Spiral mechanisms are required to account for summation of complex motion components

Stimuli from one family of complex motions are defined by their spiral pitch, where cardinal axes represent signed expansion and rotation. Intermediate spirals are represented by intermediate pitches. It is well established that vision contains mechanisms that sum over space and direction to detect these stimuli (Morrone et al., Nature 376 (1995) 507) and one possibility is that four cardinal mechanisms encode the entire family. We extended earlier work (Meese & Harris, Vision Research 41 (2001) 1901) using subthreshold summation of random dot kinematograms and a two-interval forced choice technique to investigate this possibility. In our main experiments, the spiral pitch of one component was fixed and that of another was varied in steps of 15° relative to the first. Regardless of whether the fixed component was aligned with cardinal axes or an intermediate spiral, summation to-coherence-threshold between the two components declined as a function of their difference in spiral pitch. Similar experiments showed that none of the following were critical design features or stimulus parameters for our results: superposition of signal dots, limited life-time dots, the presence of speed gradients, stimulus size or the number of dots. A simplex algorithm was used to fit models containing mechanisms spaced at a pitch of either 90° (cardinal model) or 45° (cardinal+model) and combined using a fourth-root summation rule. For both models, direction half-bandwidth was equated for all mechanisms and was the only free parameter. Only the cardinal+model could account for the full set of results. We conclude that the detection of complex motion in human vision requires both cardinal and spiral mechanisms with a half-bandwidth of approximately 46°. © 2002 Elsevier Science Ltd. All rights reserved.

Adaptation and gain pool summation:Alternative models and masking data

Foley [J. Opt. Soc. Am. A 11 (1994) 1710] has proposed an influential psychophysical model of masking in which mask components in a contrast gain pool are raised to an exponent before summation and divisive inhibition. We tested this summation rule in experiments in which contrast detection thresholds were measured for a vertical 1 c/deg (or 2 c/deg) sine-wave component in the presence of a 3 c/deg (or 6 c/deg) mask that had either a single component oriented at -45° or a pair of components oriented at ±45°. Contrary to the predictions of Foley's model 3, we found that for masks of moderate contrast and above, threshold elevation was predicted by linear summation of the mask components in the inhibitory stage of the contrast gain pool. We built this feature into two new models, referred to as the early adaptation model and the hybrid model. In the early adaptation model, contrast adaptation controls a threshold-like nonlinearity on the output of otherwise linear pathways that provide the excitatory and inhibitory inputs to a gain control stage. The hybrid model involves nonlinear and nonadaptable routes to excitatory and inhibitory stages as well as an adaptable linear route. With only six free parameters, both models provide excellent fits to the masking and adaptation data of Foley and Chen [Vision Res. 37 (1997) 2779] but unlike Foley and Chen's model, are able to do so with only one adaptation parameter. However, only the hybrid model is able to capture the features of Foley's (1994) pedestal plus orthogonal fixed mask data. We conclude that (1) linear summation of inhibitory components is a feature of contrast masking, and (2) that the main aftereffect of spatial adaptation on contrast increment thresholds can be assigned to a single site. © 2002 Elsevier Science Ltd. All rights reserved.

Contrast integration over area is extensive: a three-stage model of spatial summation

Classical studies of area summation measure contrast detection thresholds as a function of grating diameter. Unfortunately, (i) this approach is compromised by retinal inhomogeneity and (ii) it potentially confounds summation of signal with summation of internal noise. The Swiss cheese stimulus of T. S. Meese and R. J. Summers (2007) and the closely related Battenberg stimulus of T. S. Meese (2010) were designed to avoid these problems by keeping target diameter constant and modulating interdigitated checks of first-order carrier contrast within the stimulus region. This approach has revealed a contrast integration process with greater potency than the classical model of spatial probability summation. Here, we used Swiss cheese stimuli to investigate the spatial limits of contrast integration over a range of carrier frequencies (1–16 c/deg) and raised plaid modulator frequencies (0.25–32 cycles/check). Subthreshold summation for interdigitated carrier pairs remained strong (~4 to 6 dB) up to 4 to 8 cycles/check. Our computational analysis of these results implied linear signal combination (following square-law transduction) over either (i) 12 carrier cycles or more or (ii) 1.27 deg or more. Our model has three stages of summation: short-range summation within linear receptive fields, medium-range integration to compute contrast energy for multiple patches of the image, and long-range pooling of the contrast integrators by probability summation. Our analysis legitimizes the inclusion of widespread integration of signal (and noise) within hierarchical image processing models. It also confirms the individual differences in the spatial extent of integration that emerge from our approach.

