Universal scale-free representations in human visual cortex

Brice M\'enard; Michael F. Bonner; Raj Magesh Gauthaman

arxiv: 2409.06843 · v3 · submitted 2024-09-10 · 🧬 q-bio.NC · q-bio.QM

Universal scale-free representations in human visual cortex

Raj Magesh Gauthaman , Brice M\'enard , Michael F. Bonner This is my paper

Pith reviewed 2026-05-23 20:27 UTC · model grok-4.3

classification 🧬 q-bio.NC q-bio.QM

keywords scale-free representationsvisual cortexfMRIpower-law decayhyperalignmentneural codingnatural sceneshigh-dimensional representations

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The pith

Neural representations in human visual cortex show consistent scale-free variance decay as a power law across four orders of magnitude of latent dimensions, with these dimensions largely shared across people after alignment.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper aims to show that fMRI responses to natural scenes in visual cortex exhibit variance that decays systematically as a power law over a wide range of dimensions, rather than being concentrated in a few high-variance components. This pattern holds across multiple brain regions and individuals, and hyperalignment reveals that the underlying dimensions are largely common across subjects despite differences in anatomy and experience. A reader would care because it implies visual information is organized across the full high-dimensional space in a lawful way, challenging the field's usual focus on low-dimensional summaries. If correct, this points to scale-free structure as a core feature of how the brain encodes complex scenes.

Core claim

Neural representations in human visual cortex follow a remarkably consistent scale-free organization -- their variance systematically decays as a power law, detected across four orders of magnitude of latent dimensions. This scale-free structure appears consistently across multiple visual regions and across individuals, suggesting it reflects a fundamental organizing principle of visual processing. When aligned using hyperalignment, these representational dimensions are largely shared between people, revealing a universal high-dimensional spectrum of visual information that emerges despite individual differences in brain anatomy and visual experience.

What carries the argument

Scale-free organization of representational variance, identified via power-law fits to latent dimensions extracted from fMRI responses after hyperalignment across subjects.

If this is right

Traditional analyses limited to a small number of high-variance dimensions miss structured visual information distributed across the full dimensionality.
Visual information is organized systematically across the entire high-dimensional space of cortical activity.
Understanding visual representation requires examining the full spectrum rather than low-dimensional summaries.
The shared structure across individuals indicates a universal organizing principle that persists despite anatomical and experiential variation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Models of visual processing may need to incorporate scale-free statistics to match cortical population responses.
The finding could extend to test whether similar power-law structure appears in other sensory modalities or in non-human visual systems.
Individual differences in perception might arise from small perturbations on this shared high-dimensional backbone rather than wholesale reorganization.
If the pattern holds under different stimuli or tasks, it would strengthen the case that scale-free decay is intrinsic to visual coding rather than stimulus-specific.

Load-bearing premise

The latent dimensions recovered after hyperalignment capture genuine shared visual information rather than alignment artifacts or method-driven patterns.

What would settle it

Finding that power-law fits to the variance spectrum are not reliably better than exponential or other heavy-tailed alternatives, or that the aligned dimensions show no greater cross-subject consistency than random dimensions.

Figures

Figures reproduced from arXiv: 2409.06843 by Brice M\'enard, Michael F. Bonner, Raj Magesh Gauthaman.

**Figure 1.** Figure 1: Cross-decomposition procedure to estimate the cross-validated spectrum of visual representations. (Top) Each participant viewed about 10,000 natural scene images multiple times while their fMRI BOLD responses were measured. The depicted brain region is a “general” region of interest (ROI) in visual cortex that was identified by selecting all voxels whose activity was modulated by the presentation of images… view at source ↗

**Figure 2.** Figure 2: Cortical representations of natural images are scale-free and shared across subjects over many dimensions. (Left) Within-subject covariance spectra of visual cortex responses in a general region of interest including all visually responsive voxels from the Natural Scenes fMRI dataset (Allen et al., 2021). (Right) Between-subject covariance spectra for the same region of interest, characterizing shared va… view at source ↗

**Figure 3.** Figure 3: Null tests and comparison to a non-visual cortical region. The lower panels compare the activation covariance for subject 1 and the cross-covariance between subjects 1 and 2 (as shown in [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: Representational similarity in visual cortex for pairs of subjects. Correlation of neural representations as a function of ranks obtained from the ratio of cross-validated between-subject variance relative to cross-validated within-subject variance (Equation 1). These findings show that the stimulus-related variance in cortical activity is largely shared across subjects over many ranks. Furthermore, the si… view at source ↗

