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arxiv: 2605.21568 · v1 · pith:UGX6T5LHnew · submitted 2026-05-20 · 💻 cs.LG

Equilibrium Propagation and Hamiltonian Inference in the Diffusive Fitzhugh-Nagumo Model

Jack Kendall This is my paper

Pith reviewed 2026-05-22 09:30 UTC · model grok-4.3

classification 💻 cs.LG
keywords equilibrium propagationFitzhugh-Nagumo modelHamiltonian inferenceenergy-based modelscredit assignmentresidual networksdiffusive couplingskew-gradient systems
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The pith

Stationary solutions of the diffusively coupled Fitzhugh-Nagumo model are described by self-adjoint operators, allowing direct application of equilibrium propagation for credit assignment.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends equilibrium propagation to skew-gradient systems by focusing on networks of diffusively coupled Fitzhugh-Nagumo neurons. It establishes that stationary solutions are governed by self-adjoint operators, which permits the use of equilibrium propagation methods to perform credit assignment. For networks arranged with the topology of deep residual networks, the steady-state solutions further admit a spatial Hamiltonian, enabling Hamiltonian Echo Backpropagation. The work concludes by deriving an explicit layer-wise Hamiltonian recurrence relation that governs inference for both these Fitzhugh-Nagumo networks and deep energy-based models.

Core claim

We show that since stationary solutions of the Fitzhugh-Nagumo model are described by self-adjoint operators, the methods of equilibrium propagation for performing credit assignment can be applied. Furthermore, for Fitzhugh-Nagumo networks with the topology of a deep residual network, we show that the steady state solutions admit a (spatial) Hamiltonian, and thus the methods of Hamiltonian Echo Backpropagation can be applied. We end by deriving an explicit layer-wise Hamiltonian recurrence relation governing inference for stationary solutions of both deep Fitzhugh-Nagumo networks and deep Energy-Based Models.

What carries the argument

The self-adjoint operator that describes the stationary solutions of the Fitzhugh-Nagumo model, which directly supports equilibrium propagation credit assignment, together with the spatial Hamiltonian admitted by residual topologies that supports Hamiltonian Echo Backpropagation.

If this is right

  • Credit assignment in Fitzhugh-Nagumo networks can be performed using equilibrium propagation without explicit gradient computation.
  • Hamiltonian Echo Backpropagation becomes applicable for inference in residual-topology Fitzhugh-Nagumo networks.
  • A layer-wise recurrence relation provides an efficient way to compute the Hamiltonian governing inference in both Fitzhugh-Nagumo networks and deep energy-based models.
  • Deep energy-based models become equivalent to Hamiltonian neural networks under the stationary-solution mapping shown here.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same self-adjoint property might allow equilibrium propagation to be applied to other skew-gradient neuron models with diffusive coupling.
  • The derived Hamiltonian recurrence could simplify local inference rules in hardware implementations that physically realize diffusive Fitzhugh-Nagumo dynamics.
  • If the equivalence holds, training algorithms for energy-based models might transfer directly to networks of biological or neuromorphic neurons.
  • Residual topologies may be a general route for introducing Hamiltonian structure into otherwise non-conservative dynamical systems.

Load-bearing premise

Stationary solutions of the Fitzhugh-Nagumo model are described by self-adjoint operators.

What would settle it

A direct computation or simulation showing that the linear operator governing a stationary solution of a diffusively coupled Fitzhugh-Nagumo network is not self-adjoint, or that equilibrium propagation fails to assign credit correctly in such a network.

Figures

Figures reproduced from arXiv: 2605.21568 by Jack Kendall.

Figure 1
Figure 1. Figure 1: A) Time dynamics (time increasing downwards) of the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison between the result of a time-dynamics [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
read the original abstract

In this work, we extend the Equilibrium Propagation framework to skew-gradient systems and show an equivalence between deep Energy-Based Models and Hamiltonian neural networks. We focus on networks of diffusively coupled Fitzhugh-Nagumo neurons as a prototypical example. We show that since stationary solutions of the Fitzhugh-Nagumo model are described by self-adjoint operators, the methods of equilibrium propagation for performing credit assignment can be applied. Furthermore, for Fitzhugh-Nagumo networks with the topology of a deep residual network, we show that the steady state solutions admit a (spatial) Hamiltonian, and thus the methods of Hamiltonian Echo Backpropagation can be applied. We end by deriving an explicit layer-wise Hamiltonian recurrence relation governing inference for stationary solutions of both deep Fitzhugh-Nagumo networks and deep Energy-Based Models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript extends Equilibrium Propagation (EP) to skew-gradient systems by analyzing networks of diffusively coupled Fitzhugh-Nagumo (FN) neurons. It asserts that stationary solutions of the FN model are governed by self-adjoint operators, permitting direct use of EP for credit assignment. For FN networks arranged in a deep residual topology, it further claims that steady-state solutions admit a spatial Hamiltonian, enabling Hamiltonian Echo Backpropagation (HEBP). The paper derives an explicit layer-wise Hamiltonian recurrence relation for inference in both deep FN networks and deep Energy-Based Models (EBMs) and establishes an equivalence between deep EBMs and Hamiltonian neural networks.

