Alex S. Baldwin

Modeling probability and additive summation for detection across multiple mechanisms under the assumptions of signal detection theory.

Many studies have investigated how multiple stimuli combine to reach threshold. There are broadly speaking two ways this can occur: additive summation (AS) where inputs from the different stimuli add together in a single mechanism, or probability summation (PS) where different stimuli are detected independently by separate mechanisms. PS is traditionally modeled under high threshold theory (HTT); however, tests have shown that HTT is incorrect and that signal detection theory (SDT) is the better framework for modeling summation. Modeling the equivalent of PS under SDT is, however, relatively complicated, leading many investigators to use Monte Carlo simulations for the predictions. We derive formulas that employ numerical integration to predict the proportion correct for detecting multiple stimuli assuming PS under SDT, for the situations in which stimuli are either equal or unequal in strength. Both formulas are general purpose, calculating performance for forced-choice tasks with M alternatives, n stimuli, in Q monitored mechanisms, each subject to a non-linear transducer with exponent τ. We show how the probability (and additive) summation formulas can be used to simulate psychometric functions, which when fitted with Weibull functions make signature predictions for how thresholds and psychometric function slopes vary as a function of τ, n, and Q. We also show how one can fit the formulas directly to real psychometric functions using data from a binocular summation experiment, and show how one can obtain estimates of τ and test whether binocular summation conforms more to PS or AS. The methods described here can be readily applied using software functions newly added to the Palamedes toolbox.

The attenuation surface for contrast sensitivity has the form of a witch's hat within the central visual field

Over the full visual field, contrast sensitivity is fairly well described by a linear decline in log sensitivity as a function of eccentricity (expressed in grating cycles). However, many psychophysical studies of spatial visual function concentrate on the central ±4.5 deg (or so) of the visual field. As the details of the variation in sensitivity have not been well documented in this region we did so for small patches of target contrast at several spatial frequencies (0.7-4 c/deg), meridians (horizontal, vertical, and oblique), orientations (horizontal, vertical, and oblique), and eccentricities (0-18 cycles). To reduce the potential effects of stimulus uncertainty, circular markers surrounded the targets. Our analysis shows that the decline in binocular log sensitivity within the central visual field is bilinear: The initial decline is steep, whereas the later decline is shallow and much closer to the classical results. The bilinear decline was approximately symmetrical in the horizontal meridian and declined most steeply in the superior visual field. Further analyses showed our results to be scale-invariant and that this property could not be predicted from cone densities. We used the results from the cardinal meridians to radially interpolate an attenuation surface with the shape of a witchs hat that provided good predictions for the results from the oblique meridians. The witchs hat provides a convenient starting point from which to build models of contrast sensitivity, including those designed to investigate signal summation and neuronal convergence of the image contrast signal. Finally, we provide Matlab code for constructing the witchs hat.

Rejecting probability summation for radial frequency patterns, not so Quick!

Radial frequency (RF) patterns are used to assess how the visual system processes shape. They are thought to be detected globally. This is supported by studies that have found summation for RF patterns to be greater than what is possible if the parts were being independently detected and performance only then improved with an increasing number of cycles by probability summation between them. However, the model of probability summation employed in these previous studies was based on High Threshold Theory (HTT), rather than Signal Detection Theory (SDT). We conducted rating scale experiments to investigate the receiver operating characteristics. We find these are of the curved form predicted by SDT, rather than the straight lines predicted by HTT. This means that to test probability summation we must use a model based on SDT. We conducted a set of summation experiments finding that thresholds decrease as the number of modulated cycles increases at approximately the same rate as previously found. As this could be consistent with either additive or probability summation, we performed maximum-likelihood fitting of a set of summation models (Matlab code provided in our Supplementary material) and assessed the fits using cross validation. We find we are not able to distinguish whether the responses to the parts of an RF pattern are combined by additive or probability summation, because the predictions are too similar. We present similar results for summation between separate RF patterns, suggesting that the summation process there may be the same as that within a single RF.

What Do Contrast Threshold Equivalent Noise Studies Actually Measure? Noise vs. Nonlinearity in Different Masking Paradigms.

The internal noise present in a linear system can be quantified by the equivalent noise method. By measuring the effect that applying external noise to the system’s input has on its output one can estimate the variance of this internal noise. By applying this simple “linear amplifier” model to the human visual system, one can entirely explain an observer’s detection performance by a combination of the internal noise variance and their efficiency relative to an ideal observer. Studies using this method rely on two crucial factors: firstly that the external noise in their stimuli behaves like the visual system’s internal noise in the dimension of interest, and secondly that the assumptions underlying their model are correct (e.g. linearity). Here we explore the effects of these two factors while applying the equivalent noise method to investigate the contrast sensitivity function (CSF). We compare the results at 0.5 and 6 c/deg from the equivalent noise method against those we would expect based on pedestal masking data collected from the same observers. We find that the loss of sensitivity with increasing spatial frequency results from changes in the saturation constant of the gain control nonlinearity, and that this only masquerades as a change in internal noise under the equivalent noise method. Part of the effect we find can be attributed to the optical transfer function of the eye. The remainder can be explained by either changes in effective input gain, divisive suppression, or a combination of the two. Given these effects the efficiency of our observers approaches the ideal level. We show the importance of considering these factors in equivalent noise studies.

