Four Theorems on the Psychometric Function

In a 2-alternative forced-choice (2AFC) discrimination task, observers choose which of two stimuli has the higher value. The psychometric function for this task gives the probability of a correct response for a given stimulus difference, . This paper proves four theorems about the psychometric function. Assuming the observer applies a transducer and adds noise, Theorem 1 derives a convenient general expression for the psychometric function. Discrimination data are often fitted with a Weibull function. Theorem 2 proves that the Weibull “slope” parameter, , can be approximated by , where is the of the Weibull function that fits best to the cumulative noise distribution, and depends on the transducer. We derive general expressions for and , from which we derive expressions for specific cases. One case that follows naturally from our general analysis is Pelli''s finding that, when , . We also consider two limiting cases. Theorem 3 proves that, as sensitivity improves, 2AFC performance will usually approach that for a linear transducer, whatever the actual transducer; we show that this does not apply at signal levels where the transducer gradient is zero, which explains why it does not apply to contrast detection. Theorem 4 proves that, when the exponent of a power-function transducer approaches zero, 2AFC performance approaches that of a logarithmic transducer. We show that the power-function exponents of 0.4–0.5 fitted to suprathreshold contrast discrimination data are close enough to zero for the fitted psychometric function to be practically indistinguishable from that of a log transducer. Finally, Weibull reflects the shape of the noise distribution, and we used our results to assess the recent claim that internal noise has higher kurtosis than a Gaussian. Our analysis of for contrast discrimination suggests that, if internal noise is stimulus-independent, it has lower kurtosis than a Gaussian.     

Measurement and modeling of peripheral detection and discrimination thresholds

This report describes experimental measurements of threshold contrasts as a function of the angle to the visual axis (peripheral threshold contrasts). The visual tasks consist in detection (perception of presence) and discrimination (perception of a form feature) of simple visual signs during a fixation period realistic observing conditions being chosen. Proceeding from the experimental findings a model for forecasting off-axis threshold contrast functions on different visual conditions is developed based upon spatial frequency filters. Further with the aid of a known model visibility fields are calculated.     

Binocular interaction: contrast matching and contrast discrimination are predicted by the same model

How do signals from the 2 eyes combine and interact? Our recent work has challenged earlier schemes in which monocular contrast signals are subject to square-law transduction followed by summation across eyes and binocular gain control. Much more successful was a new 'two-stage' model in which the initial transducer was almost linear and contrast gain control occurred both pre- and post-binocular summation. Here we extend that work by: (i) exploring the two-dimensional stimulus space (defined by left- and right-eye contrasts) more thoroughly, and (ii) performing contrast discrimination and contrast matching tasks for the same stimuli. Twenty-five base-stimuli made from 1 c/deg patches of horizontal grating, were defined by the factorial combination of 5 contrasts for the left eye (0.3-32%) with five contrasts for the right eye (0.3-32%). Other than in contrast, the gratings in the two eyes were identical. In a 2IFC discrimination task, the base-stimuli were masks (pedestals), where the contrast increment was presented to one eye only. In a matching task, the base-stimuli were standards to which observers matched the contrast of either a monocular or binocular test grating. In the model, discrimination depends on the local gradient of the observer's internal contrast-response function, while matching equates the magnitude (rather than gradient) of response to the test and standard. With all model parameters fixed by previous work, the two-stage model successfully predicted both the discrimination and the matching data and was much more successful than linear or quadratic binocular summation models. These results show that performance measures and perception (contrast discrimination and contrast matching) can be understood in the same theoretical framework for binocular contrast vision.     

Joint Entropy for Space and Spatial Frequency Domains Estimated from Psychometric Functions of Achromatic Discrimination

We used psychometric functions to estimate the joint entropy for space discrimination and spatial frequency discrimination. Space discrimination was taken as discrimination of spatial extent. Seven subjects were tested. Gábor functions comprising unidimensionalsinusoidal gratings (0.4, 2, and 10 cpd) and bidimensionalGaussian envelopes (1°) were used as reference stimuli. The experiment comprised the comparison between reference and test stimulithat differed in grating''s spatial frequency or envelope''s standard deviation. We tested 21 different envelope''s standard deviations around the reference standard deviation to study spatial extent discrimination and 19 different grating''s spatial frequencies around the reference spatial frequency to study spatial frequency discrimination. Two series of psychometric functions were obtained for 2%, 5%, 10%, and 100% stimulus contrast. The psychometric function data points for spatial extent discrimination or spatial frequency discrimination were fitted with Gaussian functions using the least square method, and the spatial extent and spatial frequency entropies were estimated from the standard deviation of these Gaussian functions. Then, joint entropy was obtained by multiplying the square root of space extent entropy times the spatial frequency entropy. We compared our results to the theoretical minimum for unidimensional Gábor functions, 1/4π or 0.0796. At low and intermediate spatial frequencies and high contrasts, joint entropy reached levels below the theoretical minimum, suggesting non-linear interactions between two or more visual mechanisms. We concluded that non-linear interactions of visual pathways, such as the M and P pathways, could explain joint entropy values below the theoretical minimum at low and intermediate spatial frequencies and high contrasts. These non-linear interactions might be at work at intermediate and high contrasts at all spatial frequencies once there was a substantial decrease in joint entropy for these stimulus conditions when contrast was raised.     

