86 MATHEMATICAL BIOPHYSICS OF THE CENTRAL NERVOUS SYSTEM 



tion for the number of errors as a function of the number of trials 

 has been obtained on this basis and studied in relation to such para- 

 meters as strengths of reward and punishment, length of the alley, 

 and distance (time) from blind to goal (Landahl, 1941). One finds 

 that generally fewer errors will be required to eliminate a blind alley 

 if it is close to the goal. The dependence on the length of the alley of 

 the number of errors required to eliminate the blind is found to be 

 fairly complex. According to the strength of the reward, we find that 

 a blind far from the goal will be eliminated with more difficulty the 

 longer it is, while if it is near the goal it will be eliminated more 

 readily if it is long. What we wish particularly to emphasize is that 

 from relatively simple structure can be deduced fairly complex ac- 

 tivity. 



It is possible to generalize the mechanism to include a choice from 

 among any number N of stimuli by constructing a net similar to that 

 of Figure 2, but with N afferents and N(N — 1) crossing inhibitory 

 neurons (cf. chap. iii). Suppose that out of the N stimuli there is 

 but one correct stimulus S c . Hence, instead of considering the indi- 

 vidual wrong responses, we may consider their average effect. Thus 



if s c is the net value at s c due to the correct response and if s w is the 

 average value of all the e«/s we may write [Landahl, 1941, equation 

 (9)] 



NP 



\og—-^- + k(e c -8 w )=0 (11) 



N — 1 



'C 



in place of equation (8) of chapter ix. We note that for e c 



P w = (N — 1)/N as would be expected by chance, while for large 



e c — s w , P w tends toward zero. 



In the experimental situation, let a stimulus S'» (i s= 1 , 2 , • • • , M) 

 accompany a group of stimuli Sj (j '= 1 , 2 , • • • , N) of equal intensity, 

 one and only one of which will elicit its response. Among the stimuli 

 S) is a stimulus S ic , the "correct" stimulus corresponding to S'i , 

 which when chosen results in a reward ; response to any other stimu- 

 lus Sj when accompanied by S\ results in punishment, or at least no re- 

 ward. The number N may be referred to as the number of possible 

 choices, while the number M is the number of associations to be 

 learned in the experiment. After a wrong response is made, the ex- 

 perimenter may choose to assist (prompt) the subject in making the 

 correct response or he may not. He may do so each time, not at all 

 or, in general, some fraction, 1 — / , of the times. Thus / is a variable 

 under the control of the experimenter just as are M and N . We shall 

 assume that throughout any particular experiment M , N , and / are 



