14/12 DESIGN FOR A BRAIN 



they had many more direct effects bridging the gap. Construc- 

 tion of the diagram of ultimate effects provides a simple proof. 

 The explanation is that each system affects, and is affected by, 

 not only aj's value but also a?'s first, second, and higher derivatives 

 with respect to time. These act to provide a richness of func- 

 tional connection that is not evident at first glance. 



Part- functions 



14/12. In S. 14/8 we saw that if a whole system is to be divided 

 into independent parts some intervening variables must become 

 constant. It follows that if the independence is to be tem- 

 porary, being sometimes present and sometimes absent, the inter- 

 vening variables must be sometimes constant and sometimes 

 varying : they must, in short, be part-functions. This class of 

 variable will therefore now be considered. 



A part-function was defined in S. 7/1 as a variable which, 

 over some interval of observation, was constant over some finite 

 intervals and fluctuated over some finite intervals. It is not 

 implied that the constant values are all equal. The definition 

 refers solely to the variable's observed behaviour, making no 

 reference to any cause for such behaviour ; though there will 

 usually be some definite physical reason to account for this way 

 of behaving. A part-function will be said to be ' active ', or 

 4 inactive ', at a given moment according to whether it is, or 

 is not, varying. As the amount of time spent active tends 

 to 'all', or 'none', so does the part-function tend to full-, or 

 step-, function form. The part-function thus fills the gap be- 

 tween the two types, and may be expected to have intermediate 

 properties. 



14/13. Here are some examples. Like the step-function, it is 

 met with much more commonly in the real world than in books. 

 — The pressure on the brake-pedal during a car journey. The 

 current flowing through a telephone during the day. The posi- 

 tion co-ordinates of an animal, such as a frog or grasshopper, 

 that moves intermittently. The pressure on the sole of the 

 foot during walking. The activity of pain receptors, if they are 

 activated only intermittently. The rate of secretion of saliva in 

 an experiment on the conditioned reflex. The rate at which 



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