Self-Organized Criticality from Protected Mean-Field Dynamics: Loop Stability and Internal Renormalization in Reflective Neural Systems

TL;DR

This work demonstrates within the MSRJD field-theoretic framework that fluctuation effects do not destabilize the critical manifold, and shows that response-driven structural adaptation generates intrinsic parameter flows that attract the system toward this surface without external fine tuning.

Abstract

The reflective homeostatic dynamics provides a minimal mechanism for self-organized criticality in neural systems. Starting from a reduced stochastic description, we demonstrate within the MSRJD field-theoretic framework that fluctuation effects do not destabilize the critical manifold. Instead, loop corrections are dynamically regularized by homeostatic curvature, yielding a protected mean-field critical surface that remains marginally stable under coarse-graining. Beyond robustness, we show that response-driven structural adaptation generates intrinsic parameter flows that attract the system toward this surface without external fine tuning. Together, these results unify loop renormalization and adaptive response in a single framework and establish a concrete route to autonomous criticality in reentrant neural dynamics.

Figures (4)

• Figure 1: Reflective homeostatic reentry and unified MSRJD description. (a) Schematic of the reflective homeostatic reentrant network, in which reentrant amplification and fast-weight modulation confine the dynamics to a bounded activity shell. (b) Corresponding MSRJD representation of the reduced stochastic dynamics, showing that both loop renormalization of fluctuations and response-driven structural adaptation are generated by the same action. Together, these mechanisms yield a protected critical manifold that is dynamically selected without external fine tuning.
• Figure 2: Infrared regularization by homeostatic curvature. (a) Bare correlation propagator $G_C(\omega)$ of the Gaussian theory, which diverges in the infrared as $a_1\to 0$. (b) Nonlinear saturation encoded in the curvature $a_3$ generates a finite characteristic amplitude scale, $x_{\mathrm{typ}}^{2}\sim -a_1/a_3,$ which provides an effective infrared cutoff. (c) As a result, the loop integral contributing to $\delta a_1$ is regulated self-consistently, preventing destabilization of the critical surface.
• Figure 3: Response-driven flows in $(\Delta,\kappa)$ space. (a) Loop renormalization preserves a marginally stable critical manifold at $\Delta=0$. (b) Response-driven structural adaptation lifts this degeneracy and dynamically selects a unique edge-of-criticality fixed point, realizing self-organized criticality.
• Figure S1: One-loop correction to the linear coupling. (a) Cubic interaction vertex arising from the nonlinear drift term $\tilde{x} a_3 x^3$ in the MSRJD action. (b) Leading one-loop tadpole diagram contributing to the renormalization of the linear coupling $a_1$. The loop involves the correlation propagator $G_C$, while the external legs correspond to the response field $\tilde{x}$ and the state variable $x$. Diagrammatic contribution is translated into an analytic expression, $\delta a_1 \propto -a_3 \int (d\omega/2\pi)\, G_C(\omega)$, which exhibits a formal infrared divergence as $a_1\to 0$.