  We provide gradient flow interpretations for the continuous-time
continuous-state Hopfield neural network (HNN). The ordinary and stochastic
differential equations associated with the HNN were introduced in the
literature as analog optimizers, and were reported to exhibit good performance
in numerical experiments. In this work, we point out that the deterministic HNN
can be transcribed into Amari's natural gradient descent, and thereby uncover
the explicit relation between the underlying Riemannian metric and the
activation functions. By exploiting an equivalence between the natural gradient
descent and the mirror descent, we show how the choice of activation function
governs the geometry of the HNN dynamics.
  For the stochastic HNN, we show that the so-called "diffusion machine", while
not a gradient flow itself, induces a gradient flow when lifted in the space of
probability measures. We characterize this infinite dimensional flow as the
gradient descent of certain free energy with respect to a Wasserstein metric
that depends on the geodesic distance on the ground manifold. Furthermore, we
demonstrate how this gradient flow interpretation can be used for fast
computation via recently developed proximal algorithms.
