Multi-Timescale Reduction of a Neuromodulator-Gated Plasticity Model Reveals Saddle-Node Bifurcations, Hysteresis, and Critical Scaling Laws

Dynamical systems with widely separated timescales are ubiquitous in neural circuits, yet their high dimensionality often obscures the underlying bifurcation structure. This work presents a 15-dimensional cortico-subcortical model that explicitly separates four intrinsic timescales-fast thalamic and amygdalar firing (submillisecond), fast dopamine clearance (millisecond), slower prefrontal and serotonergic/noradrenergic dynamics (seconds), and very slow Hebbian synaptic plasticity (hours)-and rigorously reduces it to a one-dimensional normal form by means of geometric singular perturbation theory and center manifold theory. The reduction cascade, 15D → 6D → 4D → 2D → 1D, is justified at each step by Fenichel's persistence theorem or the implicit function theorem. The resulting core equation Ė = -E + tanh(2E + β), where β is an effective arousal parameter, exhibits two non-degenerate saddle-node bifurcations at β_crit^± ≈ ±0.5328 that partition the dynamics into low-activity locked, bistable switching, and high-activity locked regimes with inherent hysteresis. Three universal scaling laws are derived analytically from the normal form: (i) the state jump width scales as |β_crit -β|; (ii) the recovery time diverges as |β_crit -β| -1/2 (critical slowing down); and (iii) synaptic consolidation accelerates in the saturated regime due to simultaneous saturation of neural activity and neuromodulatory gating. These predictions are confirmed numerically to persist under additive noise, sigmoidal dopamine nonlinearities, and slow adaptation, demonstrating the structural stability of the reduction. The pathway-specific neuromodulatory gating matrix G(M ), which implements experimentally observed selectivity of dopamine, serotonin, and norepinephrine for cortico-subcortical pathways, is shown to be preserved through the reduction. The results provide a mathematically principled mechanism for arousal-dependent loss of cognitive flexibility, with testable predictions for prefrontal dynamics in psychiatric conditions involving perseverative states, and suggest a rigidity-monitor blueprint for continual-learning architectures based on real-time detection of critical slowing down.