DESIGN FOR A BRAIN 22/6 



The ultrastable system 



22/6. The definition and description already given in S. 7/26 

 have established the elementary properties of the ultrastable 

 system. A restatement in mathematical form, however, has the 

 advantage of rendering a misunderstanding less likely, and of 

 providing a base for quantitative studies. 



If a system is ultrastable, it is composed of main variables x t 

 and of step-functions a it so that the whole is state-determined: 



dxi/dt =fi(x\ a) (i = 1, . . . , n) 



dajdt = gi(x; a) {i = 1, 2, . . .) 



The functions g t must be given some form like that of S. 22/2. 

 The system is started with the representative point within the 

 critical surface <f>(x) = 0, contact with which makes the step- 

 functions change value. When they change, the new values of 

 a { are to be random samples from some distribution, assumed 

 given. 



Thus in the Homeostat, the equations of the main variables 

 are (S. 19/11): 



dXi/dt = a a x x + a i2 x 2 + a i3 x 3 + a^ (i = 1, 2, 3, 4) 

 The a's are step-functions, coming from a distribution of ' rect- 

 angular ' form, lying evenly between — 1 and -f- 1. The critical 



surfaces of the a's are specified approximately by | x | ± - = 0. 



Each individual step-function a^ depends only on whether x } 

 crosses the critical surface. 



As the a's change discontinuously, an analytic integration of 

 the differential equations is not, so far as I am aware, possible. 

 But the equations, the description, and the schedule of the 

 uniselector- wirings (the random samples) define uniquely the 

 behaviour of the x's and the a's. So the behaviour could be 

 computed to any degree of accuracy by a numerical method. 



22/7. How many trials will be necessary, on the average, for a 

 terminal field to be found ? If an ultrastable system has a 

 probability p that a new field of the main variables will be stable, 

 and if the fields' probabilities are independent, then the number 

 of fields occurring (including the terminal) will be, on the average, 

 \/p. 



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