Multistability of state-dependent switched neural networks with Mexican-hat-type activation functions

This paper investigates the issue of multistability in state-dependent switched neural networks (SSNNs) with Mexican-hat-type activation functions (AFs). It establishes the coexistence and stability of multiple equilibrium points (EPs). Initially, the state space is partitioned based on the geometric characteristics of the Mexican-hat-type AF, enabling to determine the positions of the EPs. Secondly, the coexistence of $9^{h_1}{7}^{h_2}{5}^{h_3}{3}^{h_4}$ EPs for $n$-neurons SSNNs under specific sufficient conditions is proved with the Brouwer’s fixed-point theorem. Next, by using diagonally dominant matrix theory and Gershgorin circle theorem, it is proven that there are $5^{h_1}{4}^{h_2}{3}^{h_3}{2}^{h_4}$ asymptotically stable EPs under some conditions, where $h_1,h_2,h_3$ and $h_4$ are nonnegative integers satisfying $0\le h_1+h_2+h_3+h_4\le n$. Therefore, we can obtain that SSNNs can have larger storage capacity by selecting the appropriate parameters $h_i,i=1,2,3,4$. Finally, the correctness of the results in this paper is verified through two numerical examples.