Amari动力神经场静态解的存在性与稳定性研究

Amari''s dynamical neural field plays an important role in cognition, neurophysiology and neurodynamics. It has received more and more attentions and been introduced to application fields, including pattern recognition and machine learning, because of its good descriptions of the interaction between neurons, as well as revealing the principles of some cognition activities like working memory and short-term memory. When applied to investigating cognition activities, the neural activity distribution in the stationary solution of Amari''s dynamical neural field usually corresponds to a recognized or memorized pattern. Therefore, the properties of Amari''s dynamical neural field’s stationary solution are very important. Since Amari''s dynamical neural field is highly nonlinear, its dynamics is very complex. As a result, the research on its stationary solution is inadequate, which significantly limits its application in more extensive fields. This project investigates the existence and stability for the stationary solution of Amari''s dynamical neural field in multi-dimensional space, aiming at finding the general conditions for the existence of the stationary solution when neural field has step threshold function, investigating the existence and stability of local excited region in the stationary solution and evaluating its area, providing a theoretical foundation for its more extensive applications.

Amari动力神经场在认知科学、神经生理学和神经动力学等领域中具有重要地位。它能够很好的描述神经细胞之间的相互作用，揭示工作记忆和短期记忆等认知活动的原理，因此越来越受到人们的关注和重视，并逐渐被应用到模式识别和机器学习等应用领域之中。当Amari动力神经场用于研究认知活动的时候，其静态解的活性分布形式往往对应了某个被认知或记忆的模式，因此对其静态解性质的研究具有非常重要的意义。由于Amari动力神经场具有高度非线性，动力学性质十分复杂，目前对于其静态解的研究并不充分，极大的限制了Amari动力神经场方程在更广泛领域中的应用。本课题针对Amari动力神经场方程多维空间中静态解的存在性和稳定性进行研究，拟找到在阶跃阈值函数条件下静态解存在的一般性条件，研究在连续阈值函数条件下静态解中局部兴奋域的存在性和稳定性，给出局部兴奋域范围的估计方法，为其在更广泛领域的应用做好理论准备。

本课题针对Amari 动力神经场方程多维空间中静态解的存在性和稳定性进行研究，找到了在Sigmoid型阈值函数条件下静态解存在的一般性条件及其局部兴奋域的存在性和稳定性；找到了一种通过将活性分布转化为梯度动力系统，从而估计局部兴奋域范围的估计方法；发现了在间断和连续阈值函数条件下，动力神经场静态解一些动力学特性；在相关理论研究的基础上，给出了一种新的基于视觉动力神经场的数据聚类方法；为研究离散化后的动力神经场，对Cayley图的一些性质进行研究，得到了一些理论成果。项目按照原计划执行，发表相关论文4篇，达到了预期研究目标。