Multistability and instability of neural networks with discontinuous nonmonotonic piecewise linear activation functions

In this paper, we discuss the coexistence and dynamical behaviors of multiple equilibrium points for recurrent neural networks with a class of discontinuous nonmonotonic piecewise linear activation functions. It is proved that under some conditions, such n-neuron neural networks can have at least 5n equilibrium points, 3n of which are locally stable and the others are unstable, based on the contraction mapping theorem and the theory of strict diagonal dominance matrix. The investigation shows that the neural networks with the discontinuous activation functions introduced in this paper can have both more total equilibrium points and more locally stable equilibrium points than the ones with continuous Mexican-hat-type activation function or discontinuous two-level activation functions. An illustrative example with computer simulations is presented to verify the theoretical analysis.