Mathematical description and computer simulation of retinal cometlike afterimages: a modified neural equation with stability analysis

A mathematical model for the spatiotemporal description of a well-known psychophysical phenomenon, the cometlike afterimage effect (CLAIE), is presented. The CLAIE occurs when a bright circular light spot moves slowly in the peripheral human retina. Under these conditions, the leading edge of the dot looks circular, but the trailing edge becomes elongated like a comet's tail whose length increases with speed and luminance, and the illusion is more prominent for photopic backgrounds. This cometlike motion smear is described on the basis of the temporal responsiveness and adaptation of rods. The model is an extension of an existing neural model of M. N. Oğuztöreli et al., with an additional term that allows prolonged saturation and long decay time following exposure to intense stimuli, and these effects are held responsible for the cometlike smear. The model predicts the response of photoreceptors through a nonlinear ordinary integrodifferential equation, which includes known biophysical terms for response dynamics, adaptation, saturation, and kinetics of intermediate components of the phototransduction process. The introduction of a saturation coefficient into the neural equation makes it possible to distinguish the different saturation thresholds of the rod-and-cone system. Numerical determination of the stationary solutions and complete linear stability analysis of the improved neural equation are given for a neuron of second order, and some computational results are presented for phase flows around different singular points in the phase field. A computer simulation based on the improved neural equation is presented for modeling the development and features of the CLAIE as a function of the speed and luminance of the stimulus and the background intensity. The computational results agree well with the psychophysical findings relating to the CLAIE.