'h(S — SNa), 



Ik = gKn^e - Ek)Ii = gi(S - Si) (12) 

 in = am(i — m) — /3„,m, . . . 



The term Ii = gi(S — £1) is the small component of 

 the total ionic current not carried by Na+ or K"*", gi 

 being a constant. 



The upper curves in figure 14 are solutions of the 

 above equations for initial depolarizations of 90, 15, 

 7, and 6 mv and for I,„ = o. The lower curves are 

 measured action potentials recorded under compara- 

 ble conditions. The two sets of curves are remarkably 

 similar, although not identical. The most obvious dif- 

 ference is that the voltage of the recorded action po- 

 tentials falls appreciably after the cessation of the 

 stimulus, but the voltage of the calculated potentials 

 falls only slightly. Other differences include sharper 

 peaks and a more abrupt development of postspike 

 hyperpolarization in the calculated action potentials. 

 Another, less obvious difference is the presence of a 

 slight hump on the falling phase of the calculated ac- 

 tion potential. Despite these minor shortcomings, the 

 agreeinent between real and computed action poten- 

 tials is excellent. However, mere resemblance is not 

 sufficient to establish the validity of equation 12 as an 

 adequate description of nerve membrane properties. 

 To be satisfactory, the formulation must predict a) 

 the existence of a threshold depolarization (fig. 14); 



b) the nature and duration of the refractory periods; 



c) after-hyperpolarization (figs. 14 and 15); d) the ex- 



