On oscillatory responses of the Limulus retina

Abstract

Published observations of the dynamic properties of lateral and self-inhibition in the Limulus retina lead to a non-linear integral equation for the response of ommatidia located near the center of a uniformly illuminated region. Coleman and Renninger (1976, 1978) showed that when the excitation is constant in time and the sum of the inhibitory coefficients for the illuminated region exceeds a critical value, the integral equation has a stable periodic solution describing a sustained, spatially synchronized, oscillatory response in which bursts of activity alternate with silent periods. Such spatially synchronized "bursting" has been observed in the Limulus retina in situ by Barlow and Fraioli (1978), using the preparation of Barlow and Kaplan (1971). Employing experimental data on the temporal dependence of lateral and self-inhibition, which were then available only for the excised eye, Coleman and Renninger calculated a value of 0.34 s for the period p of the bursting response, which is significantly above the range, 0.11---0.20 s, of values of p observed for the Limulus eye in situ. Brodie et al. (1978) have recently published measurements of the temporal dependence of lateral and self-inhibition for the in situ preparation. Here we show that when the kernel functions in Coleman and Renninger's integral equation are chosen in accord with these new data, the periodic solutions of the equation have a period of approximately 0.13s, which is in the range (0.11---0.20 s) required for agreement with experiment. Other properties of the periodic solutions, i.e., their general form and the threshold levels of inhibition required for their existence, are also in accord with published observations of the behavior of the retina in situ.