456 The Molecular Basis of Nerve Conduction /24 : 4 



where a is the axon diameter. Combining this with the earlier defini- 

 tions, and with Equations HI and H3, leads to the differential propagation 

 equation 



a 8 2 V 8V 



2^ j^ = C M — + g K n\V - V K ) + g N& m*h(V - F Na ) + g h (V - V h ) 



(H6) 



Equations H2, H4, H5, and H6 then form a set of simultaneous 

 partial differential equations which are to be solved for V as a function 

 of x and t. Although these equations do not appear excessively involved, 

 it is not possible to solve them even by numerical methods without an 

 additional assumption. Since it is known that the spikes maintain 

 their shape as they propagate along the axon, it is assumed that there 

 exists a constant 0, such that 



8x 2 6 2 8t 2 [ } 



This is to say V obeys a wave equation with propagation velocity d. 



Assuming this is true, one may eliminate the space derivatives, 

 arriving at 



WJ 2 S" = Cm Tt + ^ 4(F " Vk) + s^ z KV - r Na ) + g h (v - r L ) 



(H8) 



Now the system has been reduced to ordinary differential equations 

 which can be integrated numerically. One may conclude that if the 

 answers are reasonable, the assumption of the wave equation was correct. 



Computations show that if 6 is chosen too large, V goes to + oo for a 

 small initial stimulus, and if 6 is chosen too small, Fgoes to — oo. There 

 exists a narrow range for 6, within which finite solutions are obtained for 

 V. These values of 6 agree with the experimental values within 15 

 per cent. 



The theoretical solutions for the original experiment described, of 

 stimulating a length of membrane with a current pulse, agree well with 

 the empirical data. Likewise, the computed form and amplitude of 

 the propagated spike potential are in accord with the experimental 

 measurements. The theory and experiment agree on a number of 

 other aspects of axon behavior. The only outstanding disagreement 

 with theory is that the calculated exchange of K + ions is higher than 

 that found in cuttlefish axons. 



Figure 1 1 shows the agreement between theory and experiment for 

 the form of the transmitted spike potential. The successes of the 

 differential equations derived for data from voltage clamped axons are 



