Convergent activation dynamics in continuous time networks

The activation dynamics of nets are considered from a rigorous mathematical point of view. A net is identified with the dynamical system defined by a continuously differentiable vector field on the space of activation vectors, with fixed weights, biases, and inputs. Chaotic and oscillatory nets are briefly discussed, but the main goal is to find conditions guaranteeing that the trajectory of every (or almost every) initial activation state converges to an equilibrium. Several new results of this type areproved. These are illustrated with applications to additive nets. Cascades of nets are considered and a cascade decomposition theorem is proved. An extension of the Cohen-Grossberg convergence theorem is proved for certain nets with nonsymmetric weight matrices. Keywords-Convergence, Cascade, Liapunov function, Excitatory net, Inhibitory net, Global asymptotic stability, Chaotic dynamics.