108 MATHEMATICAL BIOPHYSICS OF THE CENTRAL NERVOUS SYSTEM 



expresses the necessary and sufficient condition for the firing of one 

 of the neurons of this net. In other words, the behavior of any non- 

 cyclic net can be described exclusively by TPE's, and conversely any 

 TPE describes the behavior of a neuron in some theoretically possible 

 non-cyclic net. 



We give, as an illustration of this, an example due to McCulloch 

 and Pitts, the construction of a neural net capable of giving the illu- 

 sion of heat produced by transient cooling. In this illusion, a cold 

 object held momentarily against the skin produces the sensation of 

 heat, whereas if this object is held for a longer time the sensation 

 is only of cold with no sensation of heat, even initially. Heat and cold 

 are served by different skin receptors ; let c a and c 2 be these neurons. 

 Let c 3 and c 4 be the neurons whose activity implies a sensation of heat 

 and cold. Then for the firing of o a it is necessary either that c* shall 

 have fired, or else that c 2 fired momentarily only, whereas for c 4 to 

 fire it is necessary that c 2 shall fire for a period of time. 



These conditions are expressible symbolically in the f orm 



N a (t) . = .S{NAt) vS[SN 2 (t) .~N 2 (*)]}, 

 NM) . = .S[SN 2 (t) .N 2 (t)]. 



For convenience we suppose that the threshold 6 = 2 for each neuron, 

 which is to say that two endfeet from active neurons must connect 

 with any neuron in order that it may be made active. To construct 

 the net we must exhibit connections between the peripheral alferents 

 &x and c 2 , and the peripheral eff erents c 3 and c 4 , by way, perhaps, of 

 internuncial neurons, of such a sort that the equivalences hold as 

 given above. We start with the sentence N 2 (t) affected by the great- 

 est number of operations S and construct a neuron c a such that 



N a (t) . = .SN 2 (t). 



This requires two endfeet from Ca upon c c . An endfoot from c 2 and 

 one from c a each upon c 4 gives 



N,(t) . = .S[N a (t) .N 2 (t)]. = .S[SN 2 (l) .N 2 (t)] 



which satisfies the second equivalence. We next introduce c 6 having 

 upon it two endfeet from c a and an inhibitory connection from c 2 , 

 giving 



N t (t) . = .S[N a (t) .~N 2 (t)]. = .SlSN 2 (t) .~tf a (*)]. 



Finally a pair of endfeet from c & upon c 3 and also a pair from d gives 

 the disjunction on the right of 



N 3 (t) . = .S[NAt)vN b (t)-\, 



