24/18 



DESIGN FOR A BRAIN 



variables are therefore represented in a?'s behaviour by aj's deriva- 

 tives, and Z?'s variables are thus not independent of A's. 



24/18. In S. 14/16 we wanted to compare two probabilities, 

 each that a system would be stable, one composed of part- 

 functions and the other of full-functions, other things being equal. 

 The method of S. 20/12 will define the individual probabilities. 

 The question of what we mean by ' other things ' may be treated 

 by postulating that, regarded as two random processes, (a) the 

 one system's full-functions and (b) the active sections of the other 

 system's part-functions are to have the same statistical properties 

 (when averaged over all lines of behaviour.) This postulate is 

 stated purely in terms of the systems' observable behaviour, so 

 that it would be easy, in a given case, to test whether the postu- 

 late was satisfied. 



Now consider a system of n variables, part-functions that on 

 the average are active over a fraction p of the time. The average 

 number of variables active at one time will be pn = k, say. 

 Suppose that, at a point Q, the average number of variables are 

 active. For convenience, re-label the variables to list the active 

 first. Add parameters ol v . . . to generate the distribution. At 

 Q we have 



dxjdt = f 1 {x 1 , . . . , x n \ <*!, . . .y 



dx k /dt =f k (x v . . . , x n ; oc v . . .) 



dxk+i/dt = 



dxn/dt = 



The differential matrix at this point will be formally of order 

 n X n, but the rows from k + 1 to n will be all zero. If now we 

 test the probability of stability at this Q we find that in fact it 

 depends on the probability that the a-combination has given (a) 

 f x = . . . =fk = 0, and (b) that the matrix 







dfk 

 _dx 1 



dx k . 



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