Feedback-Coupled Memory Systems: A Dynamical Model for Adaptive Coordination

Stefano Grassi

cs.MAMar 12, 2026

Feedback-Coupled Memory Systems: A Dynamical Model for Adaptive Coordination

Authors: Stefano Grassi

Organizations: Bangkok University, Phahonyothin Rd, Khlong Nueng, Khlong Luang District, Pathum Thani 12120, Thailand

Abstract

This paper develops a dynamical framework for adaptive coordination in systems of interacting agents referred to here as Feedback-Coupled Memory Systems (FCMS). Instead of framing coordination as equilibrium optimization or agent-centric learning, the model describes a closed-loop interaction between agents, incentives, and a persistent environment. The environment stores accumulated coordination signals, a distributed incentive field transmits them locally, and agents update in response, generating a feedback-driven dynamical system. Three main results are established. First, under dissipativity, the closed-loop system admits a bounded forward-invariant region, ensuring dynamical viability independently of global optimality. Second, when incentives depend on persistent environmental memory, coordination cannot be reduced to a static optimization problem. Third, within the FCMS class, coordination requires a bidirectional coupling in which memory-dependent incentives influence agent updates, while agent behavior reshapes the environmental state. Numerical analysis of a minimal specification identifies a Neimark-Sacker bifurcation at a critical coupling threshold ( $β_c$ ), providing a stability boundary for the system. Near the bifurcation threshold, recovery time diverges and variance increases, yielding a computable early warning signature of coordination breakdown in observable time series. Additional simulations confirm robustness under nonlinear saturation and scalability to populations of up to $N = 10^{6}$ agents making it more relevant for real-world applications. The proposed framework offers a dynamical perspective on coordination in complex systems, with potential extensions to multi-agent systems, networked interactions, and macro-level collective dynamics.

Figures & tables

Symbol	Space	Role
$\mathbf{x}_{t}=(x_{1,t},\dots,x_{N,t})$	$\mathcal{X}\subseteq\mathbb{R}^{d}$	Joint agent state vector
$S_{t}$	$\mathcal{S}\subseteq\mathbb{R}^{m}$	Persistent environmental memory
$\mathbf{G}_{t}=(G_{1,t},\dots,G_{N,t})$	$\mathbb{R}^{N}$	Incentive field
$L_{\mathrm{global}}^{t}=\mathcal{A}(\mathbf{x}_{t},S_{t})$	$\mathbb{R}$	Global coordination signal (derived)
$\Psi$	$\mathcal{S}\times\mathcal{X}\to\mathcal{S}$	Environmental update operator
$\Phi$	$\mathbb{R}\times\mathcal{X}\times\mathcal{S}\to\mathbb{R}^{N}$	Incentive distribution operator

Table 1: Summary of model components and their roles in the FCMS architecture.

Figure 1: Recursive coordination architecture. Shows feedback loops among agents, incentive field, and persistent environment.

Figure 2: Left: Disagreement dynamics $d_{t}$ at three coupling values ( $\beta=0.50$ stable, $\beta=1.41$ near $\beta_{c}$ , $\beta=1.55$ near-critical), showing progressive critical slowing down as $\beta\to\beta_{c}\approx 1.58$ . Right: Dominant eigenvalue magnitude $\lambda(\beta)$ as a function of coupling strength, approaching the unit circle at the Neimark–Sacker threshold $\beta_{c}$ .

Figure 3: Left: Variance of disagreement $d_{t}$ under bounded noise as a function of coupling strength $\beta$ , showing a sharp increase as $\beta\to\beta_{c}$ . Right: Lag-1 autocorrelation of $d_{t}$ , approaching unity near the bifurcation threshold. Both signatures are computable from spectral properties of $J$ and constitute early warning indicators of approaching coordination breakdown.

Figure 4: Phase portrait of the nonlinear system under $\tanh$ saturation. The stable spiral confirms that the coordination manifold $d=0$ remains a locally asymptotically stable attractor. Arrows indicate the direction of flow; the fixed point $(S,d)=(0,0)$ is marked.

Figure 5: Synchronization variance as a function of population size $N$ under the mean-field FCMS specification. Variance decays as $\sim 1/N$ , confirming dimension-invariance of the dissipative-feedback criterion. The scaling connects macroscopic coordination to the statistical mechanics of large coupled systems, with the stability condition $4\eta\beta^{2}<\gamma$ acting as a thermodynamic-like bound on collective fluctuations, analogous to energy constraints in physical systems.

Component	Structural Role
Coupling ( $\beta$ )	Generates collective feedback between agents and environment
Persistence ( $1-\gamma$ )	Generates environmental memory and history sensitivity
Dissipation ( $\gamma$ )	Ensures decay of accumulated signal and contributes to local boundedness

Table 2: Structural Components of Agent-Environment Dynamics

Critical Slowing Down

See all 8 papers

Feedback-Coupled Memory Systems in Continuous Time
Jun 24, 2026Stefano GrassiDynamical Systems Multi-Agent Coordination
Bangkok University, Phahonyothin Rd, Khlong Nueng, Khlong Luang District, Pathum Thani 12120, Thailand
The Axiom of Consent: Authorization, Friction, and Multi-Agent Coordination
Jan 10, 2026Murad FarzullaMulti-Agent Coordination Replicator Dynamics
Dissensus, London, UK · King’s College London, London, UK
Multi-Agent Coordination Adaptation via Structure-Guided Orchestration
May 25, 2026Haoran Li, Shulun Chen, Shaoyuan Sun +1Multi-Agent LLM Systems Multi-Agent System Optimization
Nanjing University · University of Technology Sydney, Sydney, Australia · University of New South Wales, Sydney, Australia

Jun 24, 2026J/K move · Enter open · S save

Feedback-Coupled Memory Systems in Continuous Time

Stefano Grassi

Bangkok University, Phahonyothin Rd, Khlong Nueng, Khlong Luang District, Pathum Thani 12120, Thailand

Dynamical Systems Multi-Agent Coordination Lyapunov Stability Feedback Control

Open paper AskPDF arXiv

The Feedback-Coupled Memory Systems (FCMS) architecture formalizes closed-loop coordination through four abstract operators, two of which - the agent update operator

f_i

and the environmental update operator

Ψ

- are left axiomatically undefined in the original framework. To address this,

f_i

is defined by Mechanism-Based Intelligence (MBI), where agents update locally through a decentralized price mechanism and economic principles, and

Ψ

is defined by the Coupled Memory Graph Process (CMGP), a non-Markovian framework where the environment is treated as a physical substrate that records and responds to trajectory history coherently without external forcing. The resulting continuous-time FCMS instantiation achieves Lyapunov global dissipativity governed by the computable threshold

4β^2 < 2ημγ^2

. This generalizes both the discrete FCMS stability condition

4ηβ^2 < γ

and CMGP's physical bifurcation threshold

α_c = 1/K

, confirming that memory dissipation must outpace feedback gain as a universal organizing principle. Numerical simulation with

N=2

agents and mean-field validation at

N=10^6

confirm the stability threshold and the self-reinforcing coordination cascade that emerges when it is violated.

Abstract

Critical Slowing Down

Explore similar work

Feedback-Coupled Memory Systems in Continuous Time

The Axiom of Consent: Authorization, Friction, and Multi-Agent Coordination

Multi-Agent Coordination Adaptation via Structure-Guided Orchestration