76 MATHEMATICAL BIOPHYSICS OF THE CENTRAL NERVOUS SYSTEM 



tation is different, for the S now measures, on the psychological scale, 

 the extent to which the two members of the pair are different. If the 

 two members of any pair are identical the associated S must be zero, 

 so that the additive term is determinate and the S's can in this case 

 be determined uniquely up to a common multiplicative factor. 



The ordering of the pairs along the S-scale is, however, only an 

 intermediate step since we wish to associate the individual objects 

 with points of a metric space of sufficiently many dimensions in such 

 a way that the S for any pair is the distance between the points 

 which represent the members of the pair. But the solution of this 

 problem depends upon the character of the space's metric, and the 

 metric in turn is a property of the mediating neural mechanism. 



Suppose, then, that the objects A p and B P constitute the pair C P 

 and that the afferents for the i-th modality affected by A P and B P 

 are connected by crossing inhibition. Then, if the thresholds are low, 

 beyond the locus of the crossing the corresponding <r is proportional to 



C pi =\A pi -B pi \ ( P = l,2) (3) 



along the afferent from A pi or B pi , whichever is the greater, while it 

 is negative along the other. If these two paths join at some subse- 

 quent synapse, then succeeding neurons of this chain are stimulated 

 in amounts proportional to C P i as given by equation (3). From here 

 the mechanism is just like the one previously discussed. Hence for 

 any pair (A , B) in place of equation (1) we have 



S = 2Li\A i -Bi\ . 



If the quantities A t and B l are physically measurable, or if they are 

 measurable by psychological methods independent of the method now 

 being outlined, e.g., in terms of J. N. D/s, the quantities Li have sig- 

 nifiance and it is an empirical problem to determine whether or not a 

 set Li exists satisfying all equations of this type. If the Ai and Bi 

 are not so measurable they are precisely the quantities to be deter- 

 mined from this procedure, and we may introduce units so chosen that 

 every L% = 1 . In this case, which we assume hereafter, 



S = 2\Ai-Bi\. (4) 



If it happens that for any three stimulus-objects, the S for one 

 pair is equal to the sum of those for the other two, then this experi- 

 ment provides no data for a multidimensional analysis. This does 

 not imply, however, that only a single modality is involved. If, for 

 any three objects, the S of any pair exceeds the sum of those for the 

 other two, the equations (4) are inconsistent, and further analysis is 

 impossible on the basis of the mechanism here proposed. Passing over 



