This paper aims to explore the dynamics involved in the intricate realm of chaotic synchronization in the Hindmarsh-Rose (H-R) neuronal model, which is known for its resemblance to the brain's information-processing components. Distinct from the existing studies related to the H-R neuronal model, this research focuses on addressing nonlinearities of membrane potential through a Takagi-Sugeno (T-S) fuzzy approach. A sampled-data-based controller scheme that can resolve the stabilization issues inherent to the H-R neuronal model is proposed. Compared with many existing control schemes, sampled-data control has several advantages, which include easy digital implementation and robustness against transmission delays. This research utilizes the zero-order holder (ZOH) technique to handle discrete-time control actions in a continuous-time T-S fuzzy model. Furthermore, synchronization analysis pertaining to the T-S fuzzy-based H-R model with user-designed control inputs is conducted to understand and overcome the associated spiking, chaotic, and bursting behaviors. Closed-loop dynamics of the resulting model with and without control input, namely, the error model, are analyzed by employing the Lyapunov stability theory and integral inequalities. Specifically, a suitable Lyapunov Krasovskii functional (LKF) is formulated and solved using the linear matrix inequalities (LMIs) technique to guarantee the global asymptotically stability of the error model. The developed theoretical framework is validated using numerical simulations, and the corresponding outcomes are graphically illustrated and discussed.

中文翻译：

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Hindmarsh-Rose 神经元模型的基于模糊的采样数据同步


本文旨在探索 Hindmarsh-Rose (HR) 神经元模型中混沌同步复杂领域的动力学，该模型因其与大脑信息处理组件的相似性而闻名。与 HR 神经元模型相关的现有研究不同，本研究侧重于通过 Takagi-Sugeno (TS) 模糊方法解决膜电位的非线性问题。提出了一种基于采样数据的控制器方案，可以解决 HR 神经元模型固有的稳定性问题。与许多现有的控制方案相比，采样数据控制具有多种优点，包括易于数字实现和对传输延迟的鲁棒性。本研究利用零阶保持器 (ZOH) 技术来处理连续时间 TS 模糊模型中的离散时间控制动作。此外，还利用用户设计的控制输入对基于 TS 模糊的 HR 模型进行同步分析，以理解和克服相关的尖峰、混沌和突发行为。采用李亚普诺夫稳定性理论和积分不等式分析了有和没有控制输入的结果模型（即误差模型）的闭环动力学。具体来说，使用线性矩阵不等式（LMI）技术制定和求解合适的李亚普诺夫克拉索夫斯基泛函（LKF），以保证误差模型的全局渐近稳定性。使用数值模拟验证了所开发的理论框架，并以图形方式说明和讨论了相应的结果。