Contrast summation across eyes and space is revealed along the entire dipper function by a "Swiss cheese" stimulus.

Previous contrast discrimination experiments have shown that luminance contrast is summed across ocular (T. S. Meese, M. A. Georgeson, & D. H. Baker, 2006) and spatial (T. S. Meese & R. J. Summers, 2007) dimensions at threshold and above. However, is this process sufficiently general to operate across the conjunction of eyes and space? Here we used a "Swiss cheese" stimulus where the blurred "holes" in sine-wave carriers were of equal area to the blurred target ("cheese") regions. The locations of the target regions in the monocular image pairs were interdigitated across eyes such that their binocular sum was a uniform grating. When pedestal contrasts were above threshold, the monocular neural images contained strong evidence that the high-contrast regions in the two eyes did not overlap. Nevertheless, sensitivity to dual contrast increments (i.e., to contrast increments in different locations in the two eyes) was a factor of ∼1.7 greater than to single increments (i.e., increments in a single eye), comparable with conventional binocular summation. This provides evidence for a contiguous area summation process that operates at all contrasts and is influenced little, if at all, by eye of origin. A three-stage model of contrast gain control fitted the results and possessed the properties of ocularity invariance and area invariance owing to its cascade of normalization stages. The implications for a population code for pattern size are discussed.

Spatially extensive summation of contrast-energy is revealed by contrast detection of micro-pattern textures

Vision must analyze the retinal image over both small and large areas to represent fine-scale spatial details and extensive textures. The long-range neuronal convergence that this implies might lead us to expect that contrast sensitivity should improve markedly with the contrast area of the image. But this is at odds with the orthodox view that contrast sensitivity is determined merely by probability summation over local independent detectors. To address this puzzle, I aimed to assess the summation of luminance contrast without the confounding influence of area-dependent internal noise. I measured contrast detection thresholds for novel Battenberg stimuli that had identical overall dimensions (to clamp the aggregation of noise) but were constructed from either dense or sparse arrays of micro-patterns. The results unveiled a three-stage visual hierarchy of contrast summation involving (i) spatial filtering, (ii) long-range summation of coherent textures, and (iii) pooling across orthogonal textures. Linear summation over local energy detectors was spatially extensive (as much as 16 cycles) at Stage 2, but the resulting model is also consistent with earlier classical results of contrast summation (J. G. Robson & N. Graham, 1981), where co-aggregation of internal noise has obscured these long-range interactions.

Theory and data for area summation of contrast with and without uncertainty:evidence for a noisy energy model

Contrast sensitivity improves with the area of a sine-wave grating, but why? Here we assess this phenomenon against contemporary models involving spatial summation, probability summation, uncertainty, and stochastic noise. Using a two-interval forced-choice procedure we measured contrast sensitivity for circular patches of sine-wave gratings with various diameters that were blocked or interleaved across trials to produce low and high extrinsic uncertainty, respectively. Summation curves were steep initially, becoming shallower thereafter. For the smaller stimuli, sensitivity was slightly worse for the interleaved design than for the blocked design. Neither area nor blocking affected the slope of the psychometric function. We derived model predictions for noisy mechanisms and extrinsic uncertainty that was either low or high. The contrast transducer was either linear (c1.0) or nonlinear (c2.0), and pooling was either linear or a MAX operation. There was either no intrinsic uncertainty, or it was fixed or proportional to stimulus size. Of these 10 canonical models, only the nonlinear transducer with linear pooling (the noisy energy model) described the main forms of the data for both experimental designs. We also show how a cross-correlator can be modified to fit our results and provide a contemporary presentation of the relation between summation and the slope of the psychometric function.

