College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, PR China; Hunan Women's University, Changsha, Hunan 410002, PR China.
Neural Netw. 2013 Dec;48:180-94. doi: 10.1016/j.neunet.2013.08.004. Epub 2013 Sep 5.
This paper is concerned with the periodic dynamics of a class of delayed neural networks with discontinuous neural activation functions. Under the Filippov framework, the cone expansion and compression fixed point theorems of set-valued maps are successfully employed to derive the existence of the ω-periodic positive solution. However, before the discussion of the periodicity, there still remains a fundamental issue about viability to be solved due to the presence of general mixed time-delays involving both time-varying delays and distributed delays. This difficulty can be overcome by a transformation and the continuation theorem. Then, for the discontinuous and delayed neural network system with time-periodic coefficients, the uniqueness and global exponential stability of the periodic state solution are proved by using non-smooth analysis theory with generalized Lyapunov approach. Furthermore, the global convergence in measure of the periodic output is also investigated. The obtained results are a very good extension and improvement of previous works on discontinuous dynamical neuron systems with a broad range of neuron activations dropping the assumption of boundedness or monotonicity. Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical results.
本文研究了一类具有不连续神经激活函数的时滞神经网络的周期动力学。在 Filippov 框架下，成功运用集值映射的锥扩张和压缩不动点定理，得到了 ω-周期正解的存在性。然而，在讨论周期性之前，由于存在同时包含时变时滞和分布时滞的一般混合时滞，仍然存在一个基本的可行性问题需要解决。这个困难可以通过变换和连续定理来克服。然后，对于时变系数的不连续时滞神经网络系统，通过使用广义 Lyapunov 方法的非光滑分析理论，证明了周期状态解的唯一性和全局指数稳定性。此外，还研究了周期输出的全局收敛度。所得到的结果是对具有广泛神经元激活的不连续动力神经元系统的先前工作的很好扩展和改进，放宽了有界性或单调性的假设。最后，提供了数值模拟来说明理论结果的有效性。
