Non-Markovian Collective Motion from Self-Regulated Perceptual Dynamics

Technical Summary: Non-Markovian Collective Motion from Self-Regulated Perceptual Dynamics

Problem Statement
Collective motion in active matter is traditionally modeled using instantaneous local alignment rules, such as the Vicsek model, where agents update their headings based solely on the current configuration of neighbors. These models assume no internal memory; the alignment decision is purely reactive to the present state. However, empirical evidence from biological systems (e.g., starling flocks) and engineered swarms suggests that agents often possess internal regulatory variables—such as behavioral persistence, attention, or controller gain—that evolve on timescales slower than the alignment dynamics. These variables can store information about past alignment states, potentially introducing non-Markovian (history-dependent) behavior into the collective. The paper addresses the question of how such slow internal feedback mechanisms can generate effective non-Markovian collective motion without imposing explicit time delays or phenomenological memory kernels.

Methodology
The authors introduce a minimal two-timescale model for active agents moving in a two-dimensional periodic domain. Each agent is equipped with two coupled internal registers:

1. Fast Perceptual Register: Encodes the instantaneous tendency to align with specific neighbors. This is modeled as a set of two-state channels.
2. Slow Regulatory Register: An internal state variable that integrates the recent history of the fast register and feeds back to bias future alignment decisions.

The internal dynamics are formulated using an open-system operator representation derived from the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) generator. The authors explicitly state that this formalism is used solely as a positivity-preserving effective description for bounded two-state variables, not to assume microscopic quantum dynamics. The system is described by a reduced density operator $\rho_i$ for each agent, which is projected onto a mean-field Bloch representation to derive coupled ordinary differential equations for the Bloch components ($m^z$ for alignment bias, $s$ for the slow regulatory tone).

The physical motion is determined by a heading map where the agent's desired direction is a weighted average of its current heading and the headings of its neighbors. The alignment weights are dynamically generated by the longitudinal component of the fast perceptual register, which is modulated by the slow regulatory variable.

Key Contributions

• Two-Timescale Architecture: The paper proposes a mechanism where memory emerges from the coupling between fast perceptual dynamics and a slow regulatory variable, rather than from explicit time delays in the alignment rule.
• Operator-Algebraic Framework: It adapts the GKSL formalism, typically used in open quantum systems, to describe classical active matter with bounded internal states, providing a compact algebraic framework for switching, relaxation, and noise.
• Non-Markovian Emergence: The study demonstrates that while the full system (including the slow variable) is Markovian in an enlarged state space, the effective dynamics of the collective order parameter become non-Markovian when the slow variable is not explicitly observed, manifesting as history dependence.

Numerical Results
Simulations of the model reveal four distinct signatures of internal memory:

1. Slow–Fast Relaxation: The perceptual bias and polar order parameter ($M$) rise rapidly, while the slow regulatory tone ($S$) evolves on a significantly longer timescale. Agents achieve directional order before their internal memory variable fully equilibrates.
2. Feedback-Induced Hysteresis: When the feedback strength ($\lambda_{fb}$) is cyclically swept up and down, the stationary polar order follows different branches. The system retains a memory of its previous regulatory configuration, maintaining a high-order state even when feedback is reduced, provided the slow variable has not fully relaxed.
3. Finite Hysteresis Area: The area between the forward and backward branches in the order parameter vs. feedback strength plot remains finite across a broad range of slow relaxation rates ($\gamma_s$). This area serves as a protocol-dependent measure of history dependence, confirming that the slow variable stores sufficient history to alter collective outcomes.
4. Non-Monotonic Coordination: The covariance and phase lag between the polar order ($M$) and the regulatory tone ($S$) exhibit non-monotonic behavior with respect to feedback strength. An intermediate feedback regime is identified where the dynamical locking between collective order and internal regulation is strongest.

Significance and Claims
The paper claims that effective non-Markovian collective motion can emerge naturally from local internal feedback loops without requiring explicit delayed alignment rules or imposed memory kernels. The slow internal regulatory variable acts as a memory channel, storing the history of fast perceptual alignment and biasing future decisions. This mechanism explains how collective systems can exhibit hysteresis and persistence of ordered states.

The authors position this work as a minimal extension of the Vicsek class, where the instantaneous alignment weight is replaced by a dynamical internal variable. They emphasize that the model is "substrate-neutral," applicable to biological collectives (e.g., behavioral persistence) and engineered swarms (e.g., adaptive controller gains), though quantitative calibration would be system-specific. The study concludes that internal feedback timescales can serve as control parameters for collective order, offering a new route to understanding memory-dependent self-organization in active matter.