Area summation of first- and second-order modulations of luminance

To extend our understanding of the early visual hierarchy, we investigated the long-range integration of first- and second-order signals in spatial vision. In our first experiment we performed a conventional area summation experiment where we varied the diameter of (a) luminance-modulated (LM) noise and (b) contrastmodulated (CM) noise. Results from the LM condition replicated previous findings with sine-wave gratings in the absence of noise, consistent with long-range integration of signal contrast over space. For CM, the summation function was much shallower than for LM suggesting, at first glance, that the signal integration process was spatially less extensive than for LM. However, an alternative possibility was that the high spatial frequency noise carrier for the CM signal was attenuated by peripheral retina (or cortex), thereby impeding our ability to observe area summation of CM in the conventional way. To test this, we developed the ''Swiss cheese'' stimulus of Meese and Summers (2007) in which signal area can be varied without changing the stimulus diameter, providing some protection against inhomogeneity of the retinal field. Using this technique and a two-component subthreshold summation paradigm we found that (a) CM is spatially integrated over at least five stimulus cycles (possibly more), (b) spatial integration follows square-law signal transduction for both LM and CM and (c) the summing device integrates over spatially-interdigitated LM and CM signals when they are co-oriented, but not when crossoriented. The spatial pooling mechanism that we have identified would be a good candidate component for amodule involved in representing visual textures, including their spatial extent.

RKHS-Methods at Series Summation for Software Implementation

Reproducing Kernel Hilbert Space (RKHS) and Reproducing Transformation Methods for Series Summation that allow analytically obtaining alternative representations for series in the finite form are developed.

Linear binocular combination of responses to contrast modulation:contrast-weighted summation in first- and second-order vision

Binocular combination for first-order (luminancedefined) stimuli has been widely studied, but we know rather little about this binocular process for spatial modulations of contrast (second-order stimuli). We used phase-matching and amplitude-matching tasks to assess binocular combination of second-order phase and modulation depth simultaneously. With fixed modulation in one eye, we found that binocularly perceived phase was shifted, and perceived amplitude increased almost linearly as modulation depth in the other eye increased. At larger disparities, the phase shift was larger and the amplitude change was smaller. The degree of interocular correlation of the carriers had no influence. These results can be explained by an initial extraction of the contrast envelopes before binocular combination (consistent with the lack of dependence on carrier correlation) followed by a weighted linear summation of second-order modulations in which the weights (gains) for each eye are driven by the first-order carrier contrasts as previously found for first-order binocular combination. Perceived modulation depth fell markedly with increasing phase disparity unlike previous findings that perceived first-order contrast was almost independent of phase disparity. We present a simple revision to a widely used interocular gain-control theory that unifies first- and second-order binocular summation with a single principle-contrast-weighted summation-and we further elaborate the model for first-order combination. Conclusion: Second-order combination is controlled by first-order contrast.

Target/error overlap in jargonaphasia:the case for a one-source model, lexical and non-lexical summation, and the special status of correct responses

We present three jargonaphasic patients who made phonological errors in naming, repetition and reading. We analyse target/response overlap using statistical models to answer three questions: 1) Is there a single phonological source for errors or two sources, one for target-related errors and a separate source for abstruse errors? 2) Can correct responses be predicted by the same distribution used to predict errors or do they show a completion boost (CB)? 3) Is non-lexical and lexical information summed during reading and repetition? The answers were clear. 1) Abstruse errors did not require a separate distribution created by failure to access word forms. Abstruse and target-related errors were the endpoints of a single overlap distribution. 2) Correct responses required a special factor, e.g., a CB or lexical/phonological feedback, to preserve their integrity. 3) Reading and repetition required separate lexical and non-lexical contributions that were combined at output.

