  Critical questions in dynamical neuroscience and machine learning are related
to the study of continuous-time neural networks and their stability,
robustness, and computational efficiency. These properties can be
simultaneously established via a contraction analysis.
  This paper develops a comprehensive non-Euclidean contraction theory for
continuous-time neural networks. First, for non-Euclidean
$\ell_{1}/\ell_{\infty}$ logarithmic norms, we establish quasiconvexity with
respect to positive diagonal weights and closed-form worst-case expressions
over certain matrix polytopes. Second, for locally Lipschitz maps (e.g.,
arising as activation functions), we show that their one-sided Lipschitz
constant equals the essential supremum of the logarithmic norm of their
Jacobian. Third and final, we apply these general results to classes of
continuous-time neural networks, including Hopfield, firing rate, Persidskii,
Lur'e and other models. For each model, we compute the optimal contraction rate
and corresponding weighted non-Euclidean norm via a linear program or, in some
special cases, via a Hurwitz condition on the Metzler majorant of the synaptic
matrix. Our non-Euclidean analysis establishes also absolute, connective, and
total contraction properties.

Optimization and Control;Systems and Control;Systems and Control