Multistability and instability of competitive neural networks with non-monotonic piecewise linear activation functions

This paper addresses the issue of multistability for competitive neural networks. First, a general class of continuous non-monotonic piecewise linear activation functions is introduced. Then, based on the fixed point theorem, the contraction mapping theorem and the eigenvalue properties of strict diagonal dominance matrix, it is shown that under some conditions, such n-neuron competitive neural networks have exactly 5n equilibrium points, among which 3n equilibrium points are locally exponentially stable and the others are unstable. Moreover, it is revealed that the neural networks with non-monotonic piecewise linear activation functions introduced in this paper can have greater storage capacity than the ones with Mexican-hat-type activation function and nondecreasing saturated activation function. In addition, unlike most existing multistability results of neural networks with nondecreasing activation functions, the location of those obtained 3n locally stable equilibrium points in this paper is more flexible. Finally, a numerical example is provided to illustrate and validate the theoretical findings via comprehensive computer simulations.