Liu Yang, Wang Zhen, Ma Qian, Shen Hao
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China.
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China.
Neural Netw. 2022 Aug;152:80-89. doi: 10.1016/j.neunet.2022.04.015. Epub 2022 Apr 21.
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 5 EPs and 3 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) EPs, (k+1) 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.
本文研究了一类具有分段非线性激活函数的时滞递归神经网络（DRNNs）的多重稳定性。证明了DRNNs多个平衡点（EPs）的共存性和稳定性。利用布劳威尔不动点定理和拉格朗日中值定理，得到在某些条件下，具有所提出激活函数的n神经元DRNNs至少可以有5个平衡点，其中3个是局部稳定的。与具有Sigmoid激活函数的DRNNs相比，具有这种激活函数的DRNNs可以有更多的总平衡点和更多的局部稳定平衡点。这意味着在设计使用所提出激活函数的DRNNs用于联想记忆时，它可以具有更大的存储容量。此外，得到了总平衡点/稳定平衡点的数量与所提出激活函数中正弦函数频率之间的关系。最后，将上述所得结果推广到更一般的情况。结果表明，具有扩展激活函数的DRNNs可以有（2k + 1）个平衡点，其中（k + 1）个是局部稳定的，其中k与扩展激活函数中正弦函数的频率密切相关。给出了两个仿真例子来验证理论结果的正确性。
