Summation of contrast across the visual field:A common "fourth root" rule holds from the fovea to the periphery

Abstract

Increasing the area of grating-like stimuli reduces their contrast detection thresholds. Characterising the visual system's summation rule this way provides insights into early visual architecture. Previous work in the fovea has found linear summation over short distances, consistent with integration within the receptive fields of early cortical neurons. Beyond this range, the benefit of stimulus area is reduced. Here, we investigated whether the same integration rule holds for stimulus elongations centred at different positions across the visual field. We did this for "tiger tail" strips of grating (growing orthogonally to the major axis of the early receptive fields) in the fovea, parafovea (3 deg), and periphery (10.5 deg). The interpretation of results from previous studies has been complicated by variation in local contrast sensitivity across the visual field. We addressed this here by using detailed maps of the inhomogeneity for each participant (their "witch hat") to generate "compensated" stimuli where the stimulus contrast was amplified by the reciprocal of their local sensitivity. Our results followed a common fourth-root summation rule for tiger-tails in the fovea, parafovea, and periphery. We explained this by a "noisy energy" model that combined: i) a "witch hat" sensitivity surface, ii) linear filtering by receptive fields, iii) square-law contrast transduction, and iv) an internal template to direct the observer's attention to the spatial extent of the stimulus. Fitting this model with a single global sensitivity parameter accounted for foveal and parafoveal results (56 thresholds), with one further parameter needed to model the periphery (84 thresholds). [Abstract copyright: Copyright © 2026. Published by Elsevier Ltd.]