PSYCHOPHYSICAL DISCRIMINATION 67 





ne 



Figure 5 



ent neuron form synapses with a number of neurons ne whose thresh- 

 olds differ but which are otherwise equivalent. Let all those which 

 have the same threshold be brought together to act on a single neuron. 

 Only two of an indefinite number of final neurons are shown in the 

 Figure. Thus all the neurons ne having the threshold h k are brought 

 together to act on a single neuron ne 3 . Let f(h)dhhe the number of 

 these neurons having thresholds between h and h + dh . Consider- 

 ing only the stationary-state activity, and using the linear relation- 

 ship of equation (5), chapter i, with ft = 1 , we may write 



s(S,h) = (S-h) f(h) (18) 



as the value of the excitation at s t , the terminus for the neurons ne 

 with the threshold h . 



Let equivalent inhibitory neurons of negligible thresholds origi- 

 nate at each synapse s» and terminate at every other synapse s, with- 

 out duplication. If a(S,h) is the value of a at the synapse Si at which 

 the neurons of threshold h terminate, then the value of j(S) at this 

 synapse, due to the activity of the neurons ni terminating there, is 



j(S)=Xfo(S,h) f(h)dh, (19) 



X being a constant measuring the activity of the inhibitory neurons. 

 Thus 



<j(S,h)=s(S,h) - X f a(S,h) f(h)dh. (20) 



This is a special form of equation (15), chapter iii, where the 

 /J and x' of the former equation are here replaced by X and h , respec- 

 tively, N(x', x) becomes f(h), and the former h is negligible. 



If f(h) is continuous and /(0) = 0, e(S,h) vanishes at h = 

 and h = S . Hence e(S,h) has at least one maximum in this range. 

 Assuming it to have only one, we see that e(S,h) will equal j(S) at 

 only two values of h , h x and h 2 , and in this range e(S,h) > j(S) so 

 that <r (S,h) > . That is, the neurons ne 3 corresponding to thresh- 

 olds in the range h x to h 2 are excited. Thus we may write 



