On the Green function of the killed fractional Laplacian on the periodic domain

We give a very simple proof of the positivity and unimodality of the Green function for the killed fractional Laplacian on the periodic domain. The argument relies on the Jacobi triple product and a probabilistic representation of the Green function. We also show by a contour integration that the Green function is completely monotone on the positive part of the periodic domain.

Analytical expression development for eddy field and the beam dynamics effect on the CEPC booster

An analytical magnetic field expression for a circular vacuum chamber with induced eddy on it is obtained. Two boundary conditions are discussed: with and without the iron poles; the former implies the use of the image current methods. After the current angular distribution in the vacuum chamber is calculated, a contour integration is applied to obtain the field distribution in the aperture. The application to the CEPC booster is also presented, which is one of the most important issues in the design stage.

Masking, crowding and grouping: Connecting low and mid-level vision

An important task for vision science is to build a unitary framework of low and mid-level vision. As a step on this way, our study examined differences and commonalities between masking, crowding and grouping – three processes that occur through spatial interactions between neighbouring elements. We measured contrast thresholds as functions of inter-element spacing and eccentricity for Gabor detection, discrimination, contour integration, using a common stimulus grid consisting of 9 Gabor elements. From these thresholds, we derived a) the baseline contrast necessary to perform each task and b) the spatial extent over which task performance was stable. This spatial window can be taken as an indicator of field size, where elements that fall within a putative field are readily combined. We found that contrast thresholds were universally modulated by inter-element distance, with a shallower and inverted effect for grouping compared to masking and crowding. Baseline contrasts for detecting stimuli and discriminating their properties were positively linked across the tested retinal locations (parafovea and near periphery), whereas those for integrating elements and discriminating their properties were negatively linked. Meanwhile, masking and crowding spatial windows remained uncorrelated across eccentricity, although they were correlated across participants. This suggests that the computation performed by each type of visual field operates over different distances that co-varies across observers, but not across retinal locations. Contrast-processing units may thus lie at the core of the shared idiosyncrasies across tasks reported in many previous studies, despite the fundamental differences in the extent of their spatial windows.

Light Curve Calculations for Triple Microlensing Systems

We present a method to compute the magnification of a finite source star lensed by a triple lens system based on the image boundary (contour integration) method. We describe a new procedure to obtain continuous image boundaries from solutions of the tenth-order polynomial obtained from the lens equation. Contour integration is then applied to calculate the image areas within the image boundaries, which yields the magnification of a source with uniform brightness. We extend the magnification calculation to limb-darkened stars approximated with a linear profile. In principle, this method works for all multiple lens systems, not just triple lenses. We also include an adaptive sampling and interpolation method for calculating densely covered light curves. The C++ source code and a corresponding Python interface are publicly available.

Visual Noise Effect on Contour Integration and Gaze Allocation in Autism Spectrum Disorder

Contradictory results have been obtained in the studies that compare contour integration abilities in Autism Spectrum Disorders (ASDs) and typically developing individuals. The present study aimed to explore the limiting factors of contour integration ability in ASD and verify the role of the external visual noise by a combination of psychophysical and eye-tracking approaches. To this aim, 24 children and adolescents with ASD and 32 age-matched participants with typical development had to detect the presence of contour embedded among similar Gabor elements in a Yes/No procedure. The results obtained showed that the responses in the group with ASD were not only less accurate but also were significantly slower compared to the control group at all noise levels. The detection performance depended on the group differences in addition to the effect of the intellectual functioning of the participants from both groups. The comparison of the agreement and accuracy of the responses in the double-pass experiment showed that the results of the participants with ASD are more affected by the increase of the external noise. It turned out that the internal noise depends on the level of the added external noise: the difference between the two groups was non-significant at the low external noise and significant at the high external noise. In accordance with the psychophysical results, the eye-tracking data indicated a larger gaze allocation area in the group with autism. These findings may imply higher positional uncertainty in ASD due to the inability to maintain the information of the contour location from previous presentations and interference from noise elements in the contour vicinity. Psychophysical and eye-tracking data suggest lower efficiency in using stimulus information in the ASD group that could be caused by fixation instability and noisy and unstable perceptual template that affects noise filtering.

Sparse deep predictive coding captures contour integration capabilities of the early visual system

Both neurophysiological and psychophysical experiments have pointed out the crucial role of recurrent and feedback connections to process context-dependent information in the early visual cortex. While numerous models have accounted for feedback effects at either neural or representational level, none of them were able to bind those two levels of analysis. Is it possible to describe feedback effects at both levels using the same model? We answer this question by combining Predictive Coding (PC) and Sparse Coding (SC) into a hierarchical and convolutional framework applied to realistic problems. In the Sparse Deep Predictive Coding (SDPC) model, the SC component models the internal recurrent processing within each layer, and the PC component describes the interactions between layers using feedforward and feedback connections. Here, we train a 2-layered SDPC on two different databases of images, and we interpret it as a model of the early visual system (V1 & V2). We first demonstrate that once the training has converged, SDPC exhibits oriented and localized receptive fields in V1 and more complex features in V2. Second, we analyze the effects of feedback on the neural organization beyond the classical receptive field of V1 neurons using interaction maps. These maps are similar to association fields and reflect the Gestalt principle of good continuation. We demonstrate that feedback signals reorganize interaction maps and modulate neural activity to promote contour integration. Third, we demonstrate at the representational level that the SDPC feedback connections are able to overcome noise in input images. Therefore, the SDPC captures the association field principle at the neural level which results in a better reconstruction of blurred images at the representational level.

Numerical Contour Integration

We present a straightforward implementation of contour integration by setting options for Integrate and NIntegrate, taking advantage of powerful results in complex analysis. As such, this article can be viewed as documentation to perform numerical contour integration with the existing built-in tools. We provide examples of how this method can be used when integrating analytically and numerically some commonly used distributions, such as Wightman functions in quantum field theory. We also provide an approximating technique when time-ordering is involved, a commonly encountered scenario in quantum field theory for computing second-order terms in Dyson series expansion and Feynman propagators. We believe our implementation will be useful for more general calculations involving advanced or retarded Green’s functions, propagators, kernels and so on.