**Figure 5.** Figure 5: Scale-free representations are consistently observed across multiple regions of visual cortex. These plots show within-subject (top) and between-subject (bottom) covariance spectra for retinotopic regions along the visual hierarchy from V1 to V4. In comparison with the larger region of interest shown in [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗

**Figure 6.** Figure 6: Comparison of activations between different brain regions. (Left) Activation covariance spectra for visual cortex regions V1 to V4 in subject 1. (Right) Activation cross-covariance spectra computed for between-region comparisons of visual and frontal regions. Each cell represents the covariance shared between a pair of regions across different trials within subject 1 for a decade of ranks. The amount of s… view at source ↗

**Figure 7.** Figure 7: RSA correlations are largely insensitive to high-dimensional structure. Representational similarity analysis (RSA) was used to compare the visual cortex representations of two individuals. (Left) This analysis was performed on two sets of representational similarity matrices (RSMs): one computed on the original fMRI data (“highD”) and a second computed on reduced-rank fMRI data (“low-D”, from the first 10… view at source ↗

read the original abstract

How does the human brain encode complex visual information? While previous research has characterized individual dimensions of visual representation in cortex, we still lack a comprehensive understanding of how visual information is organized across the full range of neural population activity. Here, analyzing fMRI responses to natural scenes across multiple individuals, we discover that neural representations in human visual cortex follow a remarkably consistent scale-free organization -- their variance systematically decays as a power law, detected across four orders of magnitude of latent dimensions. This scale-free structure appears consistently across multiple visual regions and across individuals, suggesting it reflects a fundamental organizing principle of visual processing. Critically, when we align neural responses across individuals using hyperalignment, we find that these representational dimensions are largely shared between people, revealing a universal high-dimensional spectrum of visual information that emerges despite individual differences in brain anatomy and visual experience. Traditional analysis approaches in cognitive neuroscience have focused primarily on a small number of high-variance dimensions, potentially missing crucial aspects of visual representation. Our results demonstrate that visual information is distributed across the full dimensionality of cortical activity in a systematic way, suggesting we need to move beyond low-dimensional characterizations to fully understand how the brain represents the visual world. This work reveals a new fundamental principle of neural coding in human visual cortex and highlights the importance of examining neural representations across their full dimensionality.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The abstract claims a universal scale-free variance spectrum in visual cortex that survives hyperalignment, but supplies zero statistical validation or controls, so the result cannot be assessed.

read the letter

Colleague, the main takeaway is that the authors report neural responses to natural scenes show variance decaying as a power law across four orders of magnitude of latent dimensions, and that this pattern is largely shared across people once the data are hyperaligned. They argue this reveals a fundamental high-dimensional structure that low-dimensional summaries miss. That framing is the core observation they present as new. The paper does draw attention to a genuine limitation in standard practice: most analyses stop at the top few dimensions and ignore the rest of the spectrum. Pointing that out is useful even if the specific claim needs more support. The setup with multiple subjects and natural scenes is a reasonable starting point for asking about shared structure. The soft spots are large and sit right at the center of the claim. The abstract states the power law is detected but gives no detail on how the fit was performed, whether the range was chosen in advance, or how the power-law model was compared to alternatives such as exponential or log-normal distributions. It also provides no check on whether the same decay appears in unaligned subject-specific data or whether the hyperalignment step itself produces the apparent universality. Those omissions matter because alignment procedures can impose low-rank or scale-free patterns as a byproduct. The stress-test note flags exactly these gaps, and the supplied text does not resolve them. Without those controls the result reads as an observation that still needs basic validation rather than a finished finding. This would interest people who work on high-dimensional representational geometry or cross-subject alignment, but only after the missing statistical steps are added. I would not bring it to a reading group yet. It does not look ready for peer review in its current state; the central claim cannot be evaluated from what is shown.

Referee Report

2 major / 2 minor

Summary. The manuscript analyzes fMRI responses to natural scenes across multiple individuals and reports that neural representations in human visual cortex exhibit a consistent scale-free organization, with variance decaying as a power law across four orders of magnitude of latent dimensions. This structure is observed across multiple visual regions and individuals. After hyperalignment, the representational dimensions are largely shared between people, indicating a universal high-dimensional spectrum of visual information that persists despite individual differences in anatomy and experience. The work argues that traditional low-dimensional analyses miss this systematic distribution of visual information.

