A class of connection patterns for neural networks with absolute stability

This paper presents a class of connection patterns for neural networks with necessary and sufficient conditions for their absolute stability. The patterns are specified by an unbounded, finitely generated, and unilaterally superposable subset in the weight matrix space. We derive the results by using a Lyapunov function, spectral analysis of weight matrices, and LaSalle´s invariance principle, without assuming the boundedness and strictly increasing properties on activation functions. The results cover some early results based on detailed balance or quasi-symmetry conditions as special cases. We also analyze an important programming neural network in the literature and show that it is in a quasi-normal weight matrix form which is a special case of the presented connection patterns. This gives a new insight into the structure and dynamics of this kind of programming neural network.