Fourth-root summation of contrast over area:no end in sight when spatially inhomogeneous sensitivity is compensated by a witch's hat

Measurements of area summation for luminance-modulated stimuli are typically confounded by variations in sensitivity across the retina. Recently we conducted a detailed analysis of sensitivity across the visual field (Baldwin et al, 2012) and found it to be well-described by a bilinear “witch’s hat” function: sensitivity declines rapidly over the first 8 cycles or so, more gently thereafter. Here we multiplied luminance-modulated stimuli (4 c/deg gratings and “Swiss cheeses”) by the inverse of the witch’s hat function to compensate for the inhomogeneity. This revealed summation functions that were straight lines (on double log axes) with a slope of -1/4 extending to ≥33 cycles, demonstrating fourth-root summation of contrast over a wider area than has previously been reported for the central retina. Fourth-root summation is typically attributed to probability summation, but recent studies have rejected that interpretation in favour of a noisy energy model that performs local square-law transduction of the signal, adds noise at each location of the target and then sums over signal area. Modelling shows our results to be consistent with a wide field application of such a contrast integrator. We reject a probability summation model, a quadratic model and a matched template model of our results under the assumptions of signal detection theory. We also reject the high threshold theory of contrast detection under the assumption of probability summation over area.

Binocular summation, binocular fusion and the transition to diplopia

The visual system combines spatial signals from the two eyes to achieve single vision. But if binocular disparity is too large, this perceptual fusion gives way to diplopia. We studied and modelled the processes underlying fusion and the transition to diplopia. The likely basis for fusion is linear summation of inputs onto binocular cortical cells. Previous studies of perceived position, contrast matching and contrast discrimination imply the computation of a dynamicallyweighted sum, where the weights vary with relative contrast. For gratings, perceived contrast was almost constant across all disparities, and this can be modelled by allowing the ocular weights to increase with disparity (Zhou, Georgeson & Hess, 2014). However, when a single Gaussian-blurred edge was shown to each eye perceived blur was invariant with disparity (Georgeson & Wallis, ECVP 2012) – not consistent with linear summation (which predicts that perceived blur increases with disparity). This blur constancy is consistent with a multiplicative form of combination (the contrast-weighted geometric mean) but that is hard to reconcile with the evidence favouring linear combination. We describe a 2-stage spatial filtering model with linear binocular combination and suggest that nonlinear output transduction (eg. ‘half-squaring’) at each stage may account for the blur constancy.

Avaliação da qualidade de vida em pacientes idosos um ano após o tratamento cirúrgico de fraturas transtrocanterianas do fêmur

OBJECTIVE: Evaluate the impact that trochanteric hip fractures produce on life quality of elderly patients, treated surgically, during one year of monitoring. METHODS: Were selected 73 elderly with transtrochanteric femur fracture, aged equal or over than 65 years and of both sexes. All participants underwent a standardized questionnaire which obtained information on lifestyle habits, functionality, physical activity, ambulation and cognitive state. Were excluded patients who died, not walking, with neurological diseases and pathological fractures. RESULTS: The average age was 80.17 ± 7.2 years and 75% was female. When comparing the summation of activities of daily living (p = 0.04) and instrumental daily living (p = 0.004) obtained before and after fracture, the patients become more dependent after fracture. Activities of daily living that worsened after fracture were bathing (p = 0.04), toileting (p = 0.02) and dressing (p = 0.04). All instrumental activities of daily living showed a significant difference, with functional dependence after fracture, increased need for walking aid (p = 0.00002), aged (p = 0.01) and not doing housework (p = 0.01). The low score on the test Minimental was associated with a greater dependence to perform activities of daily living before (p = 0.00002) and after (p = 0.01) fracture. CONCLUSION: After a year, activities of daily living that depended on the lower limbs worsened significantly, all instrumental activities of daily living were significantly worse in more than 50% of patients and more than half of patients who walked without support lost this ability.

POSITIVITY PROPERTIES OF THE FOURIER TRANSFORM AND THE STABILITY OF PERIODIC TRAVELLING-WAVE SOLUTIONS

In this paper we establish a method to obtain the stability of periodic travelling-wave solutions for equations of Korteweg-de Vries-type u(t) + u(p)u(x) - Mu(x) = 0, with M being a general pseudodifferential operator and where p >= 1 is an integer. Our approach uses the theory of totally positive operators, the Poisson summation theorem, and the theory of Jacobi elliptic functions. In particular we obtain the stability of a family of periodic travelling waves solutions for the Benjamin Ono equation. The present technique gives a new way to obtain the existence and stability of cnoidal and dnoidal waves solutions associated with the Korteweg-de Vries and modified Korteweg-de Vries equations, respectively. The theory has prospects for the study of periodic travelling-wave solutions of other partial differential equations.