Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.36404

Hierarchical connectivity sets the upper critical dimension of neural avalanches

W. A. Zúñiga-Galindo
View original source ↗Read source document ↗Export metadata ↓

Abstract

Neural avalanches with size exponent $3/2$ are usually explained by mean-field critical branching, which for short-range coupling requires dimension $N\ge4$. We show that hierarchical connectivity removes this constraint. In a $p$-adic version of the Buice--Cowan field theory, a coupling that decays by $p^{-(N+σ)}$ per hierarchical level selects the Taibleson--Vladimirov operator, and the critical theory is ultrametric directed percolation with upper critical dimension $N_c=2σ$. Exact simulations on a tree give $τ_S=3/2$ for $σ\le N/2$, decreasing continuously for $σ>N/2$.

Explore related subjects

cond-mat.dis-nnq-bio.NC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

W. A. Zúñiga-Galindo. 2026-10-26. Hierarchical connectivity sets the upper critical dimension of neural avalanches. https://arxiv.org/abs/2610.36404

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Sequential Retrieval in Dense Associative Memory: Asymptotic Dynamics and Storage Capacity

We study the retrieval dynamics of an $n$-body asymmetric dense associative memory model, in which sequential retrieval is implemented by shifting one pattern index in the higher-order Hebbian interaction. The model exhibits sequential retrieval of the stored patterns, and the retrieval state corresponds to a stable limit-cycle attractor. The storage capacity is defined as the maximum loading rate for which this limit cycle remains stable. Using generating functional analysis, we derive the dynamical equations describing the sequential retrieval state. The generating functional analysis reveals that the retarded self-interaction term vanishes in the asymmetric model, and the resulting overlap dynamics is reproduced by an Amari-Maginu-type signal-to-noise analysis. Also, we find that for the interaction order $n\ge 3$, the normalized storage capacity decreases monotonically with increasing interaction order, in contrast to symmetric Modern Hopfield models.

cond-mat.dis-nn↗
arXiv2026

A 3D-$Δ$PDF-like measure for hidden anisotropy in disordered materials

We introduce a simple correlation function for quantifying hidden local anisotropy in disordered materials. Developed in analogy with the three-dimensional difference pair distribution function (3D-$Δ$PDF), our approach identifies structural correlations that are lost through averaging over bulk symmetry. The same approach might be applied to models of disordered crystals, amorphous materials, and liquids alike.

cond-mat.dis-nn↗
arXiv2026

p-Adic statistical field theory of fluctuation effects in hierarchical neural networks

The Buice-Cowan field theory [PRE 75, 051919 (2007)] describes fluctuations and correlations of neural activity through a path integral whose mean-field limit is the Wilson-Cowan equation and whose critical behavior belongs to the directed percolation class. In its continuum limit the neurons are the points of a Euclidean space and activity spreads by ordinary diffusion, which encodes neither the hierarchical organization of cortical networks nor their small-world property. We formulate a version of the theory in which the neurons are organized in an infinite, self-similar, tree-like structure, described by the p-adic numbers. The path integral carries over unchanged, and finite networks are exact restrictions of the continuum theory. The main change is in the linear response: in the Euclidean theory every short-range kernel reduces to the Laplacian, while in the p-adic theory each connectivity kernel gives its own nonlocal p-adic diffusion operator. For couplings that decay by a fixed factor per level of the hierarchy this is the Taibleson-Vladimirov operator, and the critical theory is an ultrametric Reggeon field theory whose upper critical dimension is twice the decay exponent of the coupling. Hence the decay of the coupling up the hierarchy, and not the dimension of space, decides whether mean-field exponents apply. We obtain a Ginzburg-type criterion, loop-corrected hierarchical Wilson-Cowan equations, log-periodic corrections to scaling and power-law correlations. Exact simulations on a tree support the predictions: the avalanche-size exponent takes the mean-field value 3/2 when the coupling decays slowly and decreases toward the directed-percolation value when it decays fast. The hierarchical network is also small-world when the decay exponent is smaller than the dimension, a property that the Euclidean theory obtains only by assuming long-range coupling.

cond-mat.dis-nn↗