A LOGICAL CALCULUS OF THE 

 IDEAS IMMANENT IN NERVOUS ACTIVITY* 



Warren S. McCulloch and Walter H. Pitts 



Because of the "all-or-none" character of nervous activity, 

 neural events and the relations among them can be treated by 

 means of propositional logic. It is found that the behavior of 

 everv net can l:)e described in these terms, with the addition of 

 more complicated logical means for nets containing circles; and 

 that for any logical expression satisfying certain conditions, one 

 can find a net behax'ing in the fashion it describes. It is shown 

 that many particular choices among possible neurophysiological 

 assumptions are equivalent, in the sense that for every net be- 

 having under one assumption, there exists another net which 

 behaves under the other and gives the same results, although 

 perhaps not in the same time. Various applications of the calculus 

 are discussed. 



T. 



INTRODUCTION 



HEORETIClAL neurophysiology rests on certain cardinal as- 

 sumptions. The nervous system is a net of neurons, each having a 

 soma and an axon. Their adjunctions, or synapses, are always be- 

 tween the axon of one neuron and the soma of another. At any in- 

 stant a neuron has some threshold, which excitation must exceed to 

 initiate an impulse. This, except for the fact and the time of its 

 occurrence, is determined by the neuron, not by the excitation. 

 From the point of excitation the impulse is propagated to all parts 

 of the neuron. The velocity along the axon varies directly with its 

 diameter, from less than one meter per second in thin axons, 

 which are usually short, to more than 150 ixieters per second in 

 thick axons, which are usually long. The time for axonal conduc- 

 tion is consequently of little iinportance in determining the tiine 



*Reprinted from The Bulletin of Mathematical Biophysics, 5:115-133. 1943, with 

 permission of the Editor, N. Rashevsky. 



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