Significance. If the power-law structure and its universality after alignment are robustly established, the result would be significant for cognitive neuroscience by providing evidence for a fundamental organizing principle that extends across the full dimensionality of cortical activity rather than being confined to a small number of high-variance dimensions. The cross-subject sharing would further suggest that this spectrum reflects shared visual information rather than idiosyncratic structure.

major comments (2)

[Abstract] Abstract and Results: The central claim that variance spectra follow a power law 'detected across four orders of magnitude' requires explicit validation against alternative distributions (exponential, log-normal) via likelihood-ratio tests or procedures such as those in Clauset et al. (2009); no such comparisons, goodness-of-fit metrics, or fit-range justification are described, making it impossible to assess whether the scale-free characterization is supported or could arise from other heavy-tailed forms.
[Methods] Methods (hyperalignment section): The claim that dimensions are 'largely shared' after hyperalignment rests on the assumption that the Procrustes-style mapping reveals pre-existing structure rather than imposing low-rank or scale-free properties. No controls are reported (e.g., comparison of spectra before vs. after alignment, permutation tests on the alignment matrix, or analysis of subject-specific unaligned data) to rule out method-induced artifacts.

minor comments (2)

[Abstract] The abstract refers to 'latent dimensions' without specifying the dimensionality-reduction technique (PCA, ICA, or other) or the exact number of dimensions retained; this notation should be clarified in the main text with a methods equation or table.
[Figures] Figure legends (presumed) should include the exact range of dimensions over which the power law is fitted and the number of subjects/regions contributing to each panel to allow direct evaluation of the 'four orders of magnitude' claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which highlight important aspects of statistical rigor and methodological validation. We address each major comment below and will incorporate the suggested analyses into the revised manuscript to strengthen the evidence for our claims.

read point-by-point responses

Referee: [Abstract] Abstract and Results: The central claim that variance spectra follow a power law 'detected across four orders of magnitude' requires explicit validation against alternative distributions (exponential, log-normal) via likelihood-ratio tests or procedures such as those in Clauset et al. (2009); no such comparisons, goodness-of-fit metrics, or fit-range justification are described, making it impossible to assess whether the scale-free characterization is supported or could arise from other heavy-tailed forms.

Authors: We agree that formal statistical validation against alternative distributions is required to robustly support the power-law characterization. The current manuscript presents log-log plots of the variance spectra and reports linear regression fits indicating power-law decay over four orders of magnitude, but does not include likelihood-ratio tests or goodness-of-fit metrics as described by Clauset et al. (2009). In the revision we will add these analyses, including likelihood-ratio tests comparing the power-law model to exponential and log-normal alternatives, p-value assessments for the power-law fit, and explicit justification for the chosen fit range. This will allow a clearer evaluation of whether the scale-free description is preferred over other heavy-tailed forms. revision: yes
Referee: [Methods] Methods (hyperalignment section): The claim that dimensions are 'largely shared' after hyperalignment rests on the assumption that the Procrustes-style mapping reveals pre-existing structure rather than imposing low-rank or scale-free properties. No controls are reported (e.g., comparison of spectra before vs. after alignment, permutation tests on the alignment matrix, or analysis of subject-specific unaligned data) to rule out method-induced artifacts.

Authors: We acknowledge that controls are needed to demonstrate that the shared scale-free structure reflects pre-existing neural organization rather than being imposed by the hyperalignment procedure. The current manuscript does not report the suggested controls. In the revision we will include: (i) direct comparison of variance spectra computed before versus after hyperalignment, (ii) permutation tests that randomize the alignment matrix to evaluate whether the observed sharing exceeds chance levels, and (iii) analysis of subject-specific unaligned data to confirm that the scale-free property is present individually. These additions will help rule out method-induced artifacts and support the interpretation of universal shared dimensions. revision: yes

Circularity Check

0 steps flagged

No circularity; empirical discovery via direct measurement of variance spectra

full rationale

The paper reports an empirical observation from fMRI responses to natural scenes: after hyperalignment, the variance of latent dimensions decays as a power law across four orders of magnitude, appearing consistently across regions and individuals. No equations, derivations, or self-citations are invoked that reduce this structure to a fitted parameter renamed as a prediction or to a prior result by construction. The central claim rests on statistical measurement of data properties rather than any self-definitional or load-bearing reduction, rendering the analysis self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The claim depends on standard fMRI assumptions about BOLD linearity and on the validity of hyperalignment for recovering shared dimensions; the power-law exponent is implicitly fitted to the observed variance spectrum.