Ocular dominance plasticity: inhibitory interactions and contrast equivalence

Brief monocular occlusion results in a transient change in ocular dominance, such that the previously patched eye makes a stronger contribution to the binocular percept after occlusion. The previously unpatched eye therefore makes a correspondingly weaker contribution to the binocular sum. To shed light on the mechanism underlying this change we investigate how the relationship between the perception of fusion, suppression, and diplopia changes after short-term monocular deprivation. Results show that fusible stimuli seen by the unpatched eye are actively suppressed as a result of patching and that this can be reversed by an interocular contrast imbalance. This suggests that dichoptic inhibition plays an important role in ocular dominance changes due to short-term occlusion, possibly by altering the contrast gain prior to binocular summation. This may help explain why this form of plasticity affects the perception of both fusible and rivalrous stimuli.

The mechanism of short-term monocular deprivation is not simple: separate effects on parallel and cross-oriented dichoptic masking

Short-term deprivation of the input to one eye increases the strength of its influence on visual perception. This effect was first demonstrated using a binocular rivalry task. Incompatible stimuli are shown to the two eyes, and their competition for perceptual dominance is then measured. Further studies used a combination task, which measures the contribution of each eye to a fused percept. Both tasks show an effect of deprivation, but there have been inconsistencies between them. This suggests that the deprivation causes multiple effects. We used dichoptic masking to explore this possibility. We measured the contrast threshold for detecting a grating stimulus presented to the target eye. Thresholds were elevated when a parallel or cross-oriented grating mask was presented to the other eye. This masking effect was reduced by depriving the target eye for 150 minutes. We tested fourteen subjects with normal vision, and found individual differences in the magnitude of this reduction. Comparing the reduction found in each subject between the two masks (parallel vs. cross-oriented), we found no correlation. This indicates that there is not a single underlying effect of short-term monocular deprivation. Instead there are separate effects which can have different dependencies, and be probed by different tasks.

The equivalent internal orientation and position noise for contour integration

Contour integration is the joining-up of local responses to parts of a contour into a continuous percept. In typical studies observers detect contours formed of discrete wavelets, presented against a background of random wavelets. This measures performance for detecting contours in the limiting external noise that background provides. Our novel task measures contour integration without requiring any background noise. This allowed us to perform noise-masking experiments using orientation and position noise. From these we measure the equivalent internal noise for contour integration. We found an orientation noise of 6° and position noise of 3 arcmin. Orientation noise was 2.6x higher in contour integration compared to an orientation discrimination control task. Comparing against a position discrimination task found position noise in contours to be 2.4x lower. This suggests contour integration involves intermediate processing that enhances the quality of element position representation at the expense of element orientation. Efficiency relative to the ideal observer was lower for the contour tasks (36% in orientation noise, 21% in position noise) compared to the controls (54% and 57%).

The efficiency of second order orientation coherence detection.

Neurons in early visual cortex respond to both luminance- (1st order) and contrast-modulated (2nd order) local features in the visual field. In later extra-striate areas neurons with larger receptive fields integrate information across the visual field. For example, local luminance-defined features can be integrated into contours and shapes. Evidence for the global integration of features defined by contrast-modulation is less well established. While good performance in some shape tasks has been demonstrated with 2nd order stimuli, the integration of contours fails with 2nd order elements. Recently we developed a global orientation coherence task that is more basic than contour integration, bearing similarity to the well-established global motion coherence task. Similar to our previous 1st order result for this task, we find 2nd order coherence detection to be scale-invariant. There was a small but significant threshold elevation for 2nd order relative to 1st order. We used a noise masking approach to compare the efficiency of orientation integration for the 1st and 2nd order. We find a significant deficit for 2nd order detection at both the local and global level, however the small size of this effect stands in stark contrast against previous results from contour-integration experiments, which are almost impossible with 2nd order stimuli.

Modelling probability summation for the detection of multiple stimuli under the assumptions of signal detection theory

In general there are two ways in which multiple stimuli can sum to threshold: by probability summation or by additive summation (of which linear summation is a special case). Probability summation (PS) is still often modelled using the long-refuted High Threshold Theory (HTT), in spite of the fact that most researchers believe that Signal Detection Theory (SDT) is the better model. Studies which do model PS under SDT often use Monte Carlo simulations to perform the calculations, but this method is prohibitively slow when many thousands of calculations are required, as when fitting psychometric functions with PS models and estimating bootstrap errors on the fitted parameters and model goodnesses-of-fit. We provide numerical integration formulae for calculating, on the assumptions of SDT, the proportion correct detections for n independently detected stimuli, each subject to a non-linear transducer τ, while Q channels are being monitored, and for an M-AFC task. We show how the equations can be used to simulate psychometric functions in order to determine how parameters such as the Weibull threshold and slope vary with n, τ and Q. We also show how the equations can be used to fit actual psychometric functions from a binocular summation experiment in order to obtain estimates of τ and to determine whether probability or additive summation is the better model of the data. Meeting abstract presented at VSS 2015.