Energy model for contrast detection: spatiotemporal characteristics of threshold vision

A model for contrast detection of spatiotemporal stimuli is proposed which consists of a spatiotemporal linear filter, an energy device and a threshold device. Assuming the existence of independent intrinsic noise, the probability of stimulus detection was approximated by a Weibull function of the response energy. With this assumption, the stimulus energy is a constant at fixed detection probability. This energy model for contrast detection satisfactorily accounted for the elliptical threshold contours of line pairs at stimulus separations within the range 2–30 min and at stimulus onset asynchronies within the range 20–140 ms. The threshold contour at a large stimulus onset asynchrony (300 ms) was in the form of a rounded square. This finding was explained by assuming that the probability of seeing the line pair was determined by the joint probability that at least one stimulus had been detected. With the energy model, the temporal and spatial autocorrelation functions of the response to a flashed line were evaluated. The autocorrelation functions thus determined were used to predict the temporal contrast sensitivity function to a flickering line stimulus and the spatial contrast sensitivity function to flashed gratings, which were in agreement with the experimental data. The data obtained were fitted adequately by an impulse response approximated by a spatiotemporal Gabor-like function. Received: 08 December 1997 / Accepted in revised form: 26 January 1999     

Intrinsic uncertainty explains second responses

In the simplest form of signal-detection theory (SDT), all stimuli give rise to equal-variance Gaussian probability density functions (PDFs) of sensation, with means proportional to stimulus intensity. As this simple SDT cannot accurately describe psychometric functions for two-alternative forced-choice (2AFC) detection experiments, it is commonly modified in one of two ways: with a non-linear transducer or intrinsic uncertainty. Most results can adequately be explained by either modification, but Swets et al.'s (1961) two-response 4AFC (2R4AFC) detection experiment is an exception. Simple SDT cannot predict the relationship between first- and second-response accuracies and non-linear transduction does not help. A previously unacknowledged facet of intrinsic uncertainty is that the same uncertainty required to fit 2AFC psychometric functions also produces an excellent fit to Swets et al.'s 2R4AFC results, without requiring any additional assumptions. This result is derived within the context of a primer on SDT.     

Lateral sensitivity modulation explains the flanker effect in contrast discrimination

We used a dual-masking paradigm to study how contrast discrimination can be influenced by the presence of adjacent stimuli. The task of the observer was to detect a target superimposed on a pedestal in the presence of flankers. The flankers (i) reduce the target threshold at zero pedestal contrast, (ii) shift the target threshold versus pedestal contrast (TvC) function horizontally to the left on a log-log plot at high pedestal contrasts, and (iii) reduce the size of pedestal facilitation at low pedestal contrasts. The horizontal shift at high pedestal contrasts suggests that the flanker effect is a multiplicative factor that cannot be explained by previous models of contrast discrimination. We extend the divisive inhibition model of contrast discrimination by implementing the flanker effect as a lateral multiplicative sensitivity modulation. This extended model provides a good account of the data.     

Psychometric function for nectar volume perception of a flower-visiting bat

For foraging pollinators one aspect of floral quality is the volume of nectar available. Thus, nectar-feeding animals should be able to estimate volumes of received nectar. In this study, we determined the psychometric function for nectar volume discrimination of a Neotropical flower-visiting bat Glossophaga soricina. For this, we examined the ability of bats to discriminate between two nectar volumes in a two-alternative forced-choice paradigm. We used a Bayesian inference approach to determine psychometric functions. From the derived psychometric function we assessed the discrimination threshold at a value of 0.69. G. soricina could clearly distinguish between two volumes, when the difference between the two nectar volumes divided by their average exceeded this value. This indicates that bats possess a sense for the perception and discrimination of volumes of nectar that is better developed than in honeybees.     

