THE BOOLEAN ALGEBRA OF NEURAL NETS 109 



and with the substitution from the above equivalence for N b (t) the 

 construction is seen to be complete (Figure le). The other diagrams 

 of Figure 1 are discussed by McCulloch and Pitts (1943). 



In the case of non-cyclic nets, the necessary and sufficient condi- 

 tions for the firing of any neuron can be stated exclusively in terms 

 of the behavior of the peripheral afferents, i.e., the neurons of the net 

 to which no neuron is afferent. Morover, for a given net the requisite 

 firing times of the peripheral afferents is determinate in terms of the 

 firing times of the eff erents. The introduction of cycles, however, ren- 

 ders the problems considerably more difficult. For one thing, a cycle 

 may be of such a sort that when activity is once initiated in a suffici- 

 ent number of the neurons it will continue indefinitely. For the de- 

 tails in the discussion of this case reference must be made to Mc- 

 Culloch and Pitts (1943). However, there is in every cyclic net a 

 certain minimal number of neurons whose removal would render the 

 net non-cyclic. This number is called the order of the net, so that a 

 non-cyclic net has order zero. Then the behavior of the net is deter- 

 mined by the behavior of the set which consists of these neurons and 

 the peripheral afferents, and the problem reduces to a consideration 

 of the neurons of this set. 



The most striking difference between cyclic and non-cyclic nets, 

 with reference to the types of propositions which occur among the 

 conditions for firing of a neuron, lies in this, that universal and exist- 

 ential propositions may arise for cyclic nets. Thus the net consist- 

 ing of a neuron c 2 whose threshold is 2 , and a peripheral afferent 

 Cj in which d and c 2 each has a single endfoot upon c 2 , realizes the 

 universal sentence: 



N*(t) . = . (z)t.SN 1 (z). 



This is a symbolic formulation of the assertion that for c 2 to fire at 

 the time t it is necessary and sufficient that c x must have fired in 

 every interval z prior to t . An objection is immediate at this point, 

 to the effect that the theory provides no mechanism, therefore by 

 which the circuit can ever get started, and it must be admitted that 

 within the net in question there is no such mechanism. However, it 

 is to be understood that the theory concerns the behavior of the net 

 while free from all disturbing outside influences except specified ones 

 of a specified type (stimulation of peripheral afferents). That is, it 

 is a theory of a system in isolation, as any theory must be. Hence the 

 assertion is to be taken to mean that the circuit being once started 

 in an unspecified fashion and the system being then placed in "isola- 

 tion," the circuit can be maintained only by continued stimulation of 



