DESIGN FOR A BRAIN 22/3 



In this form, the step-function must be brought into the 

 canonical equations. A possible form is: 



dk JK , K 



dt 



= ?(f + f tanh {q(X -*)}-*) . (4) 



where K is the initial value of the variable k, and q is large and 

 positive. As q — > oo, the behaviour of & tends to the step- 

 function form. 



Another method is to use Dirac's ^-function, defined by d(u) = 

 if w y£ o, while if u = 0, (5(w) tends to infinity in such a way that 



6(u)du = 1. 



Then if du/dt = 6{(f>{u, v, . . .)}, du/dt will be usually zero; but 

 if the changes of u, v, . . . take <f> through zero, then d(u) becomes 

 momentarily infinite and u will change by a finite jump. These 

 representations are of little practical use, but they are important 

 theoretically in showing that a step-function can occur in the 

 canonical representation of a system. 



22/3. In a state-determined system, a step-function will change 

 value if, and only if, the system arrives at certain states: the 

 critical. In Figure 22/2/2, for instance, all the points in the plane 

 k = K (the upper plane) and to the right of the line x x = X are 

 critical states for the step-function k when it has the initial 

 value K. 



The critical states may, of course, be distributed arbitrarily. 

 More commonly, however, the distribution is continuous. In this 

 case there will be a critical surface 



<j>(k, a? 2 a? n ) = 



which, given k, divides the critical from the non-critical states. 

 In Figure 22/2/2, for instance, the surface intersects the plane 

 k = K at the line x x = X. (The plane k = is not intersected 

 by it, for there are no states in this system whose occurrence will 

 result in k changing from 0.) 



Commonly </> is a function of only a few of the variables of the 

 system. Thus, whether a Post Office type relay opens or shuts 

 depends only on the two variables: the current in the coil, and 

 whether the relay is already open or shut. 



Such relays and critical states occur in the Homeostat. When 



274 



