112 MATHEMATICAL BIOPHYSICS OF THE CENTRAL NERVOUS SYSTEM 



dependent, the probability that impulses will arrive concomitantly 

 (that is, within an interval 6) along any group of r of these neurons is 

 equal to the product S r JJ Vj , there being r factors Vj . Likewise the 

 probability that no impulse will arrive along any one of a group of s 

 (inhibitory) neurons c k is the product 77(1 — S v k ), there being here 

 s factors. And if the group of r neurons c, is sufficient to excite Ci , 

 the probability that c, will be excited in any interval <3 by the neurons 

 Cj is the product of r + s factors. 



d'nvjlia-dn) (1) 



if the s neurons c^ include all inhibitory afferents to c ; and if the im- 

 pulses are statistically independent. If we then form the product such 

 as (1) corresponding to every group of neurons c, sufficient to fire 

 Ci by their concomitant activity in the absence of inhibitory impulses, 

 the probability that d will fire is given approximately by summing 

 all products so formed. Thus with the same notation as that em- 

 ployed for the equivalence (2) of the preceding chapter, we find that 

 the frequency vi with which c, fires is given by the equation 



dvi = n a-svk) 2 * r(ai) n vj, (2) 



where r(cti) is the number of neurons in the set ai . But if we com- 

 pare this equation, and the manner in which it was formed, with the 

 equivalence (2), chapter xii, we see that the two sides of the equiva- 

 lence become identical with the two sides of the equation when each 

 assertion N is replaced by the corresponding dv , and each negation 

 °°N by the corresponding (1 — dv). 



However, as we have remarked, this expression is approximate 

 only. Each product 6 r 77 v } is equal to the probability that at least all 

 the neurons in the particular set of r neurons Cj will fire in the time- 

 interval S , but the possibility that additional excitatory afferents may 

 also fire is not ruled out. Duplication is therefore possible and the 

 sum would then be too large. To give a concrete example, suppose 

 there are only two afferents, c 1 and c 2 , both excitatory, and each alone 

 capable of stimulating c 3 . Then c 3 may be excited by the firing of (h 

 alone, by the firing of c 2 alone, or by the concomitant firing of c^ and 

 c 2 . Now by the formula we obtain the probability 



d v A = 6 v-l + 6 Vi , 



whereas the true probability is 



d v 3 = d V! ( 1 — d v 2 ) + <5 v 2 ( 1 ~ <5 v-i ) + b v 1 d v 2 



= 6 v-i + 6 v-2 — S 2 v-i v-2 , 



the first formulation exhibiting in detail the probability of the three 



