Multistability analysis of delayed recurrent neural networks with a class of piecewise nonlinear activation functions

This paper studies the multistability of delayed recurrent neural networks (DRNNs) with a class of piecewise nonlinear activation functions. The coexistence as well as the stability of multiple equilibrium points (EPs) of DRNNs are proved. With the Brouwer's fixed point theorem as well as the Lagrange mean value theorem, it is obtained that under some conditions, the n-neuron DRNNs with the proposed activation function can have at least 5ⁿ EPs and 3ⁿ of them are locally stable. Compared with the DRNNs with sigmoidal activation functions, DRNNs with this kind of activation function can have more total EPs and more locally stable EPs. It implies that when designing DRNNs with the proposed activation function to apply in associative memory, it can have an even larger storage capacity. Furthermore, it is obtained that there exists a relationship between the number of the total EPs/stable EPs and the frequency of the sinusoidal function in the proposed activation function. Last, the above obtained results are extended to a more general case. It is shown that, DRNNs with the extended activation function can have (2k+1)ⁿ EPs, (k+1)ⁿ of which are locally stable, therein k is closely related to the frequency of the sinusoidal function in the extended activation function. Two simulation examples are given to verify the correctness of the theoretical results.