INHIBITORY INTERACTION IN THE RETINA 275 



guesses as to the quantitative form of the spatial function of interaction (we 

 have considered only K, neglecting, for the present, the thresholds). This 

 function should be symmetric, falling off equally in opposite directions from 

 any given receptor. For simplicity, it should be isotropic in the retinal mosaic 

 although this may not be the case in actuahty (in Limulus, the inhibitory 

 influences fall off more rapidly in the dorsoventral direction from any given 

 receptor than in the anteroposterior direction). We have explored several 

 forms of functional relation: one in which the inhibition had a constant 

 non-zero value up to a given distance and was zero beyond ; one in which it 

 decayed exponentially in all directions with a given space constant. For one 

 numerical solution, we chose a function which had the form of a Gaussian 

 error curve (used by Fry (1948) in a similar treatment of "border contrast"). 

 We have considered only patterns in which the intensity varied along one 

 co-ordinate, and have dealt mostly with a simple step in intensity from a low 

 value on one side of the step to a higher one on the other. Numerical solu- 

 tions of equations (4) were obtained by an iterative method of successive 

 approximations, as is sometimes done when deahng with integral equations. 

 Indeed, a Fredholm integral equation of the second type may be considered 

 an approximation to the present set of simuhaneous equations representing 

 the interaction of discrete elements. The first step in the computation is to 

 substitute the e's in place of the r's under the summation (integration) sign; 

 this yields an approximate solution which is next substituted, and the process 

 is repeated as often as necessary. If the total inhibition on any element, 



S Kpj, is less than unity, as it must be in any actual retina, the successive 



J 



approximations converge to a solution in which maxima and minima of r 



flank the intensity step on the high and low sides, respectively. We may note 

 that Fry (1948) has made a somewhat similar calculation to explain Mach's 

 bands in human vision. His treatment, however, does not involve a mutual 

 interaction of the "recurrent" type demonstrated in the Limulus eye; the in- 

 hibition was assumed to depend only on the intensity of the stimulating light 

 rather than on the activity of the receptors. This assumption is equivalent to 

 using the e's in place of the /-"s under the summation sign in our equation (4) 

 and is, indeed, the first step in our approximation procedure. Fry's model 

 generates "Mach bands"; computation with it is much easier than with ours, 

 of course, and for many purposes it may be useful in explaining contrast 

 effects in human vision even though some evidence has been presented favor- 

 ing "recurrent" inhibition in the human visual system (Alpern and David, 

 1959). 



The form of the curves we obtained by our numerical solution differs very 

 little from that obtained by Fry; there are, however, very weak minima and 

 maxima flanking the main maxima and minima of the Mach bands — second- 

 order Mach bands, so to speak. They could not be present, of course, in Fry's 



