ABSTRACT
The analog Hopfield neural network model was introduced in Chap ter 2 and its global qualitative properties were investigated in Chap ter 3. In the present chapter we first study local qualitative proper ties under the assumption that the interconnecting structure of such networks is not necessarily symmetric. In our approach we view an analog Hopfield network as an interconnection of n free subsystems (consisting of the individual neurons with their associated dynam ics) and our results are phrased in terms of the qualitative proper ties of the subsystems and the constraints imposed by the network interconnections. We use two general approaches in our analysis, employing scalar Lyapunov functions (consisting of a weighted sum of Lyapunov functions for the free subsystems) and vector Lyapunov functions (whose components are scalar Lyapunov functions for the free subsystems), respectively. In the latter approach, we invoke the comparison principle encountered in the stability theory of differen tial equations, making use of the properties of M-matrices. Issues that we address include asymptotic stability, exponential stability, and instability of an equilibrium; estimates of the domain of attrac-
tion of an asymptotically stable equilibrium; estimates of trajectory bounds (which provide estimates of the rate of convergence of trajec tories to asymptotically stable equilibria); stability under structural perturbations in the network; and qualitative properties of analog Hopfield neural networks under the high gain limit assumption for the activation functions.
