CONDITIONING 87 



not changed. Then 



£ c = £oc + bc + b(l-f)w, (12) 



since conditioning improves with each correct response as well as with 

 a fraction (1 — /) of the wrong responses. The prompted correct 

 responses are not counted in c so that we do not change the relation 

 n = c + w . At each wrong choice, a quantity p is subtracted from 

 £ ow . This contributes only p/ (N — 1) to the average. Thus 



** = 8a»-fi/{N-l). (13) 



The parameter b gives a measure of the amount of conditioning 

 per trial. If a response to one stimulus has no effect on the condition- 

 ing at centers corresponding to other stimuli, the n is independent of 

 M . But, if the response to one stimulus results in the stimulation of 

 inhibitory neurons terminating at the various other conditioning cen- 

 ters, then b will be less when there are more items M to be learned. 

 We may account for this by introducing as a rough approximation, 

 the relation 



k 



where r\ and 'Q are two parameters replacing bk. 



Assuming e ow = e oc , substituting equations (5), (12) and (13) 

 into (11), and eliminating c by equation (7), we obtain a differential 

 equation in zv and n . For the initial condition w = for n = 0, and 

 with b eliminated by relation (14), the solution of the differential 

 equation is 



(N-l)et» N 



w = log . (15) 



Nf-f-0 (Nf- f-0) e-v m + N - Nf + f 



This equation gives the number w of errors as a function of the num- 

 ber n of trials for any number M of items, for any number of N pos- 

 sible choices, and for any fraction (1 — /) of prompting by the ex- 

 perimenter. All this involves only two parameters C and r\ . As we 

 have considered a highly over-simplified mechanism and introduced 

 a number of approximations, it is not to be expected that the predic- 

 tions of equation (15) should hold over too wide a range of values 

 of M and N . 



In Figure 4 are data obtained from a single experiment for the 

 purpose of illustrating a rather special case of the experimental pro- 

 cedure outlined above. The experiment corresponds to the case in 



