Qualitative analysis of neural networks

In the present paper the authors survey and utilize results from the qualitative theory of large scale interconnected dynamical systems in order to develop a qualitative theory for the Hopfield model of neural networks. In the present approach they view such networks as an interconnection of many single neurons. The results are phrased in terms of the qualitative properties of the individual neurons and in terms of the properties of the interconnecting structure of the neural networks. This method of analysis makes it frequently possible to circumvent difficulties usually encountered in the analysis of compoex systems with high dimension. Aspects of neural networks which the authors address include asymptotic stability, exponential stability, and instability of an equilibrium; estimates of trajectory bounds; estimates of the domain of attraction of an asymptotically stable equilibrium; and stability of neural networks under structural perturbations (arising, e.g., during adaptive learning schemes). The present results are not overly conservative. Furthermore, they are in a form which will make them highly useful as constraints in synthesis or design procedures.