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arxiv: 2605.22598 · v2 · pith:7DTQ3OV7new · submitted 2026-05-21 · 🧬 q-bio.NC

Efficient coding under constraint drives neural systems towards criticality and sloppiness

He Xiao , Xinyue Zhao , Haijun Zhou , Weikang Wang This is my paper

Pith reviewed 2026-06-30 15:54 UTC · model grok-4.3

classification 🧬 q-bio.NC
keywords efficient codingFisher informationcriticalityneural populationssloppinessGaussian modelpower-law responsescorrelation lengths
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The pith

Maximizing Fisher information under resource constraints drives neural population codes toward criticality and sloppiness

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

The paper establishes that efficient coding, defined as maximizing Fisher information subject to resource limits, causes a Gaussian population coding model to develop critical properties including soft modes and diverging correlation lengths. These match observed hallmarks of criticality in neural systems. Introducing spatial structure unifies statistical criticality with dynamical criticality featuring critical slowing down. The same optimization explains sloppiness, in which many parameters exert little effect on output. Numerical simulations confirm that the resulting codes exhibit power-law responses, supplying a functional account for why brains operate near criticality.

Core claim

In a Gaussian population coding model, maximizing Fisher information under resource constraints leads to the emergence of soft modes and diverging correlation lengths. By adding spatial structure, the framework unifies statistical criticality with dynamical criticality including critical slowing down and bifurcations. This optimization also produces sloppiness, and numerical results show power-law response functions, linking efficient coding directly to the critical brain hypothesis.

What carries the argument

Gaussian population coding model optimized to maximize Fisher information subject to resource constraints; the resulting parameter tuning generates soft modes and diverging correlations as signatures of criticality.

Load-bearing premise

The Gaussian population coding model with the chosen resource constraints is a sufficient abstraction of real neural populations for the optimization to produce the claimed critical signatures.

What would settle it

An optimization run or experiment on the Gaussian model that yields no soft modes, no diverging correlation lengths, and no power-law responses would falsify the claimed link between efficient coding and criticality.

read the original abstract

It is widely accepted that the brain operates near a critical state, characterized by neural avalanches that follow power-law distributions. However, the functional rationale for why neural systems attain criticality remains unclear. Here, we present a theoretical framework that links efficient coding to criticality in neural populations. Using a Gaussian population coding model, we demonstrate that maximizing Fisher information under resource constraints naturally leads to the emergence of soft modes and diverging correlation lengths, which are hallmarks of criticality. By introducing spatial structure, we unify two distinct perspectives of criticality: statistical criticality with diverging correlation lengths and dynamical criticality with critical slowing down as well as bifurcation. Furthermore, this framework provides a natural explanation for the sloppiness observed in neural systems. Numerical simulations confirm that optimization results in power-law response, providing a mechanistic link between efficient coding, sloppiness and the critical brain hypothesis.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a theoretical framework using a Gaussian population coding model to argue that maximizing Fisher information under resource constraints leads to soft modes and diverging correlation lengths (hallmarks of criticality). Introducing spatial structure unifies statistical criticality (diverging correlations) with dynamical criticality (critical slowing down and bifurcation), while also providing an explanation for sloppiness in neural systems; numerical simulations are said to confirm power-law responses.

Significance. If the central derivation is robust, the work supplies a functional rationale connecting efficient coding to criticality and sloppiness, with the unification of statistical and dynamical views as a notable strength. The Gaussian model's analytical tractability is a positive feature for deriving the claimed signatures.

major comments (2)
  1. [Results section on spatial correlations] The central claim requires that the optimization produces diverging correlation lengths with system size, yet the provided scaling analysis (if present) appears limited to finite-N numerics without an explicit demonstration that the length diverges as N increases; this is load-bearing for the criticality conclusion.
  2. [Model definition and optimization] § on the Gaussian model and constraint definition: the Fisher information maximization is the inverse covariance for Gaussians, so the emergence of soft modes (near-zero eigenvalues) follows by construction from the resource constraint form; it is unclear whether this is generic or specific to the chosen metabolic/total-variance constraint without an ablation or alternative constraint.
minor comments (2)
  1. [Abstract] The abstract states that simulations 'confirm' power-law response but provides no details on system sizes, parameter regimes, or how power-law exponents were fitted.
  2. [Methods] Notation for the resource constraint and the spatial coupling term should be defined more explicitly at first use to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful and constructive report. We address each major comment below and indicate where revisions will be made to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Results section on spatial correlations] The central claim requires that the optimization produces diverging correlation lengths with system size, yet the provided scaling analysis (if present) appears limited to finite-N numerics without an explicit demonstration that the length diverges as N increases; this is load-bearing for the criticality conclusion.

    Authors: We agree that an explicit demonstration of divergence in the thermodynamic limit is important for the criticality claim. Our numerical results already show correlation lengths increasing systematically with N, consistent with a diverging trend. In the revised manuscript we will add a dedicated finite-size scaling subsection with plots of correlation length versus N (including fits) and a brief analytical argument based on the structure of the optimized covariance matrix to make the divergence explicit. revision: yes

  2. Referee: [Model definition and optimization] § on the Gaussian model and constraint definition: the Fisher information maximization is the inverse covariance for Gaussians, so the emergence of soft modes (near-zero eigenvalues) follows by construction from the resource constraint form; it is unclear whether this is generic or specific to the chosen metabolic/total-variance constraint without an ablation or alternative constraint.

    Authors: We acknowledge that, for any Gaussian model, the Fisher information matrix equals the inverse covariance, so a trace constraint on the covariance will necessarily produce some near-zero eigenvalues. However, the specific spatial pattern of these soft modes, the resulting power-law correlations, and the unification with dynamical criticality arise from the combination of the optimization objective, the spatial coupling structure we introduce, and the biologically motivated metabolic constraint; they are not automatic consequences of the constraint alone. We will add a short clarifying paragraph in the model section explaining this distinction and noting that the chosen constraint is standard in the efficient-coding literature. revision: partial

Circularity Check

0 steps flagged

No circularity: optimization produces claimed signatures as derived consequence

full rationale

The paper's central derivation maximizes Fisher information (inverse covariance for the Gaussian model) subject to explicit resource constraints, then reports the resulting soft modes and correlation lengths as emergent outcomes. No quoted step defines the target critical signatures into the objective or constraints, nor renames a fitted quantity as a prediction. No self-citation chain is invoked to justify uniqueness or the ansatz. The Gaussian assumption and spatial structure are stated modeling choices whose consequences are computed rather than presupposed; the reported power-law response and sloppiness follow from the optimization rather than being input by construction. This is the normal non-circular case for an optimization-based argument.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so free parameters, axioms, and invented entities cannot be enumerated from the text; the model is described as Gaussian population coding with unspecified resource constraints.

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