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arxiv: 2605.15872 · v1 · pith:CLIAMCS2new · submitted 2026-05-15 · 🌊 nlin.CD · cond-mat.dis-nn· math.DS· nlin.AO

Symmetry breaking and high-dimensional chaos in sparse random networks of exact firing rate models

Pau Clusella This is my paper

Pith reviewed 2026-05-19 17:30 UTC · model grok-4.3

classification 🌊 nlin.CD cond-mat.dis-nnmath.DSnlin.AO
keywords exact firing rate modelsnext-generation neural mass modelssparse random networkshigh-dimensional chaossymmetry breakingrandom matrix theorydirected networksneural population dynamics
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The pith

Directed sparse networks of exact firing rate models produce high-dimensional chaos in both excitatory and inhibitory populations.

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

This paper studies networks of exact firing rate models, also called next-generation neural mass models, where populations are connected by sparse random links that are either all excitatory or all inhibitory and have row-normalized weights. Linear stability analysis around the uniform asynchronous state yields a dispersion relation that ties the growth of spatial patterns to the eigenvalues of the connectivity matrix. Bounds from random matrix theory locate the parameter regions where the uniform state loses stability. In undirected networks only inhibitory cases develop stationary heterogeneous patterns through a winner-takes-all mechanism, whereas directed networks generate high-frequency rhythmic states in both excitatory and inhibitory cases. Direct simulations show these rhythmic states are high-dimensional chaotic attractors whose properties scale extensively with network size.

Core claim

In row-normalized sparse random directed networks the homogeneous fixed point of purely excitatory or purely inhibitory next-generation neural mass models loses stability inside specific parameter intervals, producing high-frequency oscillatory states; numerical integration demonstrates that these states are high-dimensional chaotic attractors with extensive Lyapunov spectra, while the corresponding undirected networks support only stationary symmetry-broken patterns in the inhibitory case.

What carries the argument

The dispersion relation from linear stability analysis of the homogeneous asynchronous state, which maps perturbation growth rates to the eigenvalues of the row-normalized sparse connectivity matrix; random-matrix-theory bounds on those spectra demarcate the instability regions for both directed and undirected topologies.

If this is right

  • Directed connectivity produces high-frequency oscillations even when all synapses are excitatory.
  • The chaotic attractors are extensive, so their dimension increases proportionally with network size.
  • Undirected networks restrict dynamics to stationary heterogeneous patterns only under all-inhibitory coupling.
  • Spectral bounds allow prediction of instability regions without computing every eigenvalue.

Where Pith is reading between the lines

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

  • Directionality in connectivity may be a generic route to complex collective rhythms in large neural populations.
  • The winner-takes-all stationary states in undirected inhibitory networks could underlie simple decision-like computations.
  • Mixed excitatory-inhibitory extensions of the same architecture might display transitions between different chaotic regimes.

Load-bearing premise

The bounds supplied by random matrix theory correctly locate the instability thresholds of the spectra of row-normalized sparse random matrices for both directed and undirected cases.

What would settle it

Direct numerical computation of the full eigenvalue spectrum of a large row-normalized sparse random matrix to test whether the predicted instability boundaries match the observed onset of oscillations, or measurement of whether the Kaplan-Yorke dimension of the simulated attractors grows linearly with the number of populations.

Figures

Figures reproduced from arXiv: 2605.15872 by Pau Clusella.

Figure 1
Figure 1. Figure 1: FIG. 1. Two-parameter bifurcation diagram of the homogeneous sys [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Spectra of row-normalized connectivity matrices. (a,b) Scaling of the radius [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Two-parameter bifurcation diagrams of the coupled system [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Numerical validation of the bifurcation diagram. (a,b) Heatmaps of the spatial variability [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Dynamical landscape in the [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Lyapunov analysis of (1) in directed networks. (a,b,c) Twenty largest Lyapunov exponents computed numerically in a directed random [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (a-c) Average power spectra of variables [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Extensive chaos in system (1). (a,b) Full Lyapunov spectra for [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
read the original abstract

Exact firing rate models, also known as next-generation neural mass models (NG-NMMs), provide a rigorous description of the dynamics of neural populations. While in its simplest form a single population only displays fixed-point activity, multi-population models may display a range of different behaviors. In this work, we study the dynamics of all-excitatory or all-inhibitory NG-NMMs coupled through sparse random networks with row-normalized network topology. Linear stability analysis of the homogeneous states of the system, representing asynchronous neural activity, provides a dispersion relation linking the emergence of spatiotemporal dynamics to the spectra of the connectivity matrix. Using bounds from random matrix theory, we identify the parameter regions where instabilities occur. In undirected networks, only inhibitory systems produce heterogeneous stationary patterns, corresponding to a winner-takes-all mechanism. In directed networks, exotic rhythmic states with high frequencies emerge in both, excitatory and inhibitory systems. Numerical simulations reveal that these hectic oscillatory states correspond to high-dimensional chaos with extensive properties.

