Critical Scaling and Metabolic Regulation in a Ginzburg--Landau Theory of Cognitive Dynamics

Abstract

We formulate a phenomenological effective field theory in which biological intelligence emerges as a macroscopic order parameter sustained by continuous metabolic flux. By modeling cognition as a coarse-grained neural activity field governed by a variational free energy, we derive closed-form expressions for information capacity and structural susceptibility using a Gaussian maximum entropy approximation. The theory predicts a universal algebraic divergence of the susceptibility, $χ\sim K^{-3/2}$, as the structural stiffness $K$ approaches the instability threshold. The exponent $γ= 3/2$ is consistent with the mean-field branching process universality class, thereby providing a theoretical rationale for the observed avalanche size exponent $τ\approx 3/2$ in cortical dynamics without invoking microscopic equivalence. We identify adult cognition as a metabolically pinned non-equilibrium steady state maintained near the critical regime $Γ\equiv K/α\approx 1$ by continuous metabolic regulation, while pathological decline corresponds to a delocalization transition triggered by the violation of structural stability conditions. The framework generates concrete, falsifiable predictions for attention scaling, altered states of consciousness, and transcranial magnetic stimulation responses, each of which can be tested against existing neuroimaging and electrophysiological datasets.

Figures (4)

• Figure 1: Phase diagram of cognitive states. The $(\alpha,K)$ plane separates fluid ($\Gamma\ll1$), rigid ($\Gamma\gg1$), and critical ($\Gamma\approx1$) regimes. Color indicates $\log_{10}\chi$, revealing maximal susceptibility near the metabolically pinned ridge.
• Figure 2: Universal thermodynamic response of the cognitive field. (a) Structural susceptibility $\chi(K)$ at fixed cognitive temperature $\alpha$, demonstrating the universal power-law divergence $\chi \sim K^{-3/2}$ as structural stiffness approaches the instability threshold. (b) Information capacity $C(K)$, exhibiting sublinear scaling $C \sim \sqrt{K/\alpha}$ and saturating despite continued structural growth. (c) Three-dimensional surface $C(\alpha,K)$, revealing a ridge of nearly constant capacity along trajectories of constant $\Gamma = K/\alpha$, identifying the metabolically pinned regime of adult cognition. (d) Surface plot of $\chi(\alpha,K)$ highlighting the approach to the delocalization boundary as $K \rightarrow K_c$.
• Figure 3: Metabolic trade-off underlying attention. Competing $1/L^2$ and $L^2$ contributions define a stable orbit at $L^*$, balancing exploration and exploitation.
• Figure 4: Cognitive life-cycle trajectory in $(\alpha, K)$ phase space. Black trajectory (Development): Transition from the fluid phase (infancy: high $\alpha$, low $K$) toward the critical line (adulthood), representing developmental annealing where structure accumulates while noise decreases. Blue trajectory (Normal Aging): Crystallization pathway where $K$ continues to increase while $\alpha$ stagnates, maintaining structural capacity at the cost of plasticity. Red trajectory (Dementia): Topological phase collapse where both $K$ and $\alpha$ decrease toward the origin, triggered when metabolic supply fails to maintain the critical state. The yellow star marks the bifurcation point where the system diverges into either normal aging or pathological collapse, determined by whether metabolic flux $J$ can sustain structural integrity above the critical threshold $K > K_c$.