Evidence for different nonlinear summation schemes for lines and gratings at threshold

Within a wide class of multichannel models of the visual system it is suggested that spatial distributions of luminance are processed by the independent activation of grating detectors, or spatial frequency channels. Probability summation is often described in terms of Quick's nonlinear pooling model [Quick RF (1974) Kybernetik 16:65–67]. Using this model, we find evidence for the existence of different kinds of nonlinear summation at threshold; for compound gratings with well-separated spatial frequency components, the threshold functions indicate nonlinear summation which is not compatible with probability summation, while for line patterns well separated in the spatial domain the probability summation rule proves compatible with the data. Received: 24 June 1998 / Accepted in revised form: 16 March 1999     

Lack of convexity of threshold curves for compound grating: implications for modelling visual pattern detection

Threshold contrasts were determined for complex gratings made from pairs of sine-wave gratings with various spatial frequencies and contrast ratios. It was found that the plots representing the relationship between the contrast of two-component gratings with frequencies f ₁=2.0 deg^–1 and f ₂=kf ₁, (1.03 k 3.0), when the compound was at threshold (the so-called contrast threshold curves), tend to have either an elliptical (for k 1.5) or a rectangular (for k=3) form. Nevertheless, the elliptical fits as the matched-channel model predicted, and the roundedly rectangular fits, as the probability-summation models predicted, were very poor. Furthermore, statistical analysis shows that two of the six contrast threshold curves exhibiting a significant local disturbance fail to be convex.Theoretical treatment in terms of convex analysis shows that such a convexity violation of the contrast threshold curves rules out the possibility of detection being produced by a single linear channel as well as a parallel array of linear channels, either with or without probability summation, provided that the detection principle of the most sensitive channel is adopted. The adaptive matched-channel model originally proposed by Hauske cannot also account for the results obtained; however, it can be modified to be in line with them. We hypothesise that the detection in our experiment is likely to occur by means of several (no more than seven or eight) adaptive partially matched channels.     

Area summation in human vision at and above detection threshold

The initial image-processing stages of visual cortex are well suited to a local (patchwise) analysis of the viewed scene. But the world's structures extend over space as textures and surfaces, suggesting the need for spatial integration. Most models of contrast vision fall shy of this process because (i) the weak area summation at detection threshold is attributed to probability summation (PS) and (ii) there is little or no advantage of area well above threshold. Both of these views are challenged here. First, it is shown that results at threshold are consistent with linear summation of contrast following retinal inhomogeneity, spatial filtering, nonlinear contrast transduction and multiple sources of additive Gaussian noise. We suggest that the suprathreshold loss of the area advantage in previous studies is due to a concomitant increase in suppression from the pedestal. To overcome this confound, a novel stimulus class is designed where: (i) the observer operates on a constant retinal area, (ii) the target area is controlled within this summation field, and (iii) the pedestal is fixed in size. Using this arrangement, substantial summation is found along the entire masking function, including the region of facilitation. Our analysis shows that PS and uncertainty cannot account for the results, and that suprathreshold summation of contrast extends over at least seven target cycles of grating.     

A model for direction selectivity in threshold motion perception

Thresholds were measured for a moving line superimposed on moving sinusoidal gratings. When line and grating moved in the same direction significant subthreshold summation was observed over a range of spatial frequencies. For motion of the line and grating in opposite directions, summation was never observed. This supports the hypothesis that direction selective mechanisms are responsible for motion perception at threshold. Further analysis of the data produced estimates of the spatial frequency tuning of these mechanisms. A quantitative model is proposed to interpret the data, and it is suggested that flickering gratings are not decomposed into their moving components by the visual system.     

Learning a grating discrimination task broadens human spatial frequency tuning

 The effect of spatial frequency discrimination learning on spatial frequency detection tuning curves, obtained by a summation to threshold paradigm, has been investigated. Three human observers were exposed to a grating discrimination task for longer than two weeks, and their detection thresholds for compound Gabor gratings were measured before and after this time interval. Discrimination thresholds decreased continuously and substantially during the course of learning, while the spatial frequency detection tuning curves show significant broadening in the posttest. Calculating the discrimination resolution of an ensemble of sensory coding units shows that larger bandwidths lead to better spatial frequency discrimination performance if pattern discrimination rests on multidimensional comparison or one-dimensional scaling of the spatial frequency parameter. Further, it is shown that a multiple-mechanism nonlinear pooling model is capable of explaining the results if plasticity of coding unit bandwidth or adaptive weights of the coding unit responses at the stage of response integration is assumed. The alternative sources of plasticity and the consequences of the findings for psychophysical modeling are discussed. Received: 8 September 1999 / Accepted in revised form: 16 October 2000     

