THE BOOLEAN ALGEBRA OF NEURAL NETS 105 



Now consider any net of interconnecting neurons. Let these 

 neurons be assigned designations ©» in any manner and let Ni(t) 

 denote the proposition that the neuron Ci has fired during the t-th 

 time-interval. If this is any but a peripheral afferent, fired directly 

 by a receptor, then the necessary and sufficient condition for Ni(t) 

 is that some one proposition or group of propositions among, perhaps, 

 several possible such, of the form Nj(t-l) shall be true, where the 

 neurons c, are those afferents forming excitatory connections with 

 Ci , and that furthermore the propositions N k (t—1) are all false, the 

 neurons c k being the afferents forming inhibitory connections with 

 Ci . Let a; represent the class of subscripts corresponding to any set 

 of afferents which form excitatory connections with Ci and whose 

 summated impulses suffice to excite Ci and let k; represent the class of 

 all such classes a* . Let pi represent the class of all subscripts corre- 

 sponding to all afferents forming inhibitory connections with Ci . In 

 conventional logical symbolism, the negation of N k (t—1) is repre- 

 sented by ™N k (t— 1) and the joint negation for all kefii by 



n ~zv*(*-i), 



fce/3i 



where the symbol s here denotes class membership. A sufficient con- 

 dition for the firing of neuron d is then 



n ~Ak(t-i) n Nj(t-i), 



kepi jeai 



and the necessary and sufficient condition is the disjunction of all 

 such propositions, or 



n ~tf*(*-i) 2 n Njh-i), 



the symbols 77 and 2 denoting conjunction and disjunction respec- 

 tively. If we introduce the functor 5 defined by the equivalence 



SNi(t) . = .Ni(t-l), (1) 



then the activity of the neuron d is completely described by the equi- 

 valence 



Ni(t) . = .S n ™N k (t) 2 nNj(t). (2) 



To consider examples, suppose neuron c 3 has a threshold 6 = 2. 

 If Ci and c 2 each has a single enclfoot on c 3 (Figure lc), then c 3 fires 

 in any interval only if c 1 and c 2 have both fired in the preceding in- 

 terval : 



N 3 (t) . = .SNAt) .SNAt). 



On the other hand, if c x and o 2 each has two endfeet on c 3 (Figure lb) , 



