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arxiv: 2606.10032 · v1 · pith:BTSHE3T3new · submitted 2026-06-08 · 🌊 nlin.AO · cond-mat.stat-mech· nlin.CD

Collective drift and pinning in active rotator networks with Kuramoto coupling and mixed-sign feedback disorder

Arpan Dey This is my paper

Pith reviewed 2026-06-27 13:51 UTC · model grok-4.3

classification 🌊 nlin.AO cond-mat.stat-mechnlin.CD
keywords active rotatorsKuramoto couplingfeedback disorderpinningcollective driftsynchronization
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The pith

Mixed-sign local feedback alone can tip active rotator networks between pinned and drifting states even with uniform intrinsic drive.

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

The paper studies fully connected active rotator networks under Kuramoto coupling where each unit receives an additional local feedback term whose amplitude is drawn from a zero-mean Gaussian. This produces a direct competition between local pinning forces of either sign and the global tendency toward phase alignment. Numerical regime maps in the disorder-coupling plane show that weak coupling plus strong feedback disorder favors pinning while moderate disorder and stronger coupling can restore net positive drift. The same transitions appear when the common intrinsic drive is replaced by a zero-mean frequency distribution. These observations establish that sign-mixed feedback disorder is sufficient to control the pinning-drift balance without requiring heterogeneous intrinsic frequencies.

Core claim

In active rotator networks with Kuramoto coupling, feedback amplitudes drawn from a zero-mean Gaussian produce a competition between local pinning and collective phase alignment. At weak coupling, increasing feedback disorder strength suppresses late-time drift; stronger coupling restores positive drift when disorder is not too large. These regime boundaries are mapped using mean absolute late-time drift and the fractions of positive and negative drifting oscillators, and the results persist for homogeneous intrinsic drive.

What carries the argument

Competition between local pinning from mixed-sign Gaussian feedback and collective phase alignment from Kuramoto coupling, quantified by mean absolute late-time drift and fractions of drifting oscillators.

If this is right

  • Weak coupling combined with large feedback disorder suppresses collective drift.
  • Increasing coupling strength can overcome moderate feedback disorder and restore net positive drift.
  • The same transitions occur when intrinsic frequencies are replaced by a zero-mean distribution.
  • Finite-size effects do not eliminate the identified regimes.

Where Pith is reading between the lines

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

  • Local feedback sign disorder may serve as a tunable control parameter in other oscillator networks whose connectivity is not all-to-all.
  • The same mechanism could appear in excitable-media models if feedback terms are allowed to take both signs.
  • Testing whether the reported boundaries survive when the feedback distribution is changed from Gaussian would directly test the role of the zero-mean assumption.

Load-bearing premise

Feedback amplitudes are drawn from a zero-mean Gaussian distribution on a fully connected network.

What would settle it

Simulations on a non-fully-connected network or with a non-zero-mean feedback distribution that produce unchanged regime boundaries would show the reported control by mixed-sign disorder does not hold.

Figures

Figures reproduced from arXiv: 2606.10032 by Arpan Dey.

