arxiv: 2604.21832 · v1 · submitted 2026-04-23 · 🧬 q-bio.NC · physics.bio-ph

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Hierarchical organization of critical brain dynamics

Gustavo G. Cambrainha , Daniel M. Castro , Leonardo L. Gollo , Pedro V. Carelli , Mauro Copelli

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keywords criticalitybrain hierarchyrenormalization groupspiking activityvisual cortexhippocampusscaling exponentsanatomical organization

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The pith

Criticality signatures in the brain vary systematically along the anatomical hierarchy, but static and dynamic exponents point in opposite directions.

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

The paper examines the connection between the brain's structural hierarchy and signatures of criticality in its collective neural dynamics. Using phenomenological renormalization group methods on spiking activity recorded from the mouse visual cortex and hippocampus, the analysis shows these signatures change in an orderly way as one moves up or down the known anatomical hierarchy. However, the direction of this change is not the same for all measures: exponents derived from static properties trend one way while the exponent based on dynamic properties trends the opposite way. In the visual system these patterns also shift when the animal engages in a visual task, and the correlations among criticality markers across regions are strong enough during task performance to recover the anatomical hierarchy from the dynamics alone. Scaling relations among the exponents hold as predicted and track hierarchical position, tying microscopic collective behavior to large-scale brain architecture.

Core claim

Signatures of criticality are not uniform, but instead vary systematically along the known anatomical hierarchy in both brain systems. The direction along this gradient is inconsistent across different criticality exponents, revealing a nontrivial, measure-dependent organization: exponents based on static properties point to a gradient in one direction, while the exponent based on dynamic properties points in the opposite direction. Moreover, the signatures across the visual system are strongly modulated by the engagement in a visual task. Correlations among criticality markers of different brain regions during active engagement are sufficient to reconstruct the anatomical hierarchy from the

What carries the argument

Phenomenological renormalization group methods applied to large-scale neuronal spiking activity to extract criticality exponents that vary with position in the anatomical hierarchy.

Load-bearing premise

The phenomenological renormalization group methods correctly extract comparable criticality exponents from spiking data across regions, and the anatomical hierarchy is known independently of those dynamic measures.

What would settle it

Applying the same renormalization-group analysis and finding that criticality exponents remain uniform across regions or that static and dynamic exponents align in the same direction along the hierarchy would falsify the claim of systematic, measure-dependent organization.

Figures

Figures reproduced from arXiv: 2604.21832 by Daniel M. Castro, Gustavo G. Cambrainha, Leonardo L. Gollo, Mauro Copelli, Pedro V. Carelli.

**Figure 1.** Figure 1: FIG. 1. (A) Schematic of a mouse brain with the regions considered in this work highlighted. Credit: Allen Institute. (B) view at source ↗

**Figure 2.** Figure 2: FIG. 2. Distributions of Jensen-Shannon distance ( view at source ↗

**Figure 3.** Figure 3: FIG. 3. Distances to criticality across brain regions. (A) Median and quartiles of each view at source ↗

**Figure 4.** Figure 4: FIG. 4. Correlations of criticality signatures across different areas. (A),(B) and (C) Correlations of view at source ↗

**Figure 5.** Figure 5: FIG. 5. Exponents from the real space phenomenological renormalization of each brain area during different experiment view at source ↗

**Figure 6.** Figure 6: FIG. 6. Exponents relation across brain areas. (A) Rela view at source ↗

read the original abstract

The hierarchical organization of the brain is a fundamental structural principle, while brain criticality is a leading hypothesis for its collective dynamics. However, the connection between structure and signatures of criticality remains an open question. Here, we address this issue by applying phenomenological renormalization group approaches to large-scale neuronal spiking activity from the mouse visual cortex and hippocampus. We find that signatures of criticality are not uniform, but instead vary systematically along the known anatomical hierarchy in both brain systems. Strikingly, the direction along this gradient is inconsistent across different criticality exponents, revealing a nontrivial, measure-dependent organization: exponents based on static properties point to a gradient in one direction, while the exponent based on dynamic properties points in the opposite direction. Moreover, the signatures across the visual system are strongly modulated by the engagement in a visual task. We show that the correlations among criticality markers of different brain regions during active engagement are sufficient to reconstruct the anatomical hierarchy from the dynamics. Scaling exponents closely follow a theoretically predicted scaling relation among them, and covary with the hierarchical position. Our findings provide a direct link between the collective dynamics of neurons and the macroscopic architecture of the brain.

