具有突触影响的脉冲时滞神经网络的动力学分析

Neural systems consist of a large number of neurons connected to each other, which have complicated network structures and lead to rich nonlinear dynamic behaviors. This project is discussed the dynamical behavior of impulsive delay neural network system with synaptic effects, which is to be applied to the qualitative analysis theory and numerical calculation methods, Through the above research, A deep understanding of nonlinear dynamical mechanism of synaptic effects on neural network dynamics, revealing the formation mechanism of different dynamics, and explore synaptic effects caused by the complex dynamic behavior. Thus, the study of dynamical behaviors of neural network systems not only have important theoretical significance for the mechanism and evolution of neural network systems, but also have important guidance in the applications of brain science and medical engineering, artificial intelligence techniques, and so on.

神经系统是由数目众多的神经元相互连接形成的复杂的网络结构，导致丰富的非线性动力学行为。本项目拟应用定性分析理论和数值计算相结合的方法，讨论具有突触影响的脉冲时滞神经网络系统的动力学性态，通过以上研究，深刻理解突触影响的神经网络动力学的非线性动力学机制，揭示不同动力学的形成机理，探讨在突触影响下所导致的复杂的动力学行为。因此，研究神经网络系统的动力学行为，不仅对于神经网络系统的机理和演化具有重要理论意义，同时对于脑科学和医学工程、人工智能技术等领域的应用也有重要指导作用。

本项目应用定性分析理论和数值计算相结合的方法，讨论具有突触影响的脉冲时滞神经网络系统的动力学性态。按原计划完成了项目的绝大部分内容，研究结果主要为：讨论了一类具有周期突触影响的时变脉冲神经网络周期解的存在性和稳定性。利用拓扑度的Mawhin的连续性定理和Halanay-type不等式分别获得了获得了周期解的存在性和稳定性。进一步，相关结果可以应用在一些脉冲神经网络上，我们举了一个具有两个种群的对数种群网络模型作为特例，数值模拟结果验证了结果的可行性。此结果已在线发表在“Journal of Biological Dynamics”。