Summation of contrast across the visual field: a common “fourth root” rule holds from the fovea to the periphery
Bibliographic record
Abstract
A bstract 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 local 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).
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".