Remarks on the Article by X.B. Nie, J.D. Cao and S.M. Fei “Multistability and instability of delayed competitive neural networks with nondecreasing piecewise linear activation functions”

In this paper, a simple and elementary proof of the equilibrium uniqueness theorem (i.e. Part of Theorem 1 in Nie et al. [2013. Multistability and instability of delayed competitive neural networks with nondecreasing piecewise linear activation functions. Neurocomputing 119, 281–291]) is given, which shows the equilibrium uniqueness of delayed competitive neural networks (DCNNs) with the activation function possessing two corner points in a given subset. In addition, for the piecewise linear activation function with two corner points, the dynamical behaviors of all equilibrium points of n-neuron delayed Hopfield neural networks (DHNNs) are completely analyzed, and a sufficient condition is obtained to guarantee that the n-neuron DHNNs have exactly 3n equilibrium points, where 2n of them are stable, and the others are unstable. Finally, an example is provided to show the theoretical analysis.