2 MATHEMATICAL BIOPHYSICS OF THE CENTRAL NERVOUS SYSTEM 



threadlike processes, but the terminations of these processes are of 

 two sorts. A termination of the one sort we shall call an origin, one 

 of the other sort, a terminus. When appropriate stimulation of suffici- 

 ently high degree is applied at an origin, there is conducted along the 

 neuron a "nervous impulse" all the way to the various termini. This 

 nervous impulse, arriving at a terminus, may contribute to the stimu- 

 lation of any neuron which has an origin at the same place. 



Doubtless in all strictness the impulse does not simply jump from 

 neuron to neuron but passes by way of some intermediary process 

 set up in the synapse which is the junction between the two neurons. 

 From our point of view, it is largely a matter of convenience whether 

 we postulate such an additional process or not. 



The nervous impulse manifests itself as a localized change in 

 electric potential, its duration at any point is about half a millisecond, 

 and it is transmitted at a rate that, though low in some neurons, in 

 others may equal or exceed 10* cm sec -1 . Moreover, in physiological 

 stimulation, if the stimulation is maintained, the impulses are repeat- 

 ed and may reach a frequency which is of the order of 10 2 sec -1 . The 

 more intense the stimulation, the more frequent the impulses, but 

 there is an upper limit for any given neuron which varies somewhat 

 from neuron to neuron. When we have occasion to take account of 

 it, we shall suppose this upper limit to be a fixed characteristic of 

 the neuron. 



McCulloch and Pitts (1943) have developed a theory of the 

 "quantized" dynamics of the neuron which takes account of the in- 

 dividual impulses and we shall return to this later at the end of this 

 monograph. For the present, however, we shall schematize further 

 by doing some statistical averaging and by fixing our attention upon 

 the synapse rather than upon the neuron itself. We shall choose the 

 alternative of supposing that the impulses of the afferent neurons are 

 not the immediate stimuli for the efferent neuron, but that these im- 

 pulses start or maintain at the synapse an intermediate process which 

 is the immediate stimulus. To have a concrete picture, one may imag- 

 ine that some chemical substance is released by the impulses and dis- 

 sipated or destroyed as a monomolecular breakdown. However, it is 

 by no means implied that this is the case, and furthermore, we shall 

 not speak in such terms but shall speak merely of an "excitatory 

 state," and denote the state or its intensity by s . More briefly we 

 shall speak of the excitation e. 



The amount by which the impulses increase s in unit time is pre- 

 sumably proportional to the frequency of these impulses, and the fac- 

 tor of proportionality is taken to be a characteristic of the fiber. We 

 make the simplest assumption as to the rate of dissipation of e and 



