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

In this paper, we discuss the coexistence and dynamical behaviors of multiple equilibrium points for neural networks with a class of non-monotonic piecewise linear activation functions. It is proven that under some conditions, such n-neuron neural networks have exactly 5ⁿ equilibrium points, 3ⁿ of which are locally stable and the others are unstable, based on the fixed point theorem, the contraction mapping theorem and the eigenvalue properties of strict diagonal dominance matrix. The investigation shows 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. A simulation example is provided to illustrate and validate the theoretical findings.