Figure 1
Figure 1. Figure 1: Phase-circle schematics for the local dynamics ˙θ = ω(1 − a sin θ). The panels show the two-fixed-point regime |a| > 1, the critical case |a| = 1, and the drifting regime |a| < 1. Stable, unstable, and critical fixed points are marked for the corresponding positive and negative values of a. This local classification remains useful after restoring the Kuramoto coupling [PITH_FULL_IMAGE:figures/full_fig_p00… view at source ↗
Figure 2
Figure 2. Figure 2: Wrapped phase trajectories for N = 30 fully connected oscillators. The phases are initialized uniformly at random in [0, 2π) and integrated using the Euler method, with ω = 0.5 and K = 0.6. The three panels show a = 1.5, a = 1, and a = 0.5, corresponding to locked, critical, and drifting regimes. In the uncoupled limit, K = 0, the dynamics is described by Eq. (9). For |a| < 1, no fixed point exists and the… view at source ↗
Figure 3
Figure 3. Figure 3: Heatmap of D(r, κ) = ⟨N −1 P i |Ωi |/ω⟩ for N = 100, ω = 0.5, r = σA/ω ∈ [0, 6], and κ = K/ω ∈ [0, 6]. Initial phases were drawn independently and uniformly from [0, 2π). Simulations were performed on a 100 × 100 parameter grid using Euler integration with dt = 0.01 up to T = 1000, corresponding to 105 integration steps. Late-time velocities were computed over the final 30% of each trajectory, and results … view at source ↗
Figure 4
Figure 4. Figure 4: Positive and negative drifting fractions. Main panel: positive drifting fraction f+. Inset: negative drifting fraction f−. Oscillators are classified using tolerance ϵ = 10−3 applied to late-time velocity Ωi/ω. All other simulation parameters are the same as in [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Uncoupled-limit benchmark. Comparison between the analytical prediction D0(r) and numerical simulations at κ = 0. All other simulation parameters are the same as in [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Regime map of D(r, κ) for phases initialized within θi(0) ∈ [130◦ , 140◦ ]. The solid black curve shows the analytical estimate κ = r 2/2, obtained using Θ ≃ 135◦ , and the dashed vertical line marks r = p π/2. All other simulation parameters are the same as in [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Finite-size dependence of drift recovery. Left panel: Mean absolute late-time drift DN (r = 2, κ) as a function of κ for N = 20, 40, 80, 120, 200. Right panel: Drift recovery ∆DN (r) = D high N (r) − Dlow N (r) for the same system sizes, with Dlow N averaged over κ ∈ [0, 0.5] and D high N averaged over κ ∈ [5.5, 6]. The values of r range from 0.5 to 6.0 in steps of 0.5. Negative values of ∆DN (r) at large … view at source ↗
Figure 8
Figure 8. Figure 8: Mean absolute late-time drift DN as a function of N for r = 1, 2, 3, 4, 5, 6. The left panel shows the weak-coupling window κ ∈ [0, 0.5], while the right panel shows the strong-coupling window κ ∈ [5.5, 6], using the same five system sizes as in [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Zero-mean distributed intrinsic frequencies. Left panel: D = ⟨N −1 P i |Ωi |/σω⟩ for intrinsic frequencies ωi ∼ N (0, σ2 ω), with σ 2 ω = 0.5. Right panel: positive drifting fraction f+, with the negative drifting fraction f− shown as an inset. The axes are normalized by σω, giving σA/σω and K/σω. Drift signs are classified using the tolerance ϵ = 10−3 applied to late-time Ωi/σω. All other simulation param… view at source ↗
read the original abstract

Active rotator models provide a minimal phase description of excitable and oscillatory systems, and have long been used to study mutual entrainment, synchronization, and collective transitions. Here, we investigate fully connected active rotator networks with Kuramoto coupling, where a common intrinsic drive competes with local feedback amplitudes drawn from a zero-mean Gaussian distribution. This produces a competition between local pinning and collective phase alignment. Using mean absolute late-time drift and the fractions of positive and negative drifting oscillators, we construct numerical regime maps in the feedback-disorder-coupling plane. At weak coupling, increasing the feedback disorder strength suppresses drift, while stronger coupling can restore positive late-time drift when feedback disorder is not too strong. We interpret these regimes using analytical limits for the uncoupled and coherent strong-coupling cases. We also examine finite-size effects and zero-mean distributed intrinsic frequencies. Together, these results show that mixed-sign local feedback alone can reshape the balance between pinning and drifting in coupled active rotator networks, even when the intrinsic drive is homogeneous.

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 examines fully connected active rotator networks with Kuramoto coupling where a homogeneous intrinsic drive competes with local feedback amplitudes drawn from a zero-mean Gaussian distribution. Numerical regime maps are constructed in the feedback-disorder-coupling plane using observables such as mean absolute late-time drift and fractions of drifting oscillators. At weak coupling, feedback disorder suppresses drift; stronger coupling can restore positive drift when disorder is moderate. Analytical limits for uncoupled and coherent strong-coupling cases are used for interpretation, along with checks on finite-size effects and zero-mean distributed intrinsic frequencies. The central claim is that mixed-sign local feedback alone can reshape the pinning-drifting balance even with homogeneous drive.

