μ-stability and instability of multiple equilibrium points in delayed neural networks with general discontinuous activation functions

Abstract

This paper proposes a general type of discontinuous activation functions (AFs) and studies the μ-stability of multiple equilibrium points (EPs) in delayed neural networks (DNNs). By judging 4n algebraic inequalities and the nonsingularity of an M-matrix, DNNs with the general discontinuous AFs can be shown to have 5n EPs, therein 3n EPs are locally μ-stable. Moreover, these 3n EPs are located at the points of continuity of the AF. Compared with the existing general continuous AF, DNNs with the general discontinuous AF can have more total/stable EPs. Hence, DNNs with the general discontinuous AF could store a much larger number of memory patterns if they are applied to associative memory. In addition, the attraction basin (AB) of each stable EP in DNNs is estimated. Two numerical examples are shown to testify the validity of the obtained results.