Whole-Brain Hamiltonian System

• The whole-brain Hamiltonian system is a mathematical framework that models large-scale neural dynamics using extended phase spaces, conserved quantities, and symmetry constraints.
• It integrates traditional Hamiltonian mechanics with neural network approaches, tensor generalizations, and complex-valued formulations to enhance empirical modeling in neuroscience.
• The framework facilitates stability analysis and multiscale dynamics by exploiting topological constraints, Casimir invariants, and extended Poisson bracket hierarchies in network neuroscience.

A whole-brain Hamiltonian system refers to a mathematical and physical framework in which large-scale brain dynamics are described as evolution in a phase space governed by Hamiltonian mechanics. This formalism systematically relates conserved quantities, constraints, energy flows, and symmetries in complex biological neural networks, enabling principled modeling of both local and global neural behaviour at scale. Recent advances combine extended phase-space formulations, noncanonical hierarchies, data-driven neural architectures, tensor generalizations, and complex-valued augmentation, revealing both the depth of mathematical structure and potential for empirical application in neuroscience.

1. Extended Phase Space and Redundant Variables

Traditional Hamiltonian dynamics express evolution in terms of canonical variables $(q, p)$ via Hamilton's equations $\dot{q} = \partial H / \partial p$, $\dot{p} = -\partial H / \partial q$, where $H(q, p)$ is the Hamiltonian function corresponding to total (kinetic plus potential) energy. The variant "hidden Nambu" formulation (Horikoshi et al., 2013) generalizes this by embedding the original phase space in a higher-dimensional space, replacing the doublet $(q,p)$ with an $N$-tuple $(x_1, \ldots, x_N)$. This transformation introduces redundant variables and induces constraints $G_b(x_1, \ldots, x_N) = 0$ that restrict the system to reproduce the original physics.

The extended time evolution for any observable $\tilde{f}(x_1, ..., x_N)$ involves Nambu brackets and takes the form

$$\frac{d\tilde{f}}{dt} = \{\tilde{f}, \tilde{H}, \tilde{G}_1, ..., \tilde{G}_{N-2}\}_\text{NB},$$

where multiple "Hamiltonians" (the original and induced constraints) govern the extended dynamics. In this context, constraints are "first class" in the Dirac sense, reflecting symmetries or conservation laws that permeate the brain's network architecture. The extended bracket formalism enables systematic handling of overlapping variables (shared among regions or functional modules), with generalized Nambu equations encompassing a "metric" tensor for mixing variables, supporting modeling of multiplexed, interacting dynamics.

2. Topological Constraints, Casimir Invariants, and Hierarchy

Whole-brain Hamiltonian systems frequently involve noncanonical structures characterized by topological constraints, such as conservation of signal flux, synchrony, or higher-order relationships. The hierarchical framework of noncanonical Hamiltonian systems (Yoshida et al., 2014) introduces phantom fields into an extended phase space, allowing the existence and construction of Casimir invariants even in cases where the original Poisson algebra does not admit closed-form invariants. Topological constraints manifest as conserved quantities (e.g., helicities or cross-helicities) and define the system's confinement to lower-dimensional "Casimir leaves."

The extended Poisson bracket and associated hierarchy enable embedding singularities as interior submanifolds, facilitating bifurcation analysis and probing transitions between dynamical regimes. Such a hierarchical organization offers a structured viewpoint for modeling interaction between regional and global neural dynamics, with the stratification of Poisson manifolds supporting multiscale phenomena observed in neurobiology.

3. Neural Network Approaches: Symmetry, Constraints, and Conservation

Modern implementations use neural network parameterizations to learn Hamiltonian systems directly from brain data, enforcing physical structure via inductive biases:

• Autoencoder architectures extract latent phase-space coordinates and Hamiltonian functions from observed time series (Bertalan et al., 2019), supporting discovery of intrinsic dynamical modes.
• Neural networks explicitly parameterize $H(q, p)$ or the Lagrangian, with constrained architectures in Cartesian coordinates and explicit Lagrange multipliers enforcing holonomic constraints (Finzi et al., 2020). This approach improves data efficiency and accuracy in modeling chaotic and extended-body dynamics, relevant for whole-brain systems with physical or anatomical constraints.
• Equation-driven HNNs learn solutions by penalizing deviations from Hamilton's equations rather than fitting trajectories (Mattheakis et al., 2020), resulting in exact conservation of invariants and efficient error control, a property valuable in multi-regional brain modeling.
• Automatic symmetry detection with Lie algebra frameworks (Dierkes et al., 2023) allows neural networks to identify and embed symmetry group actions present in brain dynamics, improving long-horizon prediction and interpretability by ensuring the corresponding Noether-conserved quantities align with neural invariants and modes.

