8 MATHEMATICAL BIOPHYSICS OF THE CENTRAL NERVOUS SYSTEM 



Define the functions <f>i,v(o) by recursion formulas 



^>i,v+i(c7) = 0i+v+i [<f>i,v(<r)] . (1) 



Then 



Ci+v+i — </>j,v \Oi) , (2) 



so that each function <f> i>v gives the a produced at s i+v+] in terms of 

 that present at Si . If the derivatives exist, then 



0'i,v+i(<r) — </>'i+v<-i[<£i,v(<x)] </>'i + v[^);,v-i (cr)] • • • <j>' i (<r) , (3) 



where the primes denote the derivatives. Hence if, as we suppose, 

 each <j>i(o) is monotonic, so is each 4>'i, v (<r) ; and if each <£'i(cr) is de- 

 creasing, so is each </>'i, v (a). Further, if each ■/>'< vanishes asymptoti- 

 cally, so does each <f>' itV . Hence these "higher order" excitation-func- 

 tions are functions of the same type as the ordinary ones, and we 

 may always replace any such chain of neurons by a single one, at 

 least if we are interested in the asymptotic behavior alone. 

 If every </>'» (a) ^1 for all a , then the sequence 



Co , ffi , (J? , • • • , tr n 



is decreasing. In fact, if hi is the threshold of N t , then 



<7i+l < CTj ~~ Aij , 



and if h is the lowest threshold in the chain, 



<j% *C <r Irl , 



so that the number of neurons in the chain that can be excited is not 

 greater than the greatest integer in a. /h . 



If the <j>'i(a) > 1 for small values of a , this is not necessarily the 

 case. It has been shown (Householder, 1938a) that when all the neu- 

 rons are identical, and the chain is long, the a, will then either di- 

 minish to zero, or approach a certain positive limit characteristic of 

 the chain, according to whether <r. lies below or above a certain criti- 

 cal value. The limit and the critical value are the two roots of the 

 equation 



£'(<0=i. 



Note that it is legitimate to drop the subscript when all the <£* are 

 identical. 



If all of the ct's are within the range that permits of a linear ap- 

 proximation to the ^ , it is easy to obtain an analytic expression for 

 these ctj in terms of a and the subscript i (Landahl, 1938b) . We have, 

 in fact [chap, i, equation (5)] 



