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arxiv: 2510.23688 · v2 · submitted 2025-10-27 · ⚛️ physics.soc-ph · nlin.AO

Non-Markovian Collective Motion from Self-Regulated Perceptual Dynamics

Jyotiranjan Beuria This is my paper

Pith reviewed 2026-05-18 03:13 UTC · model grok-4.3

classification ⚛️ physics.soc-ph nlin.AO
keywords collective motionnon-Markoviantwo-timescale modelactive matterinternal feedbackhysteresisVicsek model
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The pith

Slow internal regulation in each agent generates non-Markovian collective motion through feedback from past alignments.

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

This paper presents a minimal model in which agents align with neighbors using a fast perceptual register while a slower internal variable integrates recent alignment history and feeds it back to influence future decisions. In the limit of fast relaxation or weak feedback, the model recovers standard Vicsek-type collective motion without memory. With active slow feedback, simulations instead show relaxation on two timescales, hysteresis in the order parameter, memory-dependent areas of hysteresis loops, and a non-monotonic link between group order and the internal regulatory level. Readers should care because real agents in nature and technology often carry such internal states that introduce history dependence, meaning purely instantaneous alignment rules may miss key features of group behavior.

Core claim

Each agent possesses a fast perceptual register encoding the tendency to align with neighbors and a slow regulatory variable that integrates recent alignment states before feeding this information back into alignment decisions. Using a positivity-preserving effective description for the internal states, the model produces non-Markovian collective motion when slow feedback operates, including slow-fast relaxation, feedback-induced hysteresis with finite memory-dependent loop area, and non-monotonic coordination between collective order and regulatory tone.

What carries the argument

The two-timescale model with a slow regulatory variable that integrates alignment history and provides feedback to alignment decisions.

If this is right

  • The model exhibits slow-fast relaxation dynamics when the regulatory variable is active.
  • Feedback induces hysteresis in the collective order parameter.
  • The area of the hysteresis loop depends on the memory time scale.
  • Coordination between collective order and regulatory tone is non-monotonic.
  • In the fast-relaxation weak-feedback limit, it reduces to standard Vicsek alignment.

Where Pith is reading between the lines

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

  • If real biological groups have similar internal regulatory processes, their collective motion should display history dependence even without explicit long-range interactions.
  • Engineered swarms could be designed with tunable internal feedback to control memory and hysteresis in their behavior.
  • Experimental tests could involve introducing controlled delays in agent responses and measuring resulting loop areas in order parameter plots.

Load-bearing premise

The slow regulatory variable integrates recent alignment states and feeds this back to alignment decisions in a manner that produces the reported non-Markovian signatures without further unstated rules.

What would settle it

If the hysteresis and memory-dependent loop areas disappear when the slow variable is removed or its integration time is set to zero, while other dynamics remain, that would confirm the role of the internal feedback.

Figures

Figures reproduced from arXiv: 2510.23688 by Jyotiranjan Beuria.

Figure 1
Figure 1. Figure 1: Self-Perception interaction framework that leads to macroscopic alignment and col [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The temporal evolution of the collective order parameter or perceptual coherence [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The evolution of normalized covariance CMS and the phase lag ΦMS between the macroscopic self variable S(t) and the collective order parameter M(t) with the variation of the feedback strength λfb. increases, both quantities exhibit a nonmonotonic trend: moderate feedback enhances corre￾lation and reduces the phase lag, indicating improved tracking of perceptual dynamics by the self variable. However, even … view at source ↗
Figure 4
Figure 4. Figure 4: Hysteresis-like relationship between the equilibrium perceptual coherence [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
read the original abstract

Collective motion in active matter is usually modelled through instantaneous local alignment, where each agent updates its heading from the current configuration of its neighbours. Many biological and engineered agents, however, possess internal regulatory variables that evolve more slowly than alignment itself and can store information about past alignment states. We introduce a minimal two-timescale model in which each agent carries a fast perceptual register and a slow regulatory variable. The fast register encodes the instantaneous tendency to align with neighbouring headings, while the slow variable integrates recent alignment and feeds back into subsequent alignment decisions. The internal dynamics are formulated using a GKSL-derived Bloch representation, used only as a positivity-preserving effective description of bounded two-state variables; no microscopic quantum dynamics is assumed. The model reduces to Vicsek-type alignment in the fast-relaxation, weak-feedback limit, but shows distinct behaviour when slow feedback is active. Simulations reveal slow-fast relaxation, feedback-induced hysteresis, finite memory-dependent loop area, and non-monotonic coordination between collective order and regulatory tone. These results show how effective non-Markovian collective motion can emerge from local internal feedback.

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 introduces a minimal two-timescale model for collective motion in which each agent has a fast perceptual register encoding instantaneous alignment tendency and a slow regulatory variable that integrates recent alignment states and feeds back into alignment decisions. Internal dynamics use a GKSL-derived Bloch representation solely as a positivity-preserving effective description of bounded variables. The model reduces to Vicsek-type alignment in the fast-relaxation, weak-feedback limit but simulations show slow-fast relaxation, feedback-induced hysteresis with memory-dependent loop area, and non-monotonic coordination between collective order and regulatory tone, indicating emergence of effective non-Markovian collective motion from local internal feedback.

