16 MATHEMATICAL BIOPHYSICS OF THE CENTRAL NERVOUS SYSTEM 



an a 21 



O.X2. tX 2 2 



>0, (2) 



for then the line o-i = h x is steeper than a 2 = h 2 , so that for large S x 

 and S 2 the magnitudes can be so related that both the inequalities, 

 ax > h x and «r 2 > hn , are satisfied. 



The case when condition (2) fails is of some interest (Rashev- 

 sky, 1938, chap. xxii). Now, if neither corner lies in the other re- 

 gion, the rays do not intersect, and it is never possible, with any Si 

 and S 2 , for both N x and N 2 to be excited at the same time. In fact, 

 even with strong stimuli, the point (S x , S 2 ) may be outside both re- 

 gions and neither N x nor 2V 2 is excited. On the other hand, if either 

 corner does lie in the other region, the rays do intersect, and there 

 is a finite region of overlap as illustrated in Figure 3. Points (Si , S 2 ) 

 beyond the intersection and between the two rays represent pairs of 

 stimuli which, though strong, fail to excite either N t or N 2 . The 

 analytic condition for this is the simultaneous fulfilment of the two 

 inequalities 



an (h 12 — &n — fei/flu) — a 21 (/^i — h 22 — fh/a 22 ) > , 



(3) 



— <x 12 (h 12 — h 22 — ftj/on) + a 22 (h 2l — /k, 2 — h>/a 22 ) > , 



together with the failure of equation (2). 



The situation here described may be thought of as that of two 

 stimuli competing for attention. When conditions (3) hold and (2) 

 fails, there are moderate stimuli which lead to excitation of both 

 2V"i and 2V 2 (awareness of both stimuli) , while with more intense stim- 

 uli, unless one is sufficiently great as compared with the other, each 

 stimulus prevents the response appropriate to the other and no re- 

 sponse occurs. 



For the special case in which the mechanism is altogether sym- 

 metric (Landahl, 1938a; cf. also chap, ix) we may set 



a 11 = a 22 = a, — a 12 = — a 21 = P, 



(4) 



If the two responses are incompatible in nature the parameters might 

 be so related that the two conditions (3) cannot be satisfied. The 

 failure of these reduces to the single inequality 



h"^a(h'-h) (5) 



which holds necessarily in case we have h' ^ h . If the relation (5) 

 is replaced by an equation, we have a kind of discriminating mechan- 