free parameters (1)

power-law exponent
The decay rate of variance across dimensions is determined by fitting to the data rather than derived from first principles.

axioms (2)

domain assumption BOLD signal changes linearly reflect underlying neural population activity
Invoked implicitly when treating fMRI voxel patterns as direct readouts of representational dimensions.
domain assumption Hyperalignment recovers anatomically and experientially invariant representational dimensions
Central to the claim that the scale-free spectrum is universal across individuals.

pith-pipeline@v0.9.0 · 5771 in / 1305 out tokens · 23458 ms · 2026-05-23T20:27:47.996672+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

neural representations in human visual cortex follow a remarkably consistent scale-free organization—their variance systematically decays as a power law, detected across four orders of magnitude of latent dimensions... these representational dimensions are largely shared between people
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

the covariance spectrum of fMRI responses... is consistent with a power-law distribution over almost four orders of magnitude... universality in how variance is spread across latent dimensions

What do these tags mean?

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Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

[1]

K., Mondal, A

Agrawal, K. K., Mondal, A. K., Ghosh, A., & Richards, B. (2022). \Alpha-req : Assessing representation quality in self-supervised learning by measuring eigenspectrum decay. In S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, & A. Oh (Eds.), Advances in neural information processing systems (pp. 17626– 17638, V ol. 35). Curran Associates, Inc. https...

work page 2022
[2]

https://doi.org/10.1016/j.neuron.2014.07.035 Bao, P., She, L., McGill, M., & Tsao, D. Y . (2020). A map of object space in primate inferotemporal cortex. Nature, 583(7814), 103–108. https://doi.org/10.1038/s41586-020-2350-5 Bialek, W. (2022). On the dimensionality of behavior. Proceedings of the National Academy of Sciences , 119(18). https://doi.org/10.1...

work page doi:10.1016/j.neuron.2014.07.035 2014
[3]

A., Denfield, G

https://doi.org/10.3389/fphys.2013.00079 Cadena, S. A., Denfield, G. H., Walker, E. Y ., Gatys, L. A., Tolias, A. S., Bethge, M., & Ecker, A. S. (2019). Deep convolutional models improve predictions of macaque v1 responses to natural images (W. Einhäuser, Ed.). PLOS Computational Biology, 15(4), e1006897. https://doi.org/10.1371/journal.pcbi.1006897 Cayco...

work page doi:10.3389/fphys.2013.00079 2013
[4]

D., & Sompolinsky, H

https://doi.org/10.3389/fphys.2012.00186 Cohen, U., Chung, S., Lee, D. D., & Sompolinsky, H. (2020). Separability and geometry of object manifolds in deep neural networks. Nature Communications, 11(1). https://doi.org/10.1038/s41467-020-14578-5 Connolly, A. C., Guntupalli, J. S., Gors, J., Hanke, M., Halchenko, Y . O., Wu, Y .-C., Abdi, H., & Haxby, J. V ...

work page doi:10.3389/fphys.2012.00186 2012
[5]

https://doi.org/10.7554/elife.56601 He, B. J. (2014). Scale-free brain activity: Past, present, and future. Trends in Cognitive Sciences , 18(9), 480–487. https://doi.org/10.1016/j.tics.2014.04.003 He, B. J., Zempel, J. M., Snyder, A. Z., & Raichle, M. E. (2010). The temporal structures and functional significance of scale-free brain activity. Neuron, 66(...

work page doi:10.7554/elife.56601 2014
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THINGS-data, a multimodal collection of large-scale datasets for investigating object representations in human brain and behavior , volume =

https://doi.org/10.7554/elife.82580 Huth, A. G., Nishimoto, S., Vu, A. T., & Gallant, J. L. (2012). A continuous semantic space describes the representation of thousands of object and action categories across the human brain. Neuron, 76(6), 1210–1224. https://doi. org/10.1016/j.neuron.2012.10.014 Jazayeri, M., & Ostojic, S. (2021). Interpreting neural com...

