IEEE Trans Neural Netw Learn Syst. 2017 Aug;28(8):1953-1958. doi: 10.1109/TNNLS.2016.2557845. Epub 2016 May 25.
This brief is mainly concerned with a series of dynamical analyses of the Hindmarsh-Rose (HR) neuron with state-dependent time delays. The dynamical analyses focus on stability, Hopf bifurcation, as well as chaos and chaos control. Through the stability and bifurcation analysis, we determine that increasing the external current causes the excitable HR neuron to exhibit periodic or chaotic bursting/spiking behaviors and emit subcritical Hopf bifurcation. Furthermore, by choosing a fixed external current and varying the time delay, the stability of the HR neuron is affected. We analyze the chaotic behaviors of the HR neuron under a fixed external current through time series, bifurcation diagram, Lyapunov exponents, and Lyapunov dimension. We also analyze the synchronization of the chaotic time-delayed HR neuron through nonlinear control. Based on an appropriate Lyapunov-Krasovskii functional with triple integral terms, a nonlinear feedback control scheme is designed to achieve synchronization between the uncontrolled and controlled models. The proposed synchronization criteria are derived in terms of linear matrix inequalities to achieve the global asymptotical stability of the considered error model under the designed control scheme. Finally, numerical simulations pertaining to stability, Hopf bifurcation, periodic, chaotic, and synchronized models are provided to demonstrate the effectiveness of the derived theoretical results.
本简报主要关注具有状态相关时滞的 Hindmarsh-Rose (HR) 神经元的一系列动力学分析。动力学分析侧重于稳定性、Hopf 分岔以及混沌和混沌控制。通过稳定性和分岔分析，我们确定增加外部电流会导致兴奋性 HR 神经元表现出周期性或混沌爆发/尖峰行为，并发出亚临界 Hopf 分岔。此外，通过选择固定的外部电流并改变时滞，HR 神经元的稳定性会受到影响。我们通过时间序列、分岔图、李雅普诺夫指数和李雅普诺夫维数分析了固定外部电流下 HR 神经元的混沌行为。我们还通过非线性控制分析了混沌时滞 HR 神经元的同步。基于具有三重积分项的适当的 Lyapunov-Krasovskii 泛函，设计了一种非线性反馈控制方案，以实现无控和被控模型之间的同步。根据线性矩阵不等式推导出了同步准则，以在所设计的控制方案下实现所考虑的误差模型的全局渐近稳定性。最后，提供了与稳定性、Hopf 分岔、周期、混沌和同步模型相关的数值模拟，以证明所得到的理论结果的有效性。
