具有一类不连续非单调激活函数的时滞递归神经网络的多稳定性分析

Multistability Analysis of Delayed Recurrent Neural Networks with a Class of Discontinuous and Non-Monotonic Activation Functions

摘要: 本文提出了一类不连续非单调激活函数，研究了具有这类激活函数的时滞递归神经网络的多稳定性。根据激活函数的几何特性和不动点定理，给出充分条件确保n维神经网络至少存在7ⁿ个平衡点，其中4ⁿ个是局部指数稳定的。然后，我们将结果推广到更一般的情况。在不增加充分条件的情况下，本文通过增加激活函数峰值点的数量k，得到n维神经网络可以具有 (2k+3)ⁿ 平衡点，其中是 (k+2)ⁿ 局部指数稳定的。与之前文献相比，总平衡点和稳定平衡点的数量大大地增加了，从而提高了递归神经网络的存储容量。最后，给出了一个例子来证明我们的理论结果。

Abstract: This paper proposes a class of discontinuous and non-monotonic activation functions and studies the multistability of delayed recurrent neural network with these activation functions. By the geometrical properties of activation function and fixed point theorem, some sufficient conditions are presented to ensure that n-dimensional neural networks can have at least 7ⁿ equilibria, where 4ⁿ equilibria are locally exponentially stable. Then, the obtained results are extended to a more general case. Without adding sufficient conditions, this paper increases the number of peak points (k) of the activation function and finds that the n-dimensional neural networks can have (2k+3)ⁿ equilibria, where (k+2)ⁿ equilibria of them are locally exponentially stable. Compared with the previous literature, the numbers of total equilibria and stable equilibria are enormously increased, thus enhancing the memory storage capacity of DRNN. Finally, one example is presented to demonstrate our theoretical results.