88 MATHEMATICAL BIOPHYSICS OF THE CENTRAL NERVOUS SYSTEM 



n-»- 



Figure 4. — Comparison of theory with experiment: simple learning. Curve, 

 theoretical from equation (15); points, experimental (Landahl, 1941). Abscissa, 

 number of trials ; ordinate, number of errors. 



which M = N and / = 1 . Then setting y\ = 1.15 and C = .098 we ob- 

 tain the three curves for which N = 4, N = 8, and N = 12 respectively. 

 The values of v\ and C may be determined from one point of each of 

 any two curves. The third curve is then without unknown parameters. 

 But another family of such curves is also determined by equation 

 (15), without any additional parameters, for the case in which 

 prompting- follows each wrong' response, i.e., f = . In fact, / can 

 be given any value in the range = / = 1 , and M and N need not be 

 equal. 



From the previous discussion, we should expect b to depend upon 

 the strength of reward. To a first approximation, we may set 

 6 = a lP /k , where p is a measure of the strength of reward. Similar- 

 ly, we can set ft ==■ a 2 /pk where p is a measure of the strength of pun- 

 ishment. Equation (15) then determines a hyper-surface in the seven 

 variables, w , n , N , M , f , p , and p in terms of three parameters 

 a-! , a 2 , and C • 



