This project combines related work in dynamical systems and neural networks. The P.I. will investigate the dynamics of certain types of difference schemes and closely related systems of nonlinear differential equations. These systems depend on parameters that fluctuate either deterministically or stochastically. These results will be used to evaluate the convergence of the weigh dynamics in several kinds of supervised learning schemes for adaptive neural networks, including Widrow-Hoff associative learning and Barto-Anandan reinforcement learning. The P.I. will also investigate a neural network model of retinal development in fluctuating environments due to Amari and Takeuchi. In competitive and cooperative systems of differential equations, the P.I. will extend earlier work to obtain convergence theorems applicable to the activation dynamics of certain types of recurrent neural networks having asymmetrical fixed weights. A separate study will try to prove that dissipative competitive systems in R3, and a restricted class of such systems in R4, can be approximated by structurally stable ones, and satisfy a C' Closing lemma for all r>2. This would establish new classes of structurally stable systems, and would be the first proof of a nontrivial C2 closing lemma in dimensions >3.
