Multistability of Hopfield neural networks with a designed discontinuous sawtooth-type activation function

In this paper, a class of discontinuous sawtooth-type activation function is designed and the multistability of Hopfield neural networks (HNNs) with such kind of activation function is studied. By virtue of the Brouwer's fixed point theorem and the property of strictly diagonally dominant matrix (SDDM), some sufficient conditions are presented to ensure that the n-neuron HNN can have at least 7(n) equilibria, among which 4(n) equilibria are locally exponentially stable and the remaining 7(n)-4(n) equilibria are unstable. Then, the obtained results are extended to a more general case. We continue to increase the number of the peaks of the sawtooth-type activation function and we find that the n-neuron HNN can have (2k + 3)(n) equilibria, (k + 2)(n) of them are locally exponentially stable and the remaining equilibria are unstable. Therein, k denotes the total number of the peaks in the designed activation function. That is to say, there is a quantitative relationship between the number of the peaks and the number of the equilibria. It implies that one can improve the storage capacity of a HNN by increasing the number of the peaks of the activation function in theory and in practice. To some extent, this method is convenient and flexible. Compared with the existing results, HNN with the designed sawtooth-type activation function can have more total equilibria as well as more locally stable equilibria. Finally, two examples are presented to demonstrate the validity of the obtained results. (C) 2021 Elsevier B.V. All rights reserved.