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

Jack Kendall

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arXiv · 2605.21568

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

Jack Kendall

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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.

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Jack Kendall. 2026-05-20. Equilibrium Propagation and Hamiltonian Inference in the Diffusive Fitzhugh-Nagumo Model. https://arxiv.org/abs/2605.21568

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