392 Information Storage and Neural Control 



Theorem VIII 



The expression (9) for neurons of the cyclic set oj a net S'X together 

 with certain TPE expressing the actions oJ other neurons in terms oj 

 them, constitute a solution of V)I. 



Consider now the question of the reahzabihty of a set of S,. A 

 first necessary condition, demonstrable by an easy induction, is that 



(z.2)zi . pAz-2) ^ p,{z,) .^.Si^ sMj (10) 



should be true, with similar statements for the other free p in Si'. 

 i.e., no nervous net can take account of future peripheral afferents. 

 Any S, satisfying this requirement can be replaced by an equi- 

 pollent S of the form 



{Ef) (z,)zy {z,)zz,r.hPr,„, 



:f{Zr,Z,,Zs = 1 . ^ ./>.3(Z2) (11) 



where zZp denotes p, by defining 



Pr„,i = /[(zi) {Z2)zi{zs)zzp : . f(zu z-i, Zs) = . v . /(zi, Zo, Zs) 

 = 1 :/(zi, Zo, Z3) = 1 . = . /),3(z,) : -^ : S,]. 



Consider now these series of classes a,, for which 



N ,{V) : = : {E<\>) {.v)t(^m)q : 4>ecxi :ISf„,{x) . = . <i>{t, x, m) = 1. 



[/ = ry + !,••• ,M] (12) 



holds for some net. These will be called prehensible classes. Let us 

 define the Boolean ring generated by a class of classes k as the 

 aggregate of the classes which can be formed from members of k 

 by repeated application of the logical operations; i.e., we put 



-^ aeX : a, ^eX . — > . — a, a . (3, aW jSeX]. 



We shall also define 



^(k) . = . (R(k) - t'p' - "'V', 



f-i\e(K) =p X[(a, /3) : atK -^ ae\ . ^ . — a, a . (3, aV (3, S "aeX 



and 



G{'\>,t) = i[{m) . cf>{t -\- l,t, m) = '!^(m)]. 



