A STATISTICAL INTERPRETATION 113 



exclusive events each sufficient for the occurrence of the firing of C3 . 



Hence a more exact formulation in place of equation (2) would 



involve including with each product d r II v, also the product of all 



factors of the form (1 — d v. ) corresponding to all excitatory aifer- 



ents c not included in the set of the c/s . However , since d is a small 



quantity, the terms omitted in equation (2) are terms of a higher or- 

 der and in general negligible. In fact, to the same degree of approxi- 

 mation we can pick out the smallest among the r(ai) and neglect all 

 terms in the sum involving a larger r . We must suppose, however, 

 that the sum of the v k is large by comparison with the sum of the v, 

 for any set a t , since otherwise the effect of the inhibition also would 

 be negligible. 



With this understanding we find, by straightforward induction, 

 that the following rules of replacement enable us to transform a logi- 

 cal equivalence with regard to neural nets into a probability-relation: 



(1) Replace each assertion N(t) by 6 v(t) with the same sub- 

 script ; 



(2) Replace each negation ^Nit) by [1 — S v(t)] giving to 

 v the subscript of the N ; 



(3) Replace logical disjunction and conjunction by arithmetic 



addition and multiplication; 



t t 



(4) Replace the operators (z)t and (Ez)t by 77 and 2 respee- 



t=0 (=0 



tively ; 



(5) Where a function of t is preceded by an operator S a , re- 

 place the argument t by t — a 6 . 



The factor d can be everywhere omitted when the period of latent 

 addition is taken as the unit of time. 



We note in conclusion that with the approximation made here the 

 equation (2) can be expanded to the form 



dn = <5 r (l-<5 2 vk) 2 n v i9 (3) 



ks^i aiEKi(r) jeai 



where r is the smallest of the r(aO, and Ki(r) includes only those 

 classes a t which contain r neurons. On removing the parentheses and 

 multiplying out we obtain two terms, the first essentially positive, the 

 second essentially negative. The first we can therefore interpret as 

 the e , the second as the j of the present theory. 



