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

Jul 16, 2026

Models human motor cortex in a wrist-extension BCI task using a port-Hamiltonian system that combines conservative gyroscopic coupling between neural phasors with a dissipative power-law energy decay represented by a GNN surrogate. A metriplectic integrator advances the phasor state while fluctuation–dissipation-consistent noise yields stochastic trajectories at body temperature. Trained on 1,109,250 real EEG cycles, the method reaches a test MSE of 1.30×10−4 and exhibits near-criticality, 1/f spectra, and long-range DFA correlations, enabling structure-preserving closed-loop neuromodulation signals.

LanguageEnglish

CategoryResearch & Report

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Pages30

Reading Time1 h 16 min

Abstract

Introduction

Learning the Brain’s Dynamics as a Port-Hamiltonian System: A GNN-Surrogate Metriplectic Twin for Non-Equilibrium Cortical Dynamics and Closed-Loop Neuromodulation Dibakar Sigdel 1, ∗ 1 Mindverse Computing LLC, Lynnwood, WA 98087 (Dated: July 14, 2026) arXiv :2607 . 10439v1 [ q-bio .NC] 11 Jul 2026 Abstract We model human motor cortex during a wrist-extension BCI task as a port-Hamiltonian system (pHS): a conservative interconnection (gyroscopic coupling between neural phasors) plus a dissipative port (power-law energy decay driven by a GNN surrogate) . A metriplectic integrator evolves thephasor state; a Fluctuation–Dissipation-consistent noise channel produces stochastic trajectories at body temperature. Training on 1,109,250 real EEG cycles (PhysioNet EEGMMIDB, 3 held-out subjects) reaches a test MSE of 1 .30 × 10−4 and passes three scale-free criticality rungs: near-critical branching ratio (σ ≈ 1), 1/f power-law spectrum, and long-range DFA correlations. The model generates closed-loop neuromodulation signals that restore phase-locking in silico when applied to de-synchronised inputs, suggesting a path toward structure-preserving BCI decoders. I. INTRODUCTION The human brain is a composite physical medium governed by the principles of energy conservation and thermodynamic dissipation. At the macroscopic scale, cortical activity—as captured by high-density Electroencephalography (EEG)—manifests as richly structured oscillations spanning five canonical frequency bands: Delta (δ, 1–4Hz), Theta (θ, 4–8Hz), Alpha (α, 8–12Hz), Beta (β, 12–30Hz), and Gamma (γ, 30–80Hz) . These are not passive spectral features. They are the electromagnetic signatures of self-sustained cortical limit cycles through which neural assemblies coordinate information transfer, consolidate working memory, and regulate attentional gating [1] . The dominant paradigm for Brain-Computer Interface (BCI) signal processing relies on discriminative machine learning architectures—Convolutional Neural Networks, Support Vector Machines, or EEGNet variants — trained to classify categorical cognitive states from raw EEG time-series [2] . These approaches treat the brain as a black box, absorbing no structural knowledge of cortical dynamics and producing no mechanistic insight. They fail to extrapolate to novel stimulation conditions, cannot predict the effect of therapeutic intervention, and offer no thermodynamic interpretation of neurological disorder. A neurophysiologically faithful model must instead treat each cortical region as a nonlinear oscillator whose dynamics are constrained by the laws of classical thermodynamics. This ∗ [devdeep137@gmail.com](devdeep137@gmail.com) demands three structural requirements: (i) a phasor coordinate system that captures the limit-cycle nature of EEG rhythms; (ii) a dissipative Hamiltonian structure that guarantees energy conservation in the absence of stimulation and bounded energy injection under external drive; and (iii) a state-dependent functional connectome that rewires dynamically as the cortex transitions between cognitive states. Learning continuous-time dynamics with an embedded energy structure is by now established: Hamiltonian Neural Networks recover conservative dynamics from data [3], and dissipative extensions incorporate friction and external control [4] . The Port-Hamiltonian (pH) framework [5] provides the exact mathematical language for these requirements, unifying conservative routing, dissipation, and actuation in a single structure. The continuous-time dynamics x˙ = 􀀂J (x) − R(x)􀀃 ∇xH (x) + Gu(t) (1) encode conservative routing (J, skew-symmetric), irreversible dissipation (R, positive semidefinite), and structured external intervention (Gu) . The resulting power balance, H˙ = −∇H⊤ R∇H + y⊤ u (t) ≤ y⊤ u (t) , (2) states that without external stimulation the regulatory energy H is non-increasing. Strict passivity is, however, only a first approximation to cortical thermodyn

FAQs

What port-Hamiltonian components are used to model cortical motor dynamics in the task?

The model uses a conservative interconnection represented by gyroscopic coupling between neural phasors and a dissipative port where energy decays via a power-law term implemented through a GNN surrogate.

How does the approach produce stochastic trajectories during simulation?

It adds a noise channel consistent with the fluctuation–dissipation principle, generating stochastic trajectories at body temperature.

What results demonstrate that the learned dynamics match non-equilibrium cortical criticality?

Training achieves low test MSE and passes three scale-free criticality checks: a near-critical branching ratio (σ≈1), a 1/f power-law spectrum, and long-range DFA correlations.

Jul 16, 2026

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