分数阶神经元耦合系统的分岔与控制研究

A major difference between fractional order differential equations (FODEs) and integer order differential equations is that FODEs possess memory, while the main feature of nervous systems involves memory. Modelling the behavior of nervous systems by the FODE has more advantages than the classical integer order modelling. The analysis of the dynamic behaviors of fractional order coupled neuronal systems has a higher precision. The bifurcation and control of fractional order coupled neuronal systems has been the frontiers study in computational neuroscience. In this project, we shall employ the theory of FODEs, the complex network theory, the theory of nonlinear dynamics, and numerical simulations to conduct the following researches: We will introduce the fractional order into neuronal model. Based on the fractional order neuronal model, we shall analyze the stability of the equilibrium point and do a bifurcation analysis to obtain the bifurcation condition by choosing the order as the bifurcation parameter. Then, a few related bifurcation problems will be dealt with for neural networks having nodes of fractional order neuronal models in this project. The effect of the fractional order, the coupling model, the coupling strength, and the coupling delay on the bifurcation of neural networks will be first treated analytically and numerically. We will construct a systemic methodology to study the bifurcation of fractional order networks. Finally, we can have clear pictures on the bifurcation of fractional order neural networks and determine the characteristic factor influencing network dynamics. Bifurcation control will be carried out by using the pinning control and state feedback control for fractional order neural networks, which is not reported before for bifurcation control problems. It is hoped that this project could help to further understand the bifurcation characteristics for nervous systems and the analytical results for bifurcation control could be useful for the diagnosis and treatment of nervous system diseases.

分数阶微分方程在描述具有记忆性的神经系统时比传统整数阶模型更具优势，基于分数阶微分方程的神经元耦合系统动力学研究可以显著提高其分析精度。探索分数阶神经元耦合系统分岔特性，研究其分岔控制的一般方法是计算神经科学领域的前沿课题之一。本项目借助分数阶微积分理论、非线性动力学理论及复杂网络理论等方法，结合计算机模拟从事以下研究：(1)建立分数阶神经元模型，分析系统稳定的阶次范围，确定系统发生分岔的阶次阈值，揭示阶次对神经元系统分岔的影响规律；(2)基于所建模型，研究以分数阶神经元为节点动力学的耦合网络系统的分岔问题，探讨分数阶次、耦合方式、耦合强度及耦合时滞对网络分岔的影响，提出一套系统的研究分数阶网络分岔的方法；(3)进而研究分数阶神经元耦合系统的分岔控制，确定影响网络动力学能力的关键特征量，特别是分数阶次对网络分岔的影响，利用状态反馈法和牵制控制思想，设计有效的分岔控制策略。

探索分数阶神经元耦合系统分岔特性，研究其分岔控制的一般方法是计算神经科学领域的前沿课题之一。本基金项目主要研究分数阶神经元耦合系统建模、分岔及优化控制，构建分数阶网络分岔及控制理论，解决分数阶网络动态分析、优化和控制理论中的部分难题。主要成果如下：1. 分数阶动力系统的分岔理论还未建立，提出了分数阶系统的分岔发生条件，研究了具记忆特性的分数阶递归神经网络的稳定性与分岔，给出了分数阶神经网络的稳定性和分岔判据。利用残留谐波平衡方法和双尺度展开法，得到了精度较高的分数阶振子的渐近解。2. 目前整数阶系统的分岔控制研究已有大量结果。针对分数阶系统，提出了基于状态反馈的分岔控制策略，为探索分数阶网络分岔控制开辟了新的道路。状态反馈控制器有效延迟了不期望的分岔，使分数阶网络在更大阶次范围仍保持稳定。3. 突破了当前神经网络分岔研究局限于低维、时滞单一的现状，研究了具星型结构的高维混合时滞（离散与分布时滞）网络的分岔，给出了分岔条件，得到了确定分岔周期解稳定性和方向的实用算法，部分解决了高维多时滞系统分岔的难题。4. 利用Lindstedt-Poincare方法，而不是常规的中心流形定理，分析了在时滞充分小(或充分大)情形下，神经网络的分岔特性，给出了分岔周期解的近似表达。设计了时滞反馈控制器，实现了分岔点延迟的控制目标。本项目取得了一系列国际领先的研究成果，在国内外同行中产生了重要影响。已在国际重要学术刊物，如IEEE Transactions on Neural Networks and Learning Systems、IEEE Transactions on Cybernetics、Neural Networks、International Journal of Computer Mathematics、Neurocomputing、Asian Journal of Control、Applied Mathematics and Computation等发表或接受SCI论文16篇，初步统计被SCI刊物他引40余次。