Spatial, colour and contrast response characteristics of mechanisms which mediate discrimination of pattern orientation and magnification

We have measured response times for the detection of a single target presented against a set of reference elements which are characterised by combinations of four different stimulus parameters; colour, contrast polarity, magnification and orientation. The aim of the experiments was to determine the response characteristics of visual mechanisms which mediate target detection through the discrimination of orientation and magnification. In the first experiments, we determined sensitivity to differences in colour and contrast polarity, and show that the mechanisms responsible for the discrimination of orientation and of magnification are both selective in their responses to colour and to contrast polarity. There are, nonetheless, residual interactions between patterns of different contrast polarities and between those of different colour, and in the latter case, weak interactions persist under equiluminance conditions. In a second set of experiments, we examined the interactions between orientation and magnification. We conclude that the responses of visual mechanisms which mediate target detection through discrimination of orientation are markedly dependent on stimulus magnification whereas those which mediate detection through discrimination of magnification are, in contrast, relatively insensitive to stimulus orientation.     

What Is the Primary Cause of Individual Differences in Contrast Sensitivity?

One of the primary objectives of early visual processing is the detection of luminance variations, often termed image contrast. Normal observers can differ in this ability by at least a factor of 4, yet this variation is typically overlooked, and has never been convincingly explained. This study uses two techniques to investigate the main source of individual variations in contrast sensitivity. First, a noise masking experiment assessed whether differences were due to the observer’s internal noise, or the efficiency with which they extracted information from the stimulus. Second, contrast discrimination functions from 18 previous studies were compared (pairwise, within studies) using a computational model to determine whether differences were due to internal noise or the low level gain properties of contrast transduction. Taken together, the evidence points to differences in contrast gain as being responsible for the majority of individual variation across the normal population. This result is compared with related findings in attention and amblyopia.     

Synergistic mixture interactions in detection of perithreshold odors by humans

Laboratory demonstrations of synergistic mixture interactions in human odor perception have been rare. The current study examined perithreshold mixture interactions between maple lactone (ML) and selected carboxylic acids. An air-dilution olfactometer allowed precise stimulus control. Experimenters measured stimulus concentrations in vapor phase using a combination of solid-phase microextraction and gas chromatography/mass spectrometry. A probability of detection versus concentration, or a psychometric, functions was measured for pure ML. Psychometric functions were also measured for ML with the addition of fixed, subthreshold concentrations of carboxylic acids. Relative to statistical independence in detection, clear synergy occurred over a range of ML concentrations. To the best of our knowledge, the current results constitute the first clear demonstration of synergy in odor detection by humans from an experiment that combined precise stimulus control, vapor-phase calibration of stimuli, and a clear statistical definition of synergy.     

Assessment of novel binocular colour, motion and contrast tests in glaucoma

The effects of glaucoma on binocular visual sensitivity for the detection of various stimulus attributes are investigated at the fovea and in four paracentral retinal regions. The study employed a number of visual stimuli designed to isolate the processing of various stimulus attributes. We measured absolute contrast detection thresholds and functional contrast sensitivity by using Landolt ring stimuli. This psychophysical Landolt C-based contrast test of detection and gap discrimination allowed us to test parafoveally at 6 ° from fixation and foveally by employing interleaved testing locations. First-order motion perception was examined by using moving stimuli embedded in static luminance contrast noise. Red/green (RG) and yellow/blue (YB) colour thresholds were measured with the Colour Assessment and Diagnosis (CAD) test, which utilises random dynamic luminance contrast noise (± 45 %) to ensure that only colour and not luminance signals are available for target detection. Subjects were normal controls (n?=?65) and glaucoma patients with binocular visual field defects (n?=?15) classified based on their Humphrey Field Analyzer mean deviation (MD) scores. The impairment of visual function varied depending on the stimulus attribute and location tested. Progression of loss was noted for all tests as the degree of glaucoma increased. For subjects with mild glaucoma (MD ?0.01 dB to ?6.00 dB) significantly more data points fell outside the normal age-representative range for RG colour thresholds than for any other visual test, followed by motion thresholds. This was particularly the case for the parafoveal data compared with the foveal data. Thus, a multifaceted measure of binocular visual performance, incorporating RG colour and motion test at multiple locations, might provide a better index for comparison with quality of life measures in glaucoma.     

A vector-magnitude model of contrast detection

A model of contrast detection is proposed in which the visibility of a low-contrast stimulus is determined by a non-Euclidean magnitude of the vector composed of the responses of a large number of independent channels. Although the vector-magnitude model is quite different from the probability-summation model which has been suggested previously, the contrast thresholds and psychometric functions predicted by the two models can be in agreement within 10% for up to 10⁵ channels in the system. Presently available experimental evidence is insufficient to establish the correctness of either model, but the computational simplicity of the vector-magnitude model makes it interesting, if only as a useful approximation to the probability-summation model.     