Significance. If the numerical results hold, the work demonstrates a concrete mechanism by which local feedback disorder alters collective phase dynamics in active rotator networks, extending beyond standard Kuramoto entrainment. The explicit use of analytical limits for boundary cases and examination of finite-size effects provide a solid foundation for the regime maps. This could inform studies of excitable media and synchronization with heterogeneous feedback.

major comments (2)
  1. [Numerical Methods] Numerical Methods section (or equivalent): The construction of regime maps relies on direct numerical integration, yet no details are provided on the integration scheme (e.g., Euler vs. Runge-Kutta), time step, total integration time, ensemble size over disorder realizations, or convergence diagnostics. These omissions make it impossible to assess the robustness of the reported boundaries between pinning and drifting regimes.
  2. [Results] Results, regime-map figures: The observables (mean absolute late-time drift and drifting fractions) are well-defined, but without error bars, standard deviations across realizations, or explicit thresholds used to classify 'drifting' vs. 'pinned' oscillators, the sharpness of the reported transitions cannot be evaluated.
minor comments (2)
  1. [Abstract] Abstract and introduction: The phrase 'mixed-sign local feedback alone' is clear in context but could be rephrased for precision to emphasize that the disorder is zero-mean Gaussian and the network is all-to-all.
  2. [Analytical limits] Analytical limits: The strong-coupling coherent state analysis is useful, but a brief derivation sketch or reference to the exact reduction would aid readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and constructive feedback. We will revise the manuscript to address the concerns regarding numerical methods and results presentation.

read point-by-point responses

  1. Referee: [Numerical Methods] Numerical Methods section (or equivalent): The construction of regime maps relies on direct numerical integration, yet no details are provided on the integration scheme (e.g., Euler vs. Runge-Kutta), time step, total integration time, ensemble size over disorder realizations, or convergence diagnostics. These omissions make it impossible to assess the robustness of the reported boundaries between pinning and drifting regimes.

    Authors: We agree with this assessment. The original manuscript omitted these details. In the revised version, we will include a new subsection in the Methods describing the numerical integration scheme (we use a fourth-order Runge-Kutta method with fixed time step of 0.01), total integration time (up to t=1000 after transients), ensemble size (100 disorder realizations), and convergence diagnostics based on the stabilization of the mean drift observable. revision: yes

  2. Referee: [Results] Results, regime-map figures: The observables (mean absolute late-time drift and drifting fractions) are well-defined, but without error bars, standard deviations across realizations, or explicit thresholds used to classify 'drifting' vs 'pinned' oscillators, the sharpness of the reported transitions cannot be evaluated.

    Authors: We concur that error bars and classification thresholds should be provided for clarity. We will add error bars (standard deviation over realizations) to the regime map figures and explicitly state the threshold for classifying an oscillator as drifting (absolute drift velocity exceeding 0.05 in our units). This will allow better evaluation of the transition sharpness. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation rests on explicit numerical construction and independent analytical limits

full rationale

The paper constructs regime maps from direct numerical integration of the model equations using well-defined observables (mean absolute late-time drift, fractions of drifting oscillators). Analytical limits are derived for the uncoupled case and the strong-coupling coherent state; these limits follow from the governing ODEs without parameter fitting or self-referential definitions. No step reduces a claimed prediction to a fitted input by construction, and no load-bearing claim depends on a self-citation chain. The zero-mean Gaussian feedback and all-to-all connectivity are stated as modeling choices, not derived results. This is the standard non-circular case of simulation plus limiting analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities can be extracted beyond the standard model assumptions stated in the summary.

pith-pipeline@v0.9.1-grok · 5705 in / 1011 out tokens · 15552 ms · 2026-06-27T13:51:35.501879+00:00 · methodology

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Reference graph

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