390 Information Storage and Neural Control 



Theorem VII 



Alterable synapses can be replaced by circles. 



This is accomplished by the method of Figure li. It is also to 

 be remarked tliat a neuron which becomes and remains spon- 

 taneously active can likewise be replaced by a circle, which is set 

 into activity by a peripheral afferent when the activity commences, 

 and inhibited by one when it ceases. 



THE THEORY: NETS WITH CIRCLES 



The treatment of nets which do not satisfy our previous assump- 

 tion of freedom from circles is very much more difficult than that 

 case. This is largely a consequence of the possibility that activity 

 may be set up in a circuit and continue reverberating around it 

 foi an indefinite period of time, so that the realizable Pr may 

 involve reference to past events of an indefinite degree of remote- 

 ness. Consider such a net VX, say of order /;, and let Ci, c-2, . . . , r^ be 

 a cyclic set of neurons of ^^l. It is first of all clear from the definition 

 that every N^ of ^^\ can be expressed as a TPE, of M, A^2, . . . , Np 

 and the absolute afferents; the solution of v)l involves then only 

 the determination of expressions for the cyclic set. This clone, we 

 shall derive a set of expressions [A\: 



Ndz,) . ^ . Pr.[S"" M(zi), *S"''-^ N,(z,), ... , S"''^ N,{z,)], (2) 



where Pr , also involves peripheral afferents. Now if n is the least 

 common multiple of the n,„ we shall, by substituting their equiva- 

 lents according to (2) in (3) for the A^„ and repeating this 

 process often enough on the result, obtain S of the form 



N,{z,) . ^ . Pr,[S"N,{zr), S"N,(zi), ... , S" Np(z,)]. (3) 



These expressions may be written in the Hilbert disjunctive nor- 

 mal form as 



N,{z,) . = . E S„ n '?" A^. (zi ) n ~ *^" ^i(2i), for suitable ^ (-i) 



where S„ is a TPE of the absolute afferents of V^I. There exist 

 some 2" different sentences formed out of the pN, by conjoining 

 to the conjunction of some set of them the conjunction of the 



