Stability analysis and synthesis algorithm of a class of discrete-time neural networks

In the present paper, we investigate the qualitative properties of a class of discrete-time neural networks described by a system of first order ordinary difference equations with a discontinuous right-hand side. The class of systems considered retains the basic structure of Hopfield type neural networks with the nonlinearities having infinite gain. For the present model, we establish qualitative results which enable us to generalize the solutions of discrete-time systems in the discontinuous surfaces and to characterize the set of system equilibrium points (distribution of equilibria in the state space and stability properties of the equilibrium points). In addition, we develop an efficient synthesis procedure utilizing the eigenstructure method for the present class of neural networks. The synthesized networks are capable of learning new patterns as well as forgetting learned patterns. The resulting neural networks can easily be implemented in digital hardware. Furthermore, when implemented by a serial processor, the present results offer extremely efficient means of simulating corresponding continuous-time neural networks (modelled by systems of ordinary differential equations). The applicability of the present results is demonstrated by means of several specific examples.