Renormalization-Group Geometry of Homeostatically Regulated Reentry Networks

TL;DR

This work introduces a minimal continuous-time formulation of a homeostatically regulated reentrant network (FHRN) and shows that its population dynamics admit an exact reduction to a one-dimensional radial flow.

Abstract

Reentrant computation-recursive self-coupling in which a network continuously reinjects and reinterprets its own internal state-plays a central role in biological cognition but remains poorly characterized in neural network architectures. We introduce a minimal continuous-time formulation of a homeostatically regulated reentrant network (FHRN) and show that its population dynamics admit an exact reduction to a one-dimensional radial flow. This reduction reveals a dynamically fixed threshold for sustained reflective activity and enables a complete renormalization-group (RG) analysis of the reentry-homeostasis interaction. We derive a closed RG system for the parameters governing structural gain, homeostatic stiffness, and reentrant amplification, and show that all trajectories are attracted to a critical surface defined by $γρ=1$, where intrinsic leak and reentrant drive exactly balance. The resulting phase structure comprises quenched, reactive, and reflective regimes and exhibits a mean-field critical onset with universal scaling. Our results provide an RG-theoretic characterization of reflective computation and demonstrate how homeostatic fields stabilize deep reentrant transformations through scale-dependent self-regulation.

Figures (5)

• Figure 1: Homeostatically regulated reentrant dynamics and reflective shell formation. (a) Schematic of the homeostatically regulated reentrant network. External input $x_t^{(\mathrm{ex})}$ is combined with a reentrant signal $\gamma W\, g(\|y_{t-1}\|)\, y_{t-1}$, producing the effective block input $x_t$. The population activity $y_t$ is computed through a fast-weight operator $A_t$, while the reentrant loop feeds the internal state back into the next step with homeostatic gain control. (b) Phase portrait of the continuous-time FHRN dynamics in the reflective regime $\gamma\rho>1$. Trajectories spiral under the reentry operator and converge toward a finite-radius shell, indicating sustained reflective activity. (c) Radial growth-rate map $s(y)=\dot r/r=-1+\gamma\rho\, g(\|y\|)$, where $r=\|y\|$. The white contour $s(y)=0$ identifies the homeostatic reflective shell. Inside the shell ($s>0$) trajectories are radially unstable and expand outward, while outside ($s<0$) they contract inward, rendering the shell globally attracting.
• Figure 2: Renormalization-group flow of the homeostatically regulated reentrant network in the full $(\gamma,\kappa,\rho)$ parameter space. Arrows indicate the normalized RG vector field $(\beta_\gamma,\beta_\kappa,\beta_\rho)$, emphasizing the topology of the flow independently of its local speed. The translucent surface marks the critical manifold $\gamma\rho=1$, which is approached transversely by the RG trajectories and is therefore infrared-attractive. This surface separates quenched and reflective regimes and unifies reentrant amplification, homeostatic regulation, and spectral scaling into a single scale-invariant geometric structure.
• Figure 3: Two-dimensional slices of the RG flow highlighting the phase structure. Arrows denote the normalized RG vector field, emphasizing flow topology rather than local speed. Background colors indicate quenched (blue), reactive (purple), and reflective (yellow) regimes. (a) RG streamlines in the $(\gamma,\kappa)$ plane at fixed $\rho=1$. The vertical red line marks the reflective critical condition $\gamma\rho=1$, while the dashed line $\gamma\rho=1-\kappa$ separates the quenched and reactive regions. Flows approach the critical line transversely, demonstrating its infrared-attractive character. (b) RG streamlines in the $(\gamma,\rho)$ plane at fixed $\kappa$. The red curve $\gamma\rho=1$ defines the reflective manifold, which again acts as an infrared attractor separating quenched and reflective phases.
• Figure 4: Critical scaling of the order parameter $m=r_\infty-1$ near $\gamma_c=1/\rho$. Numerical solutions follow the predicted mean-field exponent $\beta_{\mathrm{op}}=1$ over several decades.
• Figure S1: Dependence of the homeostatic reflective shell on the curvature parameter $\kappa$. Phase portraits of the full FHRN dynamics are shown for fixed reentry gain $\gamma=1.2$ and spectral radius $\rho=1$, while varying the homeostatic curvature $\kappa$. Dashed circles indicate the stationary shell radius $r_\infty^2 = 1 + (\gamma\rho-1)/\kappa$ predicted by the radial theory. Panels (a)--(c) show that decreasing $\kappa$ expands the reflective shell while preserving global convergence and spiral dynamics. Panel (d) illustrates that for larger $\kappa$ the shell contracts but remains stable.