20/12 DESIGN FOR A BRAIN 



an ensemble of absolute systems 



da%/dt =fi(x lt . . . , x n ; ol v . .-.) (i = 1, . . . , n) 



with parameters oy, such that each combination of a-values gives 

 an absolute system. We nominate a point Q in phase-space, and 

 then define the ' probability of stability at Q ' as the proportion 

 of a-combinations (drawn as samples from known distributions) 

 that give both (1) a resting state at Q, and (2) stable equilibrium 

 at that point. The system's general ' probability of stability ' is 

 the probability at Q averaged over all Q-points. As the proba- 

 bility will usually be zero if Q is a point, we can consider instead 

 the infinitesimal probability dp given when the point is increased 

 to an infinitesimal volume dV. 



The question is fundamental to our point of view ; for, having 

 decided that stability is necessary for homeostasis, we want to 

 get a system of 10 10 nerve-cells and a complex environment 

 stable by some method that does not demand the improbable. 

 The question cannot be treated adequately without some quan- 

 titative study. Unfortunately, the quantitative study involves 

 mathematical difficulties of a high order. Non-linear systems 

 cannot be treated generally but only individually. Here I shall 

 deal only with the linear case. It is not implied that the nervous 

 system is linear in its performance or that the answers found 

 have any quantitative application to it. The position is simply 

 that, knowing nothing of what to expect, we must collect what 

 information we can so that we shall have at least some fixed 

 points around which the argument can turn. 



The applicability of the concept of linearity is considerably 

 widened by the theorem of S. 20/4. 



The problem may be stated as follows : A matrix of order 

 n x n has elements which are real and are random samples from 

 given distributions. Find the probability that all the latent 

 roots have non-positive real parts. 



This problem seems to be still unsolved even in the special 

 cases in which all the elements have the same distributions, 

 selected to be simple, as the ' normal ' type e~ x , or the ' rect- 

 angular ' type, constant between — a and -f- a. Nevertheless, 

 some answer is desirable, so the ' rectangular ' distribution (integers 

 evenly distributed between — 9 and + 9) was tested empirically. 

 Matrices were formed from Fisher and Yates' Table of Random 



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