Spatial properties of goldfish ganglion cells

We systematically classified goldfish ganglion cells according to their spatial summation properties using the same techniques and criteria used in cat and monkey research. Results show that goldfish ganglion cells can be classified as X-, Y-, or W-like based on their responses to contrast-reversal gratings. Like cat X cells, goldfish X-like cells display linear spatial summation. Goldfish Y-like cells, like cat Y cells, respond with frequency doubling at all spatial positions when the contrast-reversal grating consists of high spatial frequencies. There is also a third class of neurons, which is neither X- nor Y-like; many of these cells' properties are similar to those of the "not-X" cells found in the eel retina. Spatial filtering characteristics were obtained for each cell by drifting sinusoidal gratings of various spatial frequencies and contrasts across the receptive field of the cell at a constant temporal rate. The spatial tuning curves of the cell depend on the temporal parameters of the stimulus; at high drift rates, the tuning curves lose their low spatial frequency attenuation. To explore this phenomenon, temporal contrast response functions were derived from the cells' responses to a spatially uniform field whose luminance varied sinusoidally in time. These functions were obtained for the center, the surround, and the entire receptive field. The results suggest that differences in the cells' spatial filtering across stimulus drift rate are due to changes in the interaction of the center and surround mechanisms; at low temporal frequencies, the center and surround responses are out-of-phase and mutually antagonistic, but at higher temporal rates their responses are in-phase and their interaction actually enhances the cell's responsiveness.     

Mechanisms for spatial integration in visual detection: a model based on lateral interactions

Recent studies of visual detection show a configuration dependent weak improvement of thresholds with the number of targets, which corresponds to a fourth-root power law. We find this result to be inconsistent with probability summation models, and account for it by a model of 'physiological' integration that is based on excitatory lateral interactions in the visual cortex. The model explains several phenomena which are confirmed by the experimental data, such as the absence of spatial and temporal uncertainty effects, temporal summation curves, and facilitation by a pedestal in 2AFC tasks. The summation exponents are dependent on the strength of the lateral interactions, and on the distance and orientation relationship between the elements.     

The spatial frequency discrimination function at low contrasts

Spatial frequency difference thresholds for sinewave gratings near contrast threshold were measured using a two-alternative forced-choice technique, and the threshold frequency differences were plotted as a proportion of standard frequency for standards from 2 to 7 cycles/degree. This function shows reliable local maxima and minima, and these features are more pronounced than they are when stimuli of 30% contrast are used. This result is consistent with the notion that at low contrasts, fewer spatial frequency channels are above threshold in the area of the visual field covered by the stimulus than when the stimulus is at high contrast.     

Space-variant visual processing: spatially limited visual channels

A multichannel model incorporating visual inhomogeneity is presented in this paper. The parameters that describe inhomogeneity have been experimentally obtained both at threshold and in several suprathreshold conditions. At threshold, probability summation is taken into account in order to determine the spatial extent of visual channels from experimental data showing an asymptotic increase in sensitivity with increasing grating area. At suprathreshold contrast, the region where luminance variations at several scales are visible has also been found. The results support a spatially limited multichannel model of early visual processing and set out a basis for studying perceptual phenomena from the viewpoint of linear space-variant visual processing.     

Spatial summation of perceived pressure,sharpness and mechanically evoked cutaneous pain

Psychophysically, spatial summation can be demonstrated as a decrease in threshold accompanying an increased field of stimulation. The present study examined to what extent different mechanically evoked percepts (pressure, sharpness, and pain) show spatial summation. Various probes were used to apply prescribed forces to the dorsal surface of the digits of 19 healthy subjects. The threshold for three perceptual qualities showed differing degrees of spatial summation: sharpness showed no statistically significant spatial summation; pain demonstrated some significant summation (46% on average); pressure showed the greatest degree of spatial summation (76% on average). The lack of significant spatial summation for sharpness threshold is consistent with the theory that perceived sharpness can be evoked by near threshold activity of a single nociceptor. The modest amount of spatial summation for pain implies that distinctly suprathreshold activation of nociceptors is required for mechanically evoked pain perception, and such input summates centrally, but not completely. The greater spatial summation observed for pressure vs. pain thresholds implies a greater degree of central summation for slowly adapting mechanoreceptors vs. nociceptors.     

