Anastomotic Bets Combating Noise 293 



fraction of the spaces in the symbol of the output neuron can 

 harmlessly be dashes? Figure 9 is Blum's diagram, based on 

 equations that are exact if N is a perfect square and fairly good 

 approximations for the rest. 



You will notice that for N less than 40, the output neuron (the 

 solid line) has fewer dashes. At '^40 they are equal, being ^^80 

 per cent of all spaces. For larger N, the output neuron has the 

 larger fraction; and, for N = 100, 90 per cent of spaces in the 

 input rank are dashes, and 98 per cent in the output neuron 

 are dashes. From this outcome, it is very clear that the output 

 neuron cannot be a majority organ like the one for N = 3. 



We all know that real nervous systems and real neurons have 

 many other useful properties. But I hope I have said enough to 

 convince you that these impoverished formal nets of formal neurons 

 can compute with an error-free capacity despite limited pertur- 

 bations of thresholds, of signal strength, and even of local synapsis, 

 provided the net is sufficiently anastomotic. If I have convinced 

 you, it has been in terms of a logic in which the functions, not 

 merely the arguments, are only probable. But even this prob- 

 abilisitic logic, for all its don't-care conditions, is adequate to 

 cope fully with noise of other kinds. Our neurons die — thousands 

 per day. Neurons, when diseased, often emit long strings of im- 

 pulses spontaneously and cannot be stopped by impulses from any 

 other neuions. And, finally, axons themselves become noisy, trans- 

 mitting a spike when none should have arisen or failing to transmit 

 one that they should have transmitted. 



To handle these problems in which the output of a neuron has 

 ceased to be any function of its input, von Neumann proposed 

 what is called "'bundling.'"' In the simplest case, one replaces a 

 simple axon from A by two axons in parallel. This alters the logic, 

 for now if all fibers in the bundle fire, A is regarded as certainly 

 true; if none fire, as certainly false; but between these limits there 

 is a region of uncertainty — call it a set of values between true and 

 false. In the simplest case, there are two such intermediate values. 

 Von Neumann found that if there are only two inputs per neuron, 

 the neurons had to be too good and the bundles too big. To com- 

 pute, say, X or ¥, with a net constructed like the net of Figure 10, 

 we find that given a probability of an error on the axon, say. 



