Multi-Timescale Reduction of a Neuromodulator-Gated Plasticity Model

This work presents a multi-timescale model of cortico-subcortical brain circuits spanning fast neural firing to slow synaptic changes occurring over hours, which is rigorously reduced step by step through geometric singular perturbation theory and center manifold theory so that at each stage mathematical theorems justify why slower variables are treated as fixed and faster variables are captured by them. The analysis reveals that an effective arousal parameter governs circuit behavior, supporting a bistable switching regime at moderate levels where the brain can flexibly transition between low and high activity states, while at extreme levels the system collapses into locked hypo-arousal or hyper-arousal regimes where only one stable state remains, and because hysteresis is present, once a locked state is entered simply returning arousal to moderate levels does not restore flexibility and an active perturbation is required to cross back. Three universal scaling laws are derived from the reduced model: the transition between states becomes increasingly sharp as the critical threshold is approached, recovery time from small perturbations grows without bound to serve as an early warning signal for impending rigidity, and synaptic consolidation accelerates when the circuit is trapped in the high-activity locked state because both neural firing and neuromodulatory gating remain saturated. These predictions are confirmed numerically under noise, sigmoidal nonlinearities, and slow adaptation to demonstrate structural stability, and the framework offers a principled mechanism for how chronic stress or trauma might narrow the window of healthy bistability and push the brain into rigid, perseverative states relevant to post-traumatic stress disorder and obsessive-compulsive disorder, while additionally the critical slowing down signature suggests a blueprint for real-time rigidity monitors in artificial continual-learning systems.