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https://doi.org/10.7554/elife.47142 Wandell, B., Dumoulin, S., & Brewer, A. (2009). Visual cortex in humans. InEncyclopedia of neuroscience (pp. 251– 257). Elsevier. https://doi.org/10.1016/b978-008045046-9.00241-2 Werner, G. (2010). Fractals in the nervous system: Conceptual implications for theoretical neuroscience. Frontiers in Physiology. https://doi....

work page doi:10.7554/elife.47142 2009
[8]

throughout the spectrum. We find that the second latent dimension appears sensitive to airplanes/trains while the third latent dimension is activated strongly by flowers in vases, and the fourth is activated by clocktower-like buildings. However, we expect that these patterns are driven by low-level features such as spatial orientation that covary strongl...

work page 2023

[1] [1]

K., Mondal, A

Agrawal, K. K., Mondal, A. K., Ghosh, A., & Richards, B. (2022). \Alpha-req : Assessing representation quality in self-supervised learning by measuring eigenspectrum decay. In S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, & A. Oh (Eds.), Advances in neural information processing systems (pp. 17626– 17638, V ol. 35). Curran Associates, Inc. https...

work page 2022

[2] [2]

https://doi.org/10.1016/j.neuron.2014.07.035 Bao, P., She, L., McGill, M., & Tsao, D. Y . (2020). A map of object space in primate inferotemporal cortex. Nature, 583(7814), 103–108. https://doi.org/10.1038/s41586-020-2350-5 Bialek, W. (2022). On the dimensionality of behavior. Proceedings of the National Academy of Sciences , 119(18). https://doi.org/10.1...

work page doi:10.1016/j.neuron.2014.07.035 2014

[3] [3]

A., Denfield, G

https://doi.org/10.3389/fphys.2013.00079 Cadena, S. A., Denfield, G. H., Walker, E. Y ., Gatys, L. A., Tolias, A. S., Bethge, M., & Ecker, A. S. (2019). Deep convolutional models improve predictions of macaque v1 responses to natural images (W. Einhäuser, Ed.). PLOS Computational Biology, 15(4), e1006897. https://doi.org/10.1371/journal.pcbi.1006897 Cayco...

work page doi:10.3389/fphys.2013.00079 2013

[4] [4]

D., & Sompolinsky, H

https://doi.org/10.3389/fphys.2012.00186 Cohen, U., Chung, S., Lee, D. D., & Sompolinsky, H. (2020). Separability and geometry of object manifolds in deep neural networks. Nature Communications, 11(1). https://doi.org/10.1038/s41467-020-14578-5 Connolly, A. C., Guntupalli, J. S., Gors, J., Hanke, M., Halchenko, Y . O., Wu, Y .-C., Abdi, H., & Haxby, J. V ...

work page doi:10.3389/fphys.2012.00186 2012

[5] [5]

https://doi.org/10.7554/elife.56601 He, B. J. (2014). Scale-free brain activity: Past, present, and future. Trends in Cognitive Sciences , 18(9), 480–487. https://doi.org/10.1016/j.tics.2014.04.003 He, B. J., Zempel, J. M., Snyder, A. Z., & Raichle, M. E. (2010). The temporal structures and functional significance of scale-free brain activity. Neuron, 66(...

work page doi:10.7554/elife.56601 2014

[6] [6]

THINGS-data, a multimodal collection of large-scale datasets for investigating object representations in human brain and behavior , volume =

https://doi.org/10.7554/elife.82580 Huth, A. G., Nishimoto, S., Vu, A. T., & Gallant, J. L. (2012). A continuous semantic space describes the representation of thousands of object and action categories across the human brain. Neuron, 76(6), 1210–1224. https://doi. org/10.1016/j.neuron.2012.10.014 Jazayeri, M., & Ostojic, S. (2021). Interpreting neural com...

work page doi:10.7554/elife.82580 2012

[7] [7]

https://doi.org/10.7554/elife.47142 Wandell, B., Dumoulin, S., & Brewer, A. (2009). Visual cortex in humans. InEncyclopedia of neuroscience (pp. 251– 257). Elsevier. https://doi.org/10.1016/b978-008045046-9.00241-2 Werner, G. (2010). Fractals in the nervous system: Conceptual implications for theoretical neuroscience. Frontiers in Physiology. https://doi....

work page doi:10.7554/elife.47142 2009

[8] [8]

throughout the spectrum. We find that the second latent dimension appears sensitive to airplanes/trains while the third latent dimension is activated strongly by flowers in vases, and the fourth is activated by clocktower-like buildings. However, we expect that these patterns are driven by low-level features such as spatial orientation that covary strongl...

work page 2023