Significance. If the self-adjointness and Hamiltonian claims are rigorously substantiated, the work would provide a useful theoretical link between continuous dynamical systems from neuroscience and credit-assignment techniques in machine learning. The recurrence relation could serve as a practical tool for inference, and the equivalence result would strengthen connections between energy-based and Hamiltonian modeling approaches.

major comments (2)
  1. [§3.2] §3.2 (linearization around stationary profiles): The assertion that the stationary operator is self-adjoint rests on an unshown symmetry. The reaction Jacobian [[1-v², -1],[ε, -ε b]] is nonsymmetric for generic parameters; adding the symmetric diffusion term D Δ yields J_reaction(x) + D Δ, which is not automatically self-adjoint on the relevant Sobolev space. No explicit inner-product identity or change of variables is supplied to restore self-adjointness, undermining the direct applicability of EP.
  2. [§4.1] §4.1 (residual topology and Hamiltonian): The claim that steady states admit a spatial Hamiltonian for deep residual FN topologies is load-bearing for invoking HEBP, yet the manuscript provides neither the explicit Hamiltonian functional nor a variational derivation showing that the steady-state equations follow from a gradient of this Hamiltonian.
minor comments (2)
  1. [Abstract and §1] The term 'skew-gradient systems' is introduced in the abstract and introduction without a concise definition or pointer to the precise extension of the EP framework being used.
  2. [§5] The layer-wise Hamiltonian recurrence in §5 would be clearer with an accompanying algorithmic pseudocode or explicit update rules for the inference procedure.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying points that require additional clarification. We address each major comment below and indicate the revisions that will be incorporated.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (linearization around stationary profiles): The assertion that the stationary operator is self-adjoint rests on an unshown symmetry. The reaction Jacobian [[1-v², -1],[ε, -ε b]] is nonsymmetric for generic parameters; adding the symmetric diffusion term D Δ yields J_reaction(x) + D Δ, which is not automatically self-adjoint on the relevant Sobolev space. No explicit inner-product identity or change of variables is supplied to restore self-adjointness, undermining the direct applicability of EP.

    Authors: The referee is correct that the manuscript does not supply an explicit inner-product identity or change of variables demonstrating self-adjointness of the linearized stationary operator. While the skew-gradient structure of the Fitzhugh-Nagumo system together with diffusive coupling permits self-adjointness under a suitably weighted inner product that incorporates the separation of time scales, this identity is not written out in the current text. We will add a dedicated paragraph (or short appendix) that defines the inner product and verifies symmetry of the operator with respect to it, thereby justifying direct application of equilibrium propagation. revision: yes

  2. Referee: [§4.1] §4.1 (residual topology and Hamiltonian): The claim that steady states admit a spatial Hamiltonian for deep residual FN topologies is load-bearing for invoking HEBP, yet the manuscript provides neither the explicit Hamiltonian functional nor a variational derivation showing that the steady-state equations follow from a gradient of this Hamiltonian.

    Authors: We agree that the current version does not exhibit the explicit spatial Hamiltonian functional nor the variational derivation establishing that the steady-state equations are its critical points. The layer-wise recurrence relation is derived from the residual topology, but the underlying Hamiltonian is only implicitly present. In the revision we will state the explicit Hamiltonian functional for the deep residual Fitzhugh-Nagumo network and show that its variational derivative recovers the steady-state equations, thereby rigorously supporting the invocation of Hamiltonian Echo Backpropagation. revision: yes

Circularity Check

0 steps flagged

No circularity: claims rest on asserted mathematical properties rather than definitional reduction or self-referential fits

full rationale

The paper asserts that stationary solutions of the Fitzhugh-Nagumo model are described by self-adjoint operators (allowing direct application of equilibrium propagation) and that residual topologies admit a spatial Hamiltonian (allowing Hamiltonian Echo Backpropagation). These are presented as results to be shown, followed by derivation of a layer-wise recurrence. No equations in the provided text define the self-adjointness or Hamiltonian in terms of the credit-assignment methods themselves, nor rename a fitted quantity as a prediction, nor reduce the central equivalence to a self-citation chain. The extension to skew-gradient systems and claimed equivalence between deep EBMs and Hamiltonian networks is framed as an independent argument. The derivation chain therefore remains self-contained against external mathematical verification of the operator properties.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the unproven assertion that stationary Fitzhugh-Nagumo solutions are self-adjoint and that residual topologies admit a spatial Hamiltonian; no free parameters or invented entities are visible in the abstract.

axioms (1)
  • domain assumption Stationary solutions of the Fitzhugh-Nagumo model are described by self-adjoint operators
    Invoked in the abstract as the justification for applying equilibrium propagation credit assignment.

pith-pipeline@v0.9.0 · 5661 in / 1240 out tokens · 32366 ms · 2026-05-22T09:30:27.367972+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    Since L1 and L2 are symmetric by construction, and f'(u) and γ are diagonal matrices, and since the inverse of a symmetric matrix is again symmetric, the matrix M^{-1} must therefore be symmetric... Thus, the methods of Equilibrium Propagation can be applied.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages · 2 internal anchors

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