62 MATHEMATICAL BIOPHYSICS OF THE CENTRAL NERVOUS SYSTEM 



neurons. If a neuron of negligible threshold having afferent synapses; 

 at s x and s 2 tends to excite this outside group of neurons, then the 

 threshold h will decrease with the absolute value of u x or <r 2 , these 

 being proportional to the absolute values of a 3 and <x 4 . Thus h(\<j\) 

 should decrease linearly for small |<r|, though h cannot become nega- 

 tive. If we set 



h = h e-°w, (6) 



we have a suitable form, with but one new parameter introduced. 



In Figure 3 is shown a comparison between theory and experi- 

 ment for the visual data by W. N. Kellogg. The curves are computed 

 from the equations by setting the standard deviation equal to 0.58 

 meter-candles, — x = —0.10 meter-candles, ho = 0.49 meter-candles 

 and 9 = 1.14 meter-candles 1 . The intensity of the standard was 21.68 

 meter-candles. In the inset of the Figure is shown a comparison be- 

 tween equation (6) and the values of h determined from the data. 

 These are symmetric about — x c . This type of relationship between 

 h and the difference between the stimuli was found for each of the 

 individuals upon whom the experiment was carried out. The deriva- 

 tive of h(S) is discontinuous at —x (l . One would not expect to ob- 

 serve such a discontinuity even if it were present. If the threshold of 

 the neuron which produces the change of h with stimulus difference,, 

 had not been neglected, the value of h would have been a constant in 

 the neighborhood of — x, . For these reasons, a dotted curve is intro- 

 duced in the Figure to indicate that the discontinuities are not ex- 

 pected to appear in the data. For further details the reader is re- 

 ferred to the paper by H. D. Landahl (1939b). 



In the case of the auditory data (Figure 4) by W. N. Kellogg 

 (1930) one finds that a further asymmetry is present. A rather accu- 

 rate representation of the data results if one assumes that the effect 

 on the threshold h due to stimuli for which S r > S 2 is not the same as 

 that for which S 2 > <S\ . Since for this modality S 2 — Si may be a. 

 rather large fraction, the first term of the expansion of equation (1) 

 leads to noticeable error. Thus the parameters are measured in terms. 

 of the logarithm of the ratio of the stimuli. If we let h. = 0.43, 

 x<, = .02 , the standard deviation 0.18 or approximately 9 (milli- 

 volts) 2 , and 9 = 2.8 for o^ > , we obtain the curves shown in Figure 

 4. The points are the experimental values obtained by averaging the 

 results from a number of subjects. The inset shows the relationship 

 between h and log Si/S 2 which is decidedly asymmetric. If this be 

 considered significant, one might attempt to correlate the asymmetry 

 with the mode of presentation or perhaps with the modality. The. 



