92 MATHEMATICAL BIOPHYSICS OF THE CENTRAL NERVOUS SYSTEM 



OP and the i-th axis. The assumption that a, depends upon 6, alone 

 is made for the sake of simplicity, and a natural generalization would 

 be to allow all three angles to enter, these angles being related by the 

 identity 



2cos 2 0i = l. (1) 



For further simplification we suppose that the dependence upon 6i is 

 through the cosine so that the connections from each primary recep- 

 tor are symmetric about the associated axis, and we suppose in addi- 

 tion that when the units for the Si are properly chosen the three func- 

 tions a, (S , r , cos 6) are identical. Neither of these restrictions is 

 essential. 



The immediate result at the color-center of the stimulation of the 

 three primary receptors is then the production at every point P of 



^(S ,P) =2 l * i (S i ,r,cosO l ). (2) 



If, finally, the functions ai are so chosen that for any 5 , a has aways 

 a maximum at a single point P , then the introduction of sufficient mu- 

 tual inhibition between synapses (cf. chap, iii) will make the resul- 

 tant a negative everywhere but in the neighborhood of P , and the 

 desired segregation of the pathways is secured. 



If the functions are properly chosen the analytical result will be 

 essentially a representation of the familiar color-pyramid, as, indeed, 

 we wish it to be, since this is found empirically to provide an accurate 

 representation of the phenomena. What we provide, and what is not 

 contained in the theory of the color-pyramid, is a neural mechanism 

 capable of mediating perceptions organized in this way. We have 

 supposed that the basis for a difference between perceptions lies in 

 the discreteness of loci of excitation and the mechanism here described 

 is capable of separating the loci. By the same rule an observed sim- 

 ilarity must have its basis in some community of the loci. Hence if 

 we suppose that synapses of the color-center which are collinear with 

 the origin are all connected to some further center common to that set 

 we have a possible basis for the identity of colors of varying intens- 

 ity; if the centers of this group corresponding to complanar rays 

 themselves lead to a common tertiary center we have a basis for iden- 

 tity of colors of varying intensity and saturation, and so on. 



The form of the predictions from this mechanism must coincide 

 with those of the simpler one (chap, ix) when only intensity, but not 

 color or saturation, is varied. Hence each function <n(S , r , 1) must 

 be proportional to r(S—r) if we suppose the /^-centers of the previous 

 mechanism to be uniformly spaced. Further, each a\ as a function of 

 Si should have a maximum at 0; . = . The simplest possible supposi- 



