The Neuro Stability Number (NSN) Framework: Non-Equilibrium Thermodynamics, Codimension-2 Bifurcations, and Optimal Control of Proteopathic Degeneration.

Abstract

Proteopathic neurodegeneration, post-traumatic axonal collapse, and chronic ischemic degradation are conventionally categorized as distinct biochemical etiologies and managed via single-target molecular therapies. Here, we present a unified mathematical framework that transcends the static, lesion-centric paradigm by treating the neuron as an open, non-equilibrium thermodynamic system governed by gradient flow dynamics. We derive the fundamental balance equations of neurodegeneration directly from a non-equilibrium Landau-Ginzburg free-energy functional, where the rate of toxic waste accumulation serves as a macroscopic order parameter. By factoring in a time-dependent, phase-sensitive glymphatic mobility operator, we demonstrate that chronic sleep-phase desynchronization induces a continuous parameter drift culminating in a codimension-2 cusp (assembly) catastrophe. Under this framework, the transition from healthy homeostasis to irreversible neurodegeneration is characterized as a structural saddle-node bifurcation. To bridge acute and ischemic pathologies, we extend the model into a spatiotemporal continuum via partial differential equations incorporating rank-2 tensor diffusion and Navier-Stokes-derived advective glymphatic bulk flows. Traumatic Brain Injury (TBI) is mathematically formulated as a discontinuous phase-space displacement driven by Dirac delta distributional impulses coupled with Heaviside-modulated mobility collapses. Chronic Ischemic Hypoxia (CIH) is modeled as an energy-starvation parametric translation of the bifurcation surface itself, driven by local perfusion deficits. Finally, we translate the empirical 30-day "Resonant Reversal" protocol into a rigorous variational problem using Pontryagin's Maximum Principle, proving that a synchronized, multi-channel control schedule can overcome thermodynamic hysteresis and force the pathological state back into the basin of healthy attraction. The model is evaluated through synthetic numerical demonstrations, and an empirical validation framework is proposed for future testing against ADNI and TRACK-TBI cohorts along with three explicit, falsifiable Popperian predictions.