Hopfield neural networks as port-Hamiltonian and gradient systems

TL;DR

This paper reinterprets continuous Hopfield networks using system theory, revealing their port-Hamiltonian and gradient system structures through a novel electrical network perspective involving nonlinear capacitors.

Contribution

It introduces a new electrical network interpretation of Hopfield networks and establishes their equivalence to port-Hamiltonian and gradient systems, providing new analytical tools.

Findings

Hopfield networks can be formulated as port-Hamiltonian systems with passivity conditions.

They can also be represented as gradient systems with a Riemannian metric from the Hessian of energy.

The energy function acts as a dissipation potential satisfying a dissipation inequality.

Abstract

The structure of continuous Hopfield networks is revisited from a system-theoretic point of view. After adopting a novel electrical network interpretation involving nonlinear capacitors, it is shown that Hopfield networks admit a port-Hamiltonian formulation provided an extra passivity condition is satisfied. Subsequently it is shown that any Hopfield network can be represented as a gradient system, with Riemannian metric given by the inverse of the Hessian matrix of the total energy stored in the nonlinear capacitors. On the other hand, the well-known 'energy' function employed by Hopfield turns out to be the dissipation potential of the gradient system, and this potential is shown to satisfy a dissipation inequality that can be used for analysis and interconnection.