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 paper finds measure-dependent gradients of criticality signatures along the anatomical hierarchy in mouse visual cortex and hippocampus, with static and dynamic exponents reversing direction and task correlations sufficient to reconstruct the hierarchy, but the methods leave open whether sampling differences create artifacts.

read the letter

The central observation is that criticality is not uniform: static exponents and the dynamic one point in opposite directions along the known hierarchy in both brain systems, task engagement modulates the markers, and correlations among them during the task let you recover the anatomical order from the dynamics. The exponents also satisfy the expected scaling relation and covary with hierarchical position. That reversal and the reconstruction step are the genuinely new empirical pieces here; prior work on brain criticality has not reported this specific pattern from RG analysis of spiking data. Applying the phenomenological RG procedure to large-scale recordings and checking the internal scaling consistency is a reasonable way to proceed, and it gives the claims some anchor beyond pure phenomenology. The soft spots are in the execution details. The abstract supplies no information on how scaling parameters were selected for the RG, how different neuron densities and sampling fractions across regions were handled, or what statistical controls were used. The stress-test concern about cross-region commensurability is worth taking seriously because visual cortex and hippocampal subfields differ in cell density and probe coverage; if those factors affect the effective system size or event rates in the coarse-graining step, the reported gradients could be partly methodological rather than biological. Without seeing the full preprocessing pipeline, error estimates, or explicit checks for sampling bias, it is difficult to judge robustness. This work is for people already thinking about criticality as an organizing principle in neural circuits and how structure shapes collective dynamics. A reader in that area will get concrete observations to test against models, even if the current write-up requires follow-up on the methods. It deserves a serious referee because the claim is specific, the scaling check is external, and the reconstruction result is falsifiable, but the review should focus on whether the PRG exponents are comparable across regions.

Referee Report

2 major / 2 minor

Summary. The paper applies phenomenological renormalization group (PRG) methods to large-scale spiking activity from mouse visual cortex and hippocampus. It claims that criticality signatures vary systematically along the known anatomical hierarchy in both systems, with static-property exponents and dynamic-property exponents exhibiting gradients in opposite directions. Task engagement modulates the signatures in the visual system, correlations among markers suffice to reconstruct the anatomical hierarchy from dynamics, and the exponents closely follow a theoretically predicted scaling relation while covarying with hierarchical position.

Significance. If the central claims hold after addressing methodological concerns, this work would provide a direct empirical link between collective neural dynamics and macroscopic brain architecture, demonstrating nontrivial measure-dependent organization of criticality signatures. The reported adherence of exponents to a theoretical scaling relation supplies a valuable internal consistency check. Such findings could inform models of how structural hierarchies shape brain function and enable inference of anatomical organization from dynamic measures.

major comments (2)

The PRG procedure applied to spiking data assumes that extracted criticality exponents are comparable across regions despite substantial differences in neuron density, laminar organization, and sampling fraction (e.g., visual cortex versus hippocampal subregions). No validation is described for whether coarse-graining or scaling analysis is insensitive to effective system size or event rate, which is load-bearing for the claims of systematic hierarchical variation and opposite gradient directions between static and dynamic exponents.
The abstract and main text supply no information on data preprocessing, statistical tests, error estimation, sample sizes, or controls for multiple comparisons. This absence prevents evaluation of the robustness of the reported gradients, task modulation, and hierarchy reconstruction from marker correlations.

minor comments (2)

The abstract states clear findings but omits basic methodological context such as the number of animals or recordings, which would help readers assess the scale of the evidence.
Notation distinguishing static versus dynamic criticality exponents should be introduced explicitly early in the text to prevent reader confusion when discussing their opposing gradients.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and for recognizing the potential significance of linking criticality signatures to brain architecture. We address each major comment below with clarifications and commitments to revision.

read point-by-point responses

Referee: The PRG procedure applied to spiking data assumes that extracted criticality exponents are comparable across regions despite substantial differences in neuron density, laminar organization, and sampling fraction (e.g., visual cortex versus hippocampal subregions). No validation is described for whether coarse-graining or scaling analysis is insensitive to effective system size or event rate, which is load-bearing for the claims of systematic hierarchical variation and opposite gradient directions between static and dynamic exponents.

