Non-Markovian Collective Motion from Self-Regulated Perceptual Dynamics

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.