Figure 1.
The Neighboring Letters “F” and “P” on the 20/20 Line of the Snellen Eye Chart, Blurred by a Gaussian of Diameter 0.5 arcmin and Projected onto an Image of the Foveal Cone Mosaic (Photoreceptor Image Modified from [92])
The 1-arcmin features that distinguish the letters extend over only a few photoreceptors. Also shown is a sample fixational eye movement trajectory for a standing subject (courtesy of [25]), sampled every 2 ms for a duration of 500 ms and then smoothed with a 4-ms boxcar filter.
Figure 2.
(A) Tiny horizontal and vertical stimuli, sized to subtend 0.5 × 1, 0.75 × 1.5, and 1 × 2 arcmin2 when viewed at a distance of 88 cm.
(B) Performance of nine human participants on this task, measured by the fraction of correct guesses out of 32 trials. Error bars represent the 68% confidence interval.
Figure 3.
Models of Spike Generation and Decoding
(A) A block diagram of the features in the model visual system; see text for details.
(B) Firing-rate profiles rS(y) induced by horizontal and vertical stimuli on the model foveal lattice. Left: 0.5 × 1 arcmin2. Right: 1 × 2 arcmin2.
(C) A graphical representation of the discrete second-derivative operator used to calculate diffusion rates.
(D) The temporal filters that model retinal ganglion cells use to convert the time-varying light intensity into the instantaneous firing rate.
Figure 4.
Simulations of the Markov Decoder (Equation 1) for a Small Stimulus Moving on a One-Dimensional Model Retina
(A–E) Spike generation by a Markov process.
(F–J) Spike generation by a non-Markov process that includes the biphasic temporal filter from Figure 3D.
(A and F) Firing rate induced by a stimulus moving on the retina with a random walk diffusion constant of 100 arcmin2/s. The stimulus shape activates three neurons in the pattern shown in the inset. The background rate is 10 Hz, and the peak stimulated rate is 100 Hz.
(B and G) Poisson retinal spike trains drawn from this instantaneous firing rate. Each row corresponds to a neuron, spaced every 0.5 arcmin.
(C and H) Evolution of the location probability for a known stimulus shape S (inset in [A]), but an unknown location x, derived from the spike trains shown in the previous panel.
(D and I) Decoder behavior when the stimulus can instead take one of two possible shapes, but the true shape is unknown. The two stimuli each activate three retinal neurons, in mirror-image patterns (inset). The spike trains now induce two spatial distributions of the posterior probability , plotted in shades of red and blue.
(E and J) Shape probability , colored red for the correct stimulus identity and blue for the incorrect one. In these trials, we see that once the decoder coalesces around the stimulus location, it first attributes a greater probability to the wrong stimulus (leftmost arrow in [D] and [I]) before accumulating enough evidence for the correct stimulus (middle arrow). The decoder can lose track of the stimulus briefly (e.g., at rightmost arrow) but continues to favor the correct stimulus until the end of the trial. Note that (E) reflects the true posterior probabilities, whereas in (J), the Markov decoder can only estimate them because the spike generation process includes temporal filtering that the decoder neglects.
Figure 5.
Model Performance on the Horizontal versus Vertical Discrimination Task Shown in Figure 2
Performance is measured by simulating retinal responses, calculating decisions based on those responses, and computing the fraction of correct decisions (see Materials and Methods). When fixational eye movements jitter the stimulus, the Markov decoder is able to perform well on the task by accounting for the eye movement statistics (black curves). Two naive decoders are also applied to this task, one that assumes the stimulus is fixed (red) and one that assumes maximum uncertainty about those movements (blue). Performance is shown as a function of stimulus duration (A), peak stimulated firing rate (B), and stimulus size (C). Where not otherwise specified, the parameters for these simulations are background firing rate of 10 Hz, a peak stimulated rate of 100 Hz, a stimulus of 1 × 2 arcmin2, a duration of 500 ms, and a diffusion constant of 100 arcmin2/s.
Figure 6.
Markov Decoder Robustness to Mismatched Parameters
(A) Discrimination performance when the decoder's estimate for the trajectory statistics is wrong: The stimulus is known to perform a random walk on the retina, but the diffusion constant is misestimated. The performance is optimal for estimated values close to the actual diffusion constant and declines gently on either side.
(B) Performance as a function of the expected stimulated firing rate, parameterized as .
(C) Performance as a function of the expected stimulus size, obtained by convolving the true stimulus shape with a spatial Gaussian of the specified radius. In each of these plots, parameters are the same as in Figure 5.
Figure 7.
Schematic for a Network Implementation of the Markov Decoder (Equation 1)
Spikes from retinal neurons (green, top layer) are collected by neurons in a hidden layer (black, middle layer) with linear receptive fields fS(y − x) and a local gain that is set by activity in the recipient neuron. Global divisive inhibition is driven by the total activity of all neurons in the hidden layer through a pooling neuron (blue, bottom neuron).
Figure 8.
Two Independent but Competing Subnetworks, Each Structured as in Figure 7, Receive Input from the Same Retinal Ganglion Cells, but Use Different Receptive Fields
The total activity in each subnetwork is pooled by two readout neurons. The more active readout neuron indicates the network's estimate of the stimulus orientation.
Figure 9.
Markov Decoder Discrimination Performance as a Function of Eye Movement Diffusion Constant
The decoder's assumed diffusion constant is either held fixed (blue) or covaried with that of the eye (black). The measured diffusion constant for eye movements is marked in red. These simulations used a biphasic filter with perfectly matched positive and negative lobes, which is the filter that most favors large eye movements. The stimulus measured 0.5 × 1 arcmin2; otherwise, parameters were as in Figure 5.