CONDUCTION OF THE NERVE IMPULSE 



surrounding fluid medium produced by the triphasic 

 membrane current just mentioned. If the space-time 

 pattern of the membrane current is given, the problem 

 of finding the potential field in a volume conductor is 

 a purely physical problem, namely, an application of 

 Ohm's law to the electrolytic conductor around the 

 axon. 



The simplest example of problems of this type is the 

 case in which a uniform axon is surrounded through- 

 out its length by a conducting fluid of a uniform thick- 

 ness (fig. 25/I). We assume that the volume of fluid 

 is not so small as to modify the spatial distribution of 

 the membrane current di.scussed above. Let s denote 

 the resistance per unit length of the surrounding fluid 

 medium; in the present case, s « ^i, where r-, is the 

 resistance per unit length of the axoplasm. We express 

 the potential diflerence across the axon membrane at 

 point .V and time / explicitly as l^x + vO, indicating 

 that the variation in the membrane potential travels 

 leftward at a constant velocity, v. Similarh, the 

 longitudinal current and the membrane current are 

 functions of (.v -{- vt). 



It is simple to show that the total current flowing 

 through the whole cross section of the surrounding 

 fluid medium at any point, .v, at any moment, t, is 

 equal and opposite to the longitudinal current in the 

 axon at the same x and /. To the present one dimen- 

 sional approximation, the current in the medium at 

 .V and time ; is given by — /i(.v + vt'). Denoting the 



^1 



Xz 



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222 



FIG. 25. A. A uniform axon immersed in a conducting Huid 

 medium of uniform diameter; the action potential recorded 

 with electrodes at xi and x-; is given by the equation (9-3). 



B. The case in which the diameter of the fluid medium changes 

 at x'; the action potential recorded is given by equation (9-4). 



C. A uniform axon immersed in a large volume of fluid; the 

 potential near the axon is given by the triphasic curve in the 

 diagram. 



potential at .vo in the medium referred to that at .vi by 

 U2-1, it is found that 



- = L 



si, fix + I'Odx 



VQx, + vt) V(,Xi + vt}. 



(9-3) 



[Note that the integral above represents a summation 

 of the IR drops along the fluid medium at a given 

 moment /.] The action potential recorded externally 

 with electrodes placed at Xi and X2, U2-1, consists of 

 two terms, one representing the activity at Xi, 

 F(Ar2 -f vt), and the other, the activity at xi, F(xi + 

 vt). The amplitude of the observed potential variation 

 is reduced by a factor of j/n- Equation (9-3) is a 

 mathematical expression of what is known as 'diphasic 

 recording' of the action potential. Because of the 

 negative sign in front of VQx^ -\- vt), it was believed 

 that the surface of the active portion of an axon was 

 'electrically negative' to the surface at rest. It should 

 be borne in mind, however, that, if the surrounding 

 inedium is not uniform, the potential on the active 

 surface is not always negative to that on the resting 

 surface. 



The next simple example of the volume conductor 

 problems is the case in which the resistance per unit 

 length of the conducting fluid medium changes at x' 

 suddenly from si to so (fig. 25Z?). Expressing j- as a 

 function of ,v, it is found that 



U,^i 



-f 



<.v)/,(.v -I- vt) d.v 



- 1 n , - SV(^x -I- ;./) ^ 



dx 



(9-4) 



_5 K(.v, + ,-0 - — ' F(.v> + vt) 



+ 



V\x' -I- vt). 



[The last step of the calculation above was accom- 

 plished by integration by parts.] The right-hand mem- 

 ber of equation (9-4) contains three terms, the first 

 term representing the activity at x-t, the second term 

 that at .V] and the third term arising from the activity 

 at .v'. The third term changes its sign, depending on 

 whether s-i < s\ or .f-.> > S\;\X vanishes when si = 5\. 

 If Si is nearly zero, i.e. if the amount of fluid around 

 the axon is very large on one side of x' , the second 

 term in equation (9-4) vanishes and the equation in- 

 dicates that the electrode at .vi effectively records the 

 potential variation at x' . 



