Spatiotemporal pattern formation in some nonlocal and local dynamical systems

一些非局部和局部动力系统中的时空模式形成

• 批准号: 1311414
• 负责人: Gregory Faye
• 金额: $ 12.73万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-06-01 至 2014-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1311414&HistoricalAwards=false
• 关键词: Spatiotemporal; pattern; formation; some; nonlocal

This proposal aims at understanding the spatiotemporal patterns of some spatially extended systems that are organized around a bifurcation. More specifically, this research will study patterns emerging spontaneously and in a self-organized fashion for problems related to visual cortical activity and convections patterns. Model systems are nonlocal and local equations defined on unbounded spatial domains. An approach is to use a neural fields setting (nonlocal) to study the spatiotemporal dynamics of the visual cortex and the Swift-Hohenberg family (local) to analyze the formation of domain boundaries in convection patterns. It is proposed to focus on three mains projects. The focus of the first project is to give some new insights into the spatiotemporal dynamics of visual hallucinations and ongoing cortical activity in the presence of hexagonal symmetry close to different types of bifurcations. A second project is to provide existence and stability results of different types of two-dimensional cortical waves such as planar traveling waves, concentric traveling waves and spiral waves. Finally, the last project is to extend the analysis of grain boundaries into three directions: i) study small angle grain boundaries, ii) show the existence of asymmetric grain boundaries and iii) investigate hexagonal grain boundaries. From a phenomenological perspective, it is proposed to investigate the role of the different parameters of the models in a systematic fashion.The methods that are develop have a wide applicability in neurosciences, fluid and crystal dynamics. Visual hallucinations, two-dimensional cortical waves and convection patterns are most of the time organized around spatiotemporal coherent structures. A detailed theoretical and conceptual analysis of their structure, stability properties, and bifurcations is essential.

这个建议的目的是了解一些空间扩展的系统，围绕一个分叉的时空模式。更具体地说，这项研究将研究自发出现的模式，并在一个自组织的方式与视觉皮层活动和对流模式的问题。模型系统是定义在无界空间域上的非局部和局部方程。一种方法是使用一个神经领域设置（非本地）来研究的时空动态的视觉皮层和Swift-Hohenberg家庭（本地）来