Learning the Brain's Dynamics as a Port-Hamiltonian System: A GNN-Surrogate Metriplectic Twin for Non-Equilibrium Cortical Dynamics and Closed-Loop Neuromodulation

We model human motor cortex, recorded during rest and motor-imagery BCI conditions, as a port-Hamiltonian system: a conservative interconnection (skew-symmetric coupling between band-limited neural phasors) together with a dissipative port whose state-dependent decay is set by a graph-neural-network surrogate. The Hamiltonian is resolved into five interpretable frequency sub-energies, and a phase-locking prior measured from the recordings gates the learned functional connectome so that coupling is admitted only where phase coherence is present. A metriplectic formulation places the resting cortex at a non-equilibrium steady state sustained by an explicit metabolic port, with a fluctuation-dissipation-consistent noise channel governed by a single arousal temperature. Fitting the model to 'FitTrainN' phasor samples from the PhysioNet EEG Motor Movement/Imagery database, under a leakage-free split with three subjects held out entirely, yields a held-out kinematic reconstruction error of 'FitTestMSE' that is stable across random seeds. We then score the free-running model against model-independent dynamical invariants it did not author: it reproduces near-critical avalanche branching (σ\approx1) but not yet the aperiodic 1/f spectral slope or the long-range temporal correlations of real cortex a concrete, falsifiable gap that we trace to specific, testable upgrades. The port-Hamiltonian structure supplies neuroanatomically grounded stimulation ports with stability guarantees, positioning the model as a physically principled, structure-preserving substrate for closed-loop neuromodulation.