Authors: We appreciate this methodological concern. The PRG implementation used identical coarse-graining rules, binning, and scaling analysis parameters across all regions to promote comparability, and the resulting gradients align closely with the known anatomical hierarchy, which serves as an external consistency check. Nevertheless, to strengthen the claims, we will add explicit validation analyses in the revision, including (i) subsampling of high-density regions to match lower-density event rates and neuron counts, (ii) systematic variation of coarse-graining thresholds to confirm exponent stability, and (iii) a supplementary table reporting effective system sizes and firing rates per region. These additions will be discussed in a new paragraph in the Methods and Results sections. revision: yes
Referee: The abstract and main text supply no information on data preprocessing, statistical tests, error estimation, sample sizes, or controls for multiple comparisons. This absence prevents evaluation of the robustness of the reported gradients, task modulation, and hierarchy reconstruction from marker correlations.

Authors: We agree that these details should be more accessible in the main text. The supplementary Methods already contain the full preprocessing pipeline (spike sorting with Kilosort2, quality metrics, and inclusion criteria), statistical procedures (linear mixed-effects models for gradients, Pearson correlations for hierarchy reconstruction, bootstrap-derived 95% confidence intervals for exponents), sample sizes (n = 5 mice for visual cortex, n = 3 for hippocampus), and multiple-comparison correction (FDR at q < 0.05). In the revised manuscript we will (i) insert a concise Methods paragraph summarizing these elements, (ii) add error bars or shaded confidence intervals to all relevant figures, and (iii) explicitly state the sample sizes and correction method in the figure legends and Results text. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent anatomical hierarchy and external theoretical scaling check.

full rationale

The paper applies phenomenological renormalization group methods to spiking data from known anatomical regions, reports gradients in criticality exponents along the independently established hierarchy, and notes that exponents follow a theoretically predicted scaling relation. The hierarchy reconstruction from marker correlations is presented as validation against an external structural reference, not as a self-derived input. No quoted steps reduce a claimed prediction or uniqueness result to a fitted parameter or self-citation by construction. The PRG procedure is treated as a standard tool whose cross-region comparability is assumed but not shown to be tautological within the paper's equations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the analysis rests on the applicability of phenomenological renormalization group techniques to spiking data and the independence of the anatomical hierarchy from the measured dynamics. No explicit free parameters, new entities, or ad-hoc axioms are stated.

axioms (1)

domain assumption Phenomenological renormalization group methods can be applied to neuronal spiking activity to extract comparable criticality exponents across brain regions
The paper's central analysis depends on this assumption to interpret the data as signatures of criticality.

pith-pipeline@v0.9.0 · 5509 in / 1402 out tokens · 50989 ms · 2026-05-08T13:03:02.428461+00:00 · methodology

discussion (0)

Reference graph

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Each recording session lasted approximately 2 hours and 25 minutes

Data For this study, we used a publicly available dataset pro- vided by the Allen Brain Institute [41], which consists of large-scale neuronal recordings from multiple brain ar- eas in mice using Neuropixels 1.0 probes. Each recording session lasted approximately 2 hours and 25 minutes. The experiment was divided into three main parts. In the first part, ...
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This procedure eliminates correlations between spikes while preserving firing statistics, providing reference values for compari- son

Surrogate data To evaluate how our results differ from chance, we gen- erated surrogate data by shuffling the inter-spike intervals of each unit within the 30-s time window. This procedure eliminates correlations between spikes while preserving firing statistics, providing reference values for compari- son
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Starting from the covariance matrix, Cij =⟨σ iσj⟩,(A1) whereσ i(t) denotes the time series of activity of uniti, we compute its ranked eigenvaluesλ 1 > λ2 >

Momentum-space phenomenological renormalization To coarse-grain the activity and obtain normalized dis- tributions, we employ the phenomenological renormaliza- tion procedure [27, 29, 45]. Starting from the covariance matrix, Cij =⟨σ iσj⟩,(A1) whereσ i(t) denotes the time series of activity of uniti, we compute its ranked eigenvaluesλ 1 > λ2 > . . . > λ N...
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Jensen-Shannon distance correlations Applying the PRG procedure within 30-s windows yields a time series ofD JS values for each experiment. To investigate the dynamics of brain activity, we computed the Pearson correlation between theD JS time series of two different brain areas within the same session: Corr(Di JS (t), Dj JS (t)) = Cov(Di JS , Dj JS ) σDi...
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5 and Fig

Phenomenological renormalization exponents To extract PRG scaling exponents shown in Fig. 5 and Fig. 6, we employ an alternative phenomenologi- cal renormalization scheme based on real-space coarse- graining [28]. The procedure proceeds by iteratively merging neural activity variables according to their pair- wise correlations. At each step, the two neuro...
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