Significance. If the reported behaviors prove robust, the work would be significant for active matter by showing how internal regulatory variables can generate memory effects and non-Markovian signatures without explicit long-range or history terms, extending standard Vicsek models in a controlled way. The explicit reduction to Vicsek in the appropriate limit is a clear strength, as it is a direct consequence of the stated equations rather than a fitted outcome. The Bloch representation choice for effective bounded dynamics is a reasonable modeling device.

major comments (2)
  1. Abstract, paragraph describing the two-timescale model: the central claim that the slow regulatory variable's integration of recent alignment states plus feedback produces effective non-Markovian signatures (hysteresis, finite memory-dependent loop area, non-monotonic order-regulatory coordination) requires the precise update rules, coupling function, and normalizations in the GKSL-derived Bloch map to be specified; without them it is unclear whether these behaviors are generic to local internal feedback or artifacts of particular implementation choices.
  2. Simulations section: the reported outcomes for hysteresis and non-monotonic coordination provide no quantitative error bars, parameter tables, or explicit checks that the behaviors survive changes in integration rules or neighbor definitions, which is necessary to support that they constitute distinct behaviour beyond the Vicsek limit.
minor comments (2)
  1. The free parameters (timescale separation ratio and feedback coupling strength) should be listed with their ranges and default values in a dedicated table to aid reproducibility.
  2. Clarify in the model section whether the Bloch representation introduces any additional normalizations beyond positivity preservation, and how these affect the feedback mapping.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the positive assessment of the work's significance and for the detailed comments that will help improve the manuscript. We address the major comments point by point below, indicating the revisions we plan to implement.

read point-by-point responses

  1. Referee: Abstract, paragraph describing the two-timescale model: the central claim that the slow regulatory variable's integration of recent alignment states plus feedback produces effective non-Markovian signatures (hysteresis, finite memory-dependent loop area, non-monotonic order-regulatory coordination) requires the precise update rules, coupling function, and normalizations in the GKSL-derived Bloch map to be specified; without them it is unclear whether these behaviors are generic to local internal feedback or artifacts of particular implementation choices.

    Authors: We agree that the abstract, being a concise summary, does not include the full mathematical details. However, the manuscript's Section II provides the explicit update rules for the fast perceptual register (using the Bloch vector representation derived from the GKSL equation) and the slow regulatory variable, including the coupling function f(·) and normalization constants that ensure positivity and boundedness. To make this clearer in the abstract, we will revise it to briefly state that the internal dynamics follow a positivity-preserving Bloch map with specific relaxation rates and feedback strength, and we will add a sentence noting that the non-Markovian effects arise generically from the separation of timescales and the feedback loop, as confirmed by the reduction to the Vicsek model in the appropriate limit. This revision will be made. revision: yes

  2. Referee: Simulations section: the reported outcomes for hysteresis and non-monotonic coordination provide no quantitative error bars, parameter tables, or explicit checks that the behaviors survive changes in integration rules or neighbor definitions, which is necessary to support that they constitute distinct behaviour beyond the Vicsek limit.

    Authors: The referee is correct that the current version of the simulations section lacks error bars and robustness checks. We will revise the manuscript to include: (i) error bars computed from 10 independent runs with different random seeds for each parameter set; (ii) a table listing all key parameters (e.g., alignment strength, feedback rate, relaxation times, system size, density); (iii) additional figures or text demonstrating that the hysteresis loops and non-monotonic coordination persist under variations in the numerical integration scheme (e.g., Euler vs. Runge-Kutta) and neighbor definitions (metric distance vs. fixed number of nearest neighbors). These additions will confirm that the observed non-Markovian signatures are robust features of the model rather than numerical artifacts. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained: explicit rules yield emergent signatures without reduction to inputs

full rationale

The paper defines a two-timescale model via explicit dynamical equations for the fast perceptual register and slow regulatory variable, using a GKSL-derived Bloch map solely as a positivity-preserving effective description. The reduction to Vicsek alignment is stated as a direct mathematical limit of those equations in the fast-relaxation weak-feedback regime. Reported behaviors (hysteresis, memory-dependent loop area, non-monotonic coordination) are obtained from numerical integration of the stated update rules rather than from any fitted parameter or self-referential prediction. No self-citations are invoked as load-bearing uniqueness theorems, and no ansatz is smuggled via prior work. The central claim therefore rests on the independent content of the model equations and their simulation outputs, not on any circular re-expression of the inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 1 invented entities

The central claim rests on introducing a slow regulatory variable whose integration rule is not fully specified in the abstract and on treating a GKSL-derived Bloch representation as a positivity-preserving effective description without microscopic justification.

free parameters (2)
  • timescale separation ratio
    Ratio of fast perceptual relaxation to slow regulatory evolution; controls the separation between the two timescales.
  • feedback coupling strength
    Strength with which the slow variable modulates subsequent alignment decisions.
axioms (1)
  • domain assumption The internal state dynamics admit a positivity-preserving Bloch representation derived from GKSL form, used solely as an effective description of bounded two-state variables.
    Invoked to keep the regulatory variable bounded without assuming actual quantum evolution.
invented entities (1)
  • slow regulatory variable no independent evidence
    purpose: Stores integrated information about recent alignment states and provides feedback to alignment decisions.
    Postulated to generate memory-dependent non-Markovian effects; no independent falsifiable prediction outside the model is given.

pith-pipeline@v0.9.0 · 5718 in / 1554 out tokens · 36075 ms · 2026-05-18T03:13:22.967330+00:00 · methodology

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