84 MATHEMATICAL BIOPHYSICS OF THE CENTRAL NERVOUS SYSTEM 



equation (4). To a first approximation, equation (4) becomes in this 

 case Ae c = n c b (S c , t c ) where b = Aa depends on S c and t c , and where 

 c is the number of repetitions of S c and Ri together and thus replaces 

 n . If S c and t c are constant, the total e at s c is given by 



e c = s c + be . (8) 



We can obtain a similar expression for e w at s w . This is the case when 

 the final response is pleasant, but if the final response is unpleasant 

 we should expect the effect of the conditioning to be the opposite. 

 That is, the centers C are such that they have inhibitory fibers lead- 

 ing to s w . Then the coefficient corresponding to b will be some nega- 

 tive quantity — /S . Thus 



e w = e ow — pw, (9) 



as w is the number of repetitions of wrong response leading to R 2 . 



Let us apply these results to the particular experimental situation 

 which arises when Lashley's jumping-apparatus is used. Here an ani- 

 mal is forced to jump toward either of two stimuli. Choice of one 

 leads to reward, choice of the other may lead to punishment. For 

 simplicity, we assume that the times t c and t w , respectively, from 

 presentation of the stimuli to the reward and punishment, are con- 

 stants. If we then introduce equations (8) and (9) into equation (8) 

 of chapter ix, and eliminate P w and c by means of equations (6) and 

 (7), we obtain a differential equation in w and n . From this, with 

 the initial condition w = for n = , we obtain 



1 2be kle ° c ~ e ° w) 



w — fog QO) 



k(b - ft) & 2&e* (e "- £ °«> - (b - /S) (1 - e- kbn ) 



for ft^jS. For 6 = >5 , the result is a rising curve which approaches a 

 limit exponentially. In terms of the mechanism, we may consider the 

 experiment as requiring a discrimination between two stimuli whose 

 values, in effect, change in successive trials. The correct stimulus be- 

 comes effectively larger due to the conditioning while the wrong de- 

 creases. Thus, the probability of a wrong response diminishes. 



In Figure 3 is shown a comparison between the theory and the 

 experimental data by H. Gulliksen (1934). The lower and upper 

 curves were obtained respectively by setting e oc — e ow — , kb = .0121 , 

 ft = and k(e oc — c ow ) = -.46 , kb = .0229, § = . Besides giving a 

 quantitative relation between w and n, our considerations actually 

 give a great deal more. From the results of the preceding paragraphs, 

 we can obtain a function b (S c , Ri , t c ), that is, b is a function of the 

 intensity of the stimulus, the strength of the reward, and the time t c . 



