IEEE Trans Neural Netw Learn Syst. 2012 Sep;23(9):1473-85. doi: 10.1109/TNNLS.2012.2205703.
Recent papers have pointed out the interest to study convergence in the presence of multiple equilibrium points (EPs) (multistability) for neural networks (NNs) with nonsymmetric cooperative (nonnegative) interconnections and neuron activations modeled by piecewise linear (PL) functions. One basic difficulty is that the semiflows generated by such NNs are monotone but, due to the horizontal segments in the PL functions, are not eventually strongly monotone (ESM). This notwithstanding, it has been shown that there are subclasses of irreducible interconnection matrices for which the semiflows, although they are not ESM, enjoy convergence properties similar to those of ESM semiflows. The results obtained so far concern the case of cooperative NNs without delays. The goal of this paper is to extend some of the existing results to the relevant case of NNs with delays. More specifically, this paper considers a class of NNs with PL neuron activations, concentrated delays, and a nonsymmetric cooperative interconnection matrix A and delay interconnection matrix A(τ). The main result is that when A+A(τ) satisfies a full interconnection condition, then the generated semiflows, which are monotone but not ESM, satisfy a limit set dichotomy analogous to that valid for ESM semiflows. It follows that there is an open and dense set of initial conditions, in the state space of continuous functions on a compact interval, for which the solutions converge toward an EP. The result holds in the general case where the NNs possess multiple EPs, i.e., is a result on multistability, and is valid for any constant value of the delays.
最近的一些论文指出，研究具有非对称合作（非负）连接和分段线性（PL）函数建模的神经元激活的神经网络（NN）在存在多个平衡点（EP）（多稳定性）时的收敛性很有意义。一个基本的困难是，由这样的 NN 产生的半流是单调的，但由于 PL 函数中的水平段，它们不是最终强单调的（ESM）。尽管如此，已经表明存在不可约连接矩阵的子类，对于这些半流，尽管它们不是 ESM，但它们具有类似于 ESM 半流的收敛性质。迄今为止获得的结果涉及没有延迟的合作 NN 的情况。本文的目标是将一些现有结果扩展到具有延迟的相关 NN 情况。更具体地说，本文考虑了一类具有 PL 神经元激活、集中延迟和非对称合作连接矩阵 A 和延迟连接矩阵 A(τ)的 NN。主要结果是，当 A+A(τ)满足全连接条件时，生成的半流是单调的但不是 ESM 的，满足类似于 ESM 半流的极限集二分法。因此，在紧区间上连续函数的状态空间中，存在一个初始条件的开集和稠密集，使得解收敛到 EP。该结果在 NN 具有多个 EP 的一般情况下有效，即多稳定性的结果，并且对于延迟的任何常数值都是有效的。
