  This paper investigates stability conditions of continuous-time Hopfield and
firing-rate neural networks by leveraging contraction theory. First, we present
a number of useful general algebraic results on matrix polytopes and products
of symmetric matrices. Then, we give sufficient conditions for strong and weak
Euclidean contractivity, i.e., contractivity with respect to the $\ell_2$ norm,
of both models with symmetric weights and (possibly) non-smooth activation
functions. Our contraction analysis leads to contraction rates which are
log-optimal in almost all symmetric synaptic matrices. Finally, we use our
results to propose a firing-rate neural network model to solve a quadratic
optimization problem with box constraints.
