Summation of contrast across the visual field: a common “fourth root” rule holds from the fovea to the periphery

Increasing the area of a grating stimulus reduces the “threshold” contrast at which it can be detected. By measuring thresholds for stimuli of different sizes, we can characterise the summation “rule” that links these two factors, providing insight into how contrast signals are integrated over space. Previous work has shown a linear summation rule over short distances, consistent with integration within the receptive fields of early cortical neurons. Beyond this range, the benefit of summation is reduced. We investigated whether the same summation rule held in peripheral vision as in the fovea. We measured thresholds for detecting “tiger tail” strips of grating (that grow orthogonal to the major axis of those presumed early receptive fields) in the fovea, parafovea (3 deg), and periphery (10.5 deg). The interpretation of previous studies has been complicated by the variation in local contrast sensitivity across the visual field. In our study, we have mapped a detailed “Witch Hat” attenuation surface of this inhomogeneity for each participant. This was used both as a component in our summation model, and to generate “compensated” stimuli where the local stimulus contrast was amplified by the reciprocal of the local sensitivity. Our results follow a common fourth root summation rule in the fovea, parafovea, and periphery, which we explain by a “noisy energy” model that combines: i) the Witch Hat surface, ii) linear filtering by early receptive fields, iii) square-law contrast transduction, and iv) the application of an internal template that devotes the participant’s attention to the extent of the stimulus. Fitting this model with one global sensitivity parameter (per participant) accounts for the foveal and parafoveal data (56 thresholds), with one further parameter allowing us to also model the periphery (84 thresholds).