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

1 major / 2 minor

Summary. The manuscript analyzes the dynamics of next-generation neural mass models (exact firing rate models) on sparse random networks, considering all-excitatory and all-inhibitory cases in both directed and undirected topologies. Linear stability analysis of the homogeneous asynchronous fixed point produces a dispersion relation tied to the spectrum of the row-normalized connectivity matrix. Random matrix theory bounds are applied to locate instability thresholds. Undirected inhibitory networks yield heterogeneous stationary patterns via winner-takes-all symmetry breaking, while directed networks produce exotic high-frequency rhythmic states in both excitatory and inhibitory cases. Numerical simulations are presented as evidence that these states are high-dimensional chaotic attractors possessing extensive properties.

Significance. If the numerical results rigorously establish high-dimensional extensive chaos, the work would contribute to understanding how microscopic population models generate macroscopic chaotic dynamics in sparse networks, with potential relevance to cortical rhythms and symmetry breaking. The use of random matrix theory to obtain explicit instability bounds without free parameters is a methodological strength, and the focus on row-normalized sparse topologies aligns with biologically motivated connectivity.

major comments (1)
  1. [Numerical results section] Numerical results section: the claim that the hectic oscillatory states are high-dimensional chaos with extensive properties rests on qualitative descriptions of irregular time series and power spectra. To support extensivity, the manuscript must demonstrate that the number of positive Lyapunov exponents (or the Kaplan-Yorke dimension) scales linearly with network size N; without the full spectrum or finite-size scaling analysis, the distinction from low-dimensional chaos or long transients is not established and the central numerical claim remains unconfirmed.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'high frequencies' for the rhythmic states should be quantified (e.g., relative to the intrinsic frequency of an isolated population) to allow readers to assess the claimed exotic character.
  2. [Linear stability analysis] Linear stability analysis: specify whether the dispersion relation is derived in the thermodynamic limit or includes 1/N corrections, given that sparse random matrices exhibit pronounced finite-size fluctuations that could shift the instability boundaries.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. The major comment identifies a valid limitation in the current numerical evidence for high-dimensional extensive chaos. We address this point directly below and outline the revisions we will implement.

read point-by-point responses

  1. Referee: [Numerical results section] Numerical results section: the claim that the hectic oscillatory states are high-dimensional chaos with extensive properties rests on qualitative descriptions of irregular time series and power spectra. To support extensivity, the manuscript must demonstrate that the number of positive Lyapunov exponents (or the Kaplan-Yorke dimension) scales linearly with network size N; without the full spectrum or finite-size scaling analysis, the distinction from low-dimensional chaos or long transients is not established and the central numerical claim remains unconfirmed.

    Authors: We agree that the existing numerical results rely primarily on qualitative observations of irregular time series and power spectra, which is insufficient to rigorously establish extensivity or to exclude low-dimensional chaos and long transients. In the revised manuscript we will add a finite-size scaling analysis of the Lyapunov spectrum. Specifically, we will compute the full spectrum of Lyapunov exponents for several network sizes N and demonstrate that the number of positive exponents (or the Kaplan-Yorke dimension) scales linearly with N. This quantitative evidence will confirm the extensive character of the chaotic attractors and strengthen the central numerical claim. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected; derivations rely on independent standard methods

full rationale

The paper's core chain applies linear stability analysis to homogeneous states to obtain a dispersion relation tied to the connectivity matrix spectra, then invokes bounds from random matrix theory to delineate instability regions. These steps draw on external, pre-existing mathematical tools rather than reducing to self-defined quantities or parameters fitted within the paper. Numerical simulations are presented as direct observations of chaotic states, without any described reduction of predictions to fitted inputs or self-citation chains that would force the high-dimensional extensive chaos claim by construction. The approach remains self-contained against external benchmarks such as standard RMT and stability theory, with no load-bearing self-citations or ansatz smuggling evident from the provided derivation outline.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on linear stability analysis of homogeneous states and application of random matrix theory bounds to connectivity spectra; no free parameters or invented entities are explicitly introduced in the abstract description.

axioms (2)
  • standard math Linear stability analysis provides a dispersion relation linking spatiotemporal dynamics to the spectra of the connectivity matrix.
    Invoked in the abstract as the basis for identifying instabilities in homogeneous states.
  • domain assumption Bounds from random matrix theory can be used to identify parameter regions where instabilities occur in sparse random networks.
    Directly stated in the abstract for both undirected and directed cases.

pith-pipeline@v0.9.0 · 5708 in / 1273 out tokens · 36199 ms · 2026-05-19T17:30:22.760909+00:00 · methodology

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Works this paper leans on

91 extracted references · 91 canonical work pages · 1 internal anchor

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