These methodologies, when applied to whole-brain system identification, yield interpretable, physically-consistent models capable of capturing both energetics and symmetry properties on multiple scales.

4. Tensor-Based and Generalized Hamiltonian Systems

The extension to tensor-based polynomial Hamiltonian systems (Cui et al., 27 Mar 2025) leverages cubical tensors that generalize Hamiltonian matrices, allowing characterization and verification of Hamiltonian structure in nonlinear, multivariate, and high-degree dynamical systems. The defining condition for a tensor $A$ (order $k$) with structure matrix $J$,

$(J^T A)^{T_\sigma} + J A = 0,$

for all index permutations $\sigma$, ensures that the system is Hamiltonian if and only if all system tensors are Hamiltonian cubical tensors.

Polynomial Hamiltonians $H(x) = B_k x^k + \dots + B_2 x^2$ with supersymmetric tensors $B_j$ permit efficient stability analysis via tensor calculus. This facilitates Lyapunov stability tests:

$$\frac{\partial^2 H}{\partial x^2}(x^*) = k(k-1) B_k x^{*^{k-2}} + \dots + 2 B_2,$$

with positive definiteness indicating stability. For whole-brain dynamics, the tensor approach supports compact, high-dimensional representations of neural couplings, interactions, and energy landscapes, opening avenues for stability and bifurcation analysis in network neuroscience models.

5. Complex-Valued Augmentation and Analytic Signal Formulation

Embedding brain network dynamics in a complex-valued field using the Hilbert transform (Zhang et al., 29 Sep 2025) augments observed activation signals (generalized coordinates $q$) with latent "dark signals" (conjugate momenta $p$) to construct an analytic signal $\psi = q + i p$. The evolution is given by a Schrödinger-like equation:

$i \frac{d\psi}{dt} + H \psi = 0,$

where $H$ is a coupling matrix encoding inter-regional interactions. This approach models whole-brain activity as conservative dynamics in a higher-dimensional complex space, capturing both amplitude and phase information, and providing a framework for energy conservation via the function $E(\psi) = \psi^* H \psi$.

Empirically, the complex-valued model demonstrated a significant increase in short-horizon prediction accuracy (from 0.12 to 0.82 in linear regime, 0.47 to 0.88 in nonlinear regime) compared to real-valued formulations. Additionally, this paradigm recovers hierarchical intrinsic timescales, yields biologically plausible directed effective connectivity, and clarifies structure–function coupling, thereby offering a principled mechanism for interpreting age-related and task-related network reconfiguration.

6. Partition Functions, Ensemble Descriptions, and Statistical Mechanics

In the extended Nambu and Hamiltonian frameworks, partition functions formalize ensemble dynamics for systems with many degrees of freedom. The general expression,

$$Z_n^{(N)} = \prod_{k=1}^n \int dx_{(k)} dy_{(k)} dz_{(k)} \delta(\tilde{G}_{(k)}) e^{-\beta \tilde{H}},$$

or its Fourier representation, incorporates redundant variables and projects out nonphysical states, ensuring statistical summation over the physical manifold. For whole-brain systems, this formalism supports statistical or quantum-mechanical treatments of large-scale neural ensembles, encoding constraints arising from anatomy, conservation laws, or connectivity.

7. Unified Theories, Generalized Probabilistic Frameworks, and Future Research

Generalized Hamiltonian system models (Jiang et al., 29 Feb 2024) unify classical and quantum mechanics via probabilistic evolution equations in phase space, connecting time evolution, conserved energy observable, and measurement structure. The equation of motion, e.g.,

$\frac{\partial f}{\partial t} = \sum_i \mathcal{E}_i\,\mathrm{Im}\Biggl\{ \int f(q+l,p+j)\, g_i(q+y,p+z)\, K(k)\, e^{-i\frac{2(jy-lz)}{k}\, d\Omega\Biggr\},$

reduces to the quantum Moyal bracket or classical Liouville equation with appropriate choices of the dynamics kernel $K(k)$. This comprehensive perspective highlights the connections between coordinate symmetries, inner product invariance, and ensemble measurement, facilitating comparative and hybrid modeling across physical regimes expected to be relevant for multi-modal neural data.

A plausible implication is that such unifying frameworks could be applied to explore quantum-to-classical transitions in neural computation, statistical mechanics of neural assemblies, and the consequences of restricted information or non-associative dynamics in biological settings.

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Whole-brain Hamiltonian systems thus provide a rigorous, extensible paradigm for analyzing, simulating, and interpreting complex brain dynamics, integrating multilayered mathematical structures and empirical advances. The approach encompasses extended variable and constraint handling, hierarchical topological invariants, symmetry-enforced learning, tensor-based generalizations, complex-valued augmentation, and ensemble statistical mechanics—offering both principled theoretical underpinnings and robust empirical performance for advancing network neuroscience.