46 MATHEMATICAL BIOPHYSICS OF THE CENTRAL NERVOUS SYSTEM 



r when r is just less than one-half (LDR just less than unity) . 



More generally, if instead of alternating a stimulus 5 with no 

 stimulus, we had alternated S + AS with S , we should have obtained 

 the same results with log S/lh. replaced by log(l + A S/S) unless 

 S < h x . From this it is clear that for a constant S + A S , an increase 

 in S > h^ decreases the critical frequency f*. Similarly, an equal in- 

 crease in both S and S + A S decreases f*. That is, an illumination 

 added to both phases, as from stray light, decreases the critical flick- 

 er-frequency. 



Although we have referred to visual phenomena only, one may 

 well expect that analogous properties of some other modalities also 

 could be accounted for roughly by just such a simple mechanism as 

 the one considered here. 



We have assumed throughout that a > b and <p = xp . For a con- 

 stant stimulus this gives a a which rises rapidly to a maximum and 

 then subsides more slowly to zero. This resembles the "on" activity 

 of the "on-off" fibers of the retina. Had we chosen a < b , we should 

 find a < upon application of a constant stimulus, but on cessation 

 of the stimulus a would increase to a maximum and subside to zero. 

 This resembles the activity of the "off" fibers. We should have ob- 

 tained results entirely similar to these above but with (1—r) re- 

 placed by r , If, however, we suppose elements of both types to be 

 present at two different positions with different parameters and with 

 some simple interaction, the complexity of our results increases great- 

 ly. Again, if we remove the restriction $ = y> , and let R$ = xp , with 

 R a fraction having a value between zero and one, we find, that for 

 constant S , o- increases to a maximum and decreases to a constant 

 value (1—R)$. This corresponds to behavior of the continuously 

 acting elements of the retina. We proceed to consider some properties 

 exhibited by a neural element of this latter type. 



We have now a chain of two neurons, the afferent member of 

 which is of the mixed type with <f> — xp/R , < R < 1 . Let the stimu- 

 lation again be intermittent, of frequency f = 1/T and fractional 

 stimulation r . Let the intermittent stimulation be continued indef- 

 initely. The value of a at the end of each period of stimulation, that 

 is, <r(t) for t -> oo and t = rT(mod T) , can be determined in the 

 manner described above. If 6 is that value of a divided by a ( co ) for 

 r = 1 , 6 is essentially the ratio of a at the end of a period of stimula- 

 tion for a particular / and r , to the value which a would have if a 

 stimulus of the same intensity were applied continuously. would 

 be better defined as the ratio of (a - ^)/(tr 00 - h), but we shall neg- 

 lect the threshold as compared to a . We may then write 