Contrast discrimination and choice reaction times at near-threshold pedestals

Choice reaction times (CRTs) to contrast differences were measured and compared with contrast increment thresholds obtained from concurrently measured psychometric functions at pedestal contrasts in the vicinity of detection threshold. Contrast discrimination functions had a classical dipper shape. The main finding was that CRTs were shorter at low pedestal contrasts but longer at higher pedestal contrasts compared to detection, reflecting the behaviour of increment thresholds. Even when equalized for response accuracy, CRTs varied with pedestal contrast in a similar manner to the contrast increment thresholds. The finding that CRTs and contrast increment thresholds depended on pedestal contrast in a similar manner suggests that both share a common origin. This common origin is proposed to lie in the variability of the sensory effect which determines the variability of the information accumulation process, which in turn affects the response criterion and contrast increment thresholds. At low pedestals, a decrease in variability lowers thresholds and results in a lower response criterion, thereby accelerating reaction times. At high pedestals, increasing signal-dependent noise inflates variability and thus raises thresholds and the response criterion, which results in slower CRTs.     

Distributed representation of perceptual categories in the auditory cortex

Categorical perception is a process by which a continuous stimulus space is partitioned to represent discrete sensory events. Early experience has been shown to shape categorical perception and enlarge cortical representations of experienced stimuli in the sensory cortex. The present study examines the hypothesis that enlargement in cortical stimulus representations is a mechanism of categorical perception. Perceptual discrimination and identification behaviors were analyzed in model auditory cortices that incorporated sound exposure-induced plasticity effects. The model auditory cortex with over-representations of specific stimuli exhibited categorical perception behaviors for those specific stimuli. These results indicate that enlarged stimulus representations in the sensory cortex may be a mechanism for categorical perceptual learning.     

Attentional Effects on Phenomenological Appearance: How They Change with Task Instructions and Measurement Methods

It has been reported that exogenous cues accentuate contrast appearance. The empirical finding is controversial because non-veridical perception challenges the idea that attention prioritizes processing resources to make perception better, and because philosophers have used the finding to challenge representational accounts of mental experience. The present experiments confirm that when evaluated with comparison paradigms exogenous cues increase the apparent contrast. In addition, contrast appearance was also changed by simply changing the purpose of a secondary task. When comparison and discrimination reports were combined in a single experiment there was a behavioral disassociation: contrast enhanced for comparison responses, but did not change for discrimination judgments, even when participants made both types of judgment for a single stimulus. That a single object can have multiple simultaneous appearances leads inescapably to the conclusion that our unitary mental experience is illusory.     

Visibility reflects dynamic changes of effective connectivity between V1 and fusiform cortex

Identifying the neural basis of visibility is central to understanding conscious visual perception. Visibility of basic features such as brightness is often thought to reflect activity in just early visual cortex. But here we show under metacontrast masking that fMRI activity in stimulus-driven areas of early visual cortex did not reflect parametric changes in the visibility of a brightness stimulus. The psychometric visibility function was instead correlated with activity in later visual regions plus parieto-frontal areas, and surprisingly, in representations of the unstimulated stimulus surround for primary visual cortex. Critically, decreased stimulus visibility was associated with a regionally-specific decoupling between early visual cortex and higher visual areas. This provides evidence that dynamic changes in effective connectivity can closely reflect visual perception.     

Time course of perceptual discrimination and single neuron reliability

The reliability of identification of a visual target increases with time available for inspection of the stimulus. We suggest that the neural basis of this improvement is the existence of a mechanism for integrating a noisy firing rate over some period, leading to a reduction in mean firing rate variance with available processing time. We have determined the experimental time course of the improvement in reliability in a parallel search task where the available inspection time is limited by the presentation of a mask at various times after a brief stimulus. We compare the resulting psychometric functions with the predictions of a model based on Signal Detection Theory. The model is based on the assumption that the reliability of the observer's response is limited by the variability of the responses of individual neurons. The reliability of the discrimination between two stimuli at the neuronal level is then directly related to the ratio of the difference between their integrated mean responses (over many trials) to the response standard deviation. This reliability increases with inspection time. To demonstrate application of the model to electrophysiological data, neurometric functions are derived from the firing rates of a monkeyV1 cortical neuron. The data were obtained while the animal was active in a discrimination task. The results correspond qualitatively to our observed human psychometric functions.     

Perception of successive brief objects as a function of stimulus onset asynchrony: model experiments based on two-stage synchronization of neuronal oscillators

Recently we introduced a new version of the perceptual retouch model incorporating two interactive binding operations—binding features for objects and binding the bound feature-objects with a large scale oscillatory system that acts as a mediary for the perceptual information to reach consciousness-level representation. The relative level of synchronized firing of the neurons representing the features of an object obtained after the second-stage synchronizing modulation is used as the equivalent of conscious perception of the corresponding object. Here, this model is used for simulating interaction of two successive featured objects as a function of stimulus onset asynchrony (SOA). Model output reproduces typical results of mutual masking—with shortest and longest SOAs first and second object correct perception rate is comparable while with intermediate SOAs second object dominates over the first one. Additionally, with shortest SOAs misbinding of features to form illusory objects is simulated by the model.