On the variety of spatial frequency selectivities shown by neurons in area 17 of the cat

The amplitudes of the responses of over 300 neurons in area 17 of the cat were examined as a function of the spatial frequency of moving sinusoidal gratings. The optimal spatial frequency and the bandwidth of the tuning curves were determined. The bandwidth varied considerably from neuron to neuron. Neurons optimally responsive to high spatial frequencies tended to have narrower tuning curves than those responsive to lower frequencies. Neurons with narrow spatial frequency tuning curves also tended to have narrow orientation tuning curves. These observations suggest that linear spatial summation tends to occur over a relatively constant area of visual field despite marked differences in each neuron's optimal spatial frequency, a prediction of one model of visual analysis. There was little difference in either the optimal spatial frequencies or the bandwidths of tuning for different functional classes of neuron. Neurons with broad tuning curves tended to be restricted to lamina IV and its environs, being concentrated in the deep part of lamina II-III and the upper part of lamina IV ab. Neurons with very low optimal spatial frequencies were uncommon and tended to be found either at the border of laminae II-III and IV or in lamina V. These laminar distributions are discussed with respect to the laminar differences in the projection of l.g.m. X- and Y-cells to the visual cortex.     

Can the data of Campbell and Robson be explained without assuming fourier analysis?

Certain experiments on the detection of low-contrast gratings, occasionally cited as evidence of Fourier analysis within the visual system, are interpreted without the assumption of Fourier analysis. Theoretical curves are obtained and compared with the published experimental points, showing mostly satisfactory agreement. The computations utilize Gaussian receptive fields (on-center and off-center) for the retinal ganglion cells, spatial summation, center-surround antagonism, quasilinear response at low contrasts (X-cells), and the assumption that the first significant convergence is primarily between cells of like response type and like receptive field geometry.     

The physics of visual perception

By measuring the contrast threshold for gratings of different waveform and spatial frequency, Campbell & Robson suggested in 1968 that there may be 'channels' tuned to different spatial frequencies. By using the technique of adapting to a high contrast grating, it was possible to measure the band-pass characteristics of these channels. Similar techniques were used to establish the orientational tuning of the channels. Reasons are put forward why it is advantageous to organize the visual system in this manner.     

The eel retina. Ganglion cell classes and spatial mechanisms

We have been able to separate optic fibers in the eye of the eel Anguilla rostrata into two distinct classes on the basis of spatial summation properties. X fibers, the first class, are like X ganglion cells in the cat: they have null positions for contrast reversal sine gratings; they respond at the modulation frequency; and many have a strong surround mechanism. X fibers, the second class, respond with an "on-off" response to local stimulation, to diffuse light modulation, to coarse drifting gratings, and to contrast reversal gratings. We have put forward a model for the receptive field of X fibers which involves two subunits, with rectification before the subunits add their signals. This model accounts for many of the quirks of X fibers.     

Spatial summation in the tactile sensory system: probability summation and neural integration

Psychophysical thresholds for the detection of a 300-Hz burst of vibration applied to the thenar eminence were measured for stimuli applied to the skin through 1.5 cm2 and through 0.05 cm2 contactors. Thresholds were approximately 13 dB lower when the area of the contactor was 1.5 cm2 than when it was 0.05 cm2. The difference between the thresholds measured with the large and small contactors was significantly reduced when only the lowest thresholds obtained in the testing sessions were considered. This result supports the hypothesis that one component of spatial summation in the P channel is probability summation. In addition, threshold measurements within a session were less variable when measured with the 1.5 cm2 contactor. We conclude that spatial summation in the P channel is a joint function of two processes that occur as the areal extent of the stimulus increases: probability summation in which the probability of exceeding the psychophysical detection threshold increases as the number of receptors of varying sensitivities increases, and neural integration in which neural activity originating from separate receptors is combined within the central nervous system rendering the channel more sensitive to the stimulus.     

Response histogram shapes and tuning curves: The predicted responses of several cortical cell types to drifting gratings stimuli

The responses to visual stimuli of simple cortical cells show linear spatial summation within and between their receptive field subunits. Complex cortical cells do not show this linearity. We analyzed the simulated responses to drifting sinusoidal grating stimuli of simple and of several types of complex cells. The complex cells, whose responses are seen to be half-wave rectified before pooling, have receptive fields consisting of two or more DOG (difference-of-Gaussians) shaped subunits. In both cases of stimulation by contrast-reversal gratings or drifting gratings, the cells' response as a function of spatial frequency is affected by the subunit distances 2 and the stimulation frequency . Furthermore, an increased number of subunits (a larger receptive field) yields a narrower peak tuning curve with decreased modulation depth for many of the spatial frequencies. The average and the peak response tuning curves are compared for the different receptive field types.     

Pattern-specific contrast threshold elevation

Visual channels are defined psychophysically; stimuli that interact share information in the same channel, and those that do not interact are processed in different channels. Channels are often investigated by means of adaptation to one stimulus, testing contrast threshold elevation with one (or more) others. Much recent work has tested the tuning of channels for orientation and spatial frequency, using simple line gratings. This study examined the pattern-specificity of such adaptation, testing the hypothesis that the fundamental operators of the Lie Transformation Group Theory of Neuropsychology (LTG/NP) define psychophysical channels. In Experiment I the three basic pattern pairs of LTG/NP were used as adaptation and test stimuli in a conventional contrast threshold-elevation experiment. Threshold elevation was pattern-specific, thus supporting the hypothesis. In subsequent experiments various 'fractured' patterns, and patterns generated by combinations of Lie operators were used both for adaptation and test. The results were mixed; some supported the original hypothesis, but many did not. Relations between local contour orientations in adaptation and test patterns could explain some results, but not all. The hypothesis that adaptation occurs to the oriented spatial-frequency components of the test patterns, on the other hand, gave a good fit to the data. It is concluded that there is pattern-specificity in contrast threshold elevation, but it is a form of specificity that can be explained without recourse to a model of geometrical pattern processing, at least for the simple patterns used here.     

Shading and texture: separate information channels with a common adaptation mechanism?

We outline a scheme for the way in which early vision may handle information about shading (luminance modulation, LM) and texture (contrast modulation, CM). Previous work on the detection of gratings has found no sub-threshold summation, and no cross-adaptation, between LM and CM patterns. This strongly implied separate channels for the detection of LM and CM structure. However, we now report experiments in which adapting to LM (or CM) gratings creates tilt aftereffects of similar magnitude on both LM and CM test gratings, and reduces the perceived strength (modulation depth) of LM and CM gratings to a similar extent. This transfer of aftereffects between LM and CM might suggest a second stage of processing at which LM and CM information is integrated. The nature of this integration, however, is unclear and several simple predictions are not fulfilled. Firstly, one might expect the integration stage to lose identity information about whether the pattern was LM or CM. We show instead that the identity of barely detectable LM and CM patterns is not lost. Secondly, when LM and CM gratings are combined in-phase or out-of-phase we find no evidence for cancellation, nor for 'phase-blindness'. These results suggest that information about LM and CM is not pooled or merged--shading is not confused with texture variation. We suggest that LM and CM signals are carried by separate channels, but they share a common adaptation mechanism that accounts for the almost complete transfer of perceptual aftereffects.     

Frequency-contrast sensitivity of visual stimulus perception in patients with schizophrenia treated with atypical and typical antipsychotics

The effects of typical and atypical antipsychotics on the perception of visual stimuli that preferentially activate magnocellular or parvocellular visual channels have been studied. The threshold contrast sensitivity in healthy individuals and patients with schizophrenia has been recorded. The Gabor-like sinusoidal brightness gratings and spatial frequencies of 0.4, 3.6, and 17.9 cycles/degree were presented to the tested objects. The patients were divided into two groups, one receiving the therapy with atypical antipsychotics, and the other, with typical ones. The contrast sensitivity for low and medium spatial frequencies (i.e., for the stimuli corresponding to magnocellular channels) decreased compared to the norm. Note that the decrease in the contrast sensitivity for the low spatial frequencies in the patients treated with atypical antipsychotics is significantly more pronounced compared to the cohort that received typical antipsychotics.     

Attenuation of vibrotactile spatial summation.

Masked and quiet thresholds at several frequencies of vibratory stimuli were measured as a function of contactor area. The test site was the left index finger; the masking site was the left little finger. The quiet threshold data were consistent with previous investigations: Low-frequency stimuli showed no spatial summation, whereas high-frequency stimuli did. In the presence of a masker, spatial summation was reduced or eliminated for high-frequency stimuli, i.e., the masked threshold was, under some conditions, independent of contactor area. Low-frequency stimuli continued to show no spatial summation in the presence of a masker. The attenuation of spatial summation appears to be a direct function of the intensity of the masking stimulus. Additional measurements with the left thenar eminence as the test site showed that spatial summation could be attenuated by a masker placed on a contralateral body site. The implications of the results for quantifying the effectiveness of a masking stimulus, for the duplex mechanoreceptor hypothesis, and for the nature of spatial summation on the skin are discussed.