Although the magnitude of lateral inhibition in the retina of Limulus polyphemus depends strongly on the distance between ommatidia, the time of delay r between the response of one ommatidium and the consequent inhibition of another is independent of the distance between them and is approximately 0.1 sec. Moreover, experimental observations indicate that for intact undamaged retinae in intense spatially uniform illumination, the total inhibitory influence at a given ommatidium can exceed the excitation of that ommatidium. A simple theory combining these two known facts yields the following conclusions: The response of a healthy Limulus retina to an intense, temporally constant, and spatially uniform excitation e (in the usual Hartline-Ratliff units) should be a sustained synchronous oscillation of period 2T, with the mean response over one complete cycle equal to approximately 1/2 e. Under a broad class of circumstances the sustained oscillations should take the form of a succession of "bursts" and "rest periods", each of duration r. The oscillations can have, however, a fine structure in which there is repeated information about the duration and sequence of any short pulses of excitation which occurred within a time-interval of length r before the onset of steady excitation. Because it is repeated every 2r seconds, this information remains available in the retina for interaction with subsequent changes of excitation.The usually considered functional purposes for the much studied lateral inhibitory influences in the retina of the compound eye of Limulus polyphemus (1-21) are (i) enhancement of spatial contrast to facilitate edge detection (6,8,(10)(11)(12), (ii) tuning for significant spatial and temporal frequencies (13,14), and (iii) generation of special burst responses to "light off" and "on-off" stimuli (4). We here describe, in a preliminary way, theoretical considerations which indicate that, at least for the visual system of Limulus, lateral sensory inhibition in the retina of compound eyes has another purpose.Consider a compound eye of Limulus in a time-dependent field of illumination, and number 1,2,Y. . . ,M the approximately one thousand ommatidia of the eye. The response, r,(t), (at p at time t) is the rate of discharge at time t of impulses in the axon issuing from the eccentric cell of the p'th ommatidium.Following Hartline and Ratliff (3, 5) we define the excitatWion ep(t) to be the response which the p'th ommatidium would show if it were illuminated by itself in a standard state with the flux of light actually falling on it at time t -T, where T is the latent period between a change of illumination and the consequent change in the rate of firing; the standard state referred to here is one in which the p'th ommatidium is free from the effects of lateral inhibition but not the effects of light-adaptation and self-inhibition. [One may, for example, visualize the standard state as that obtained by light-adapting the entire eye and then holding all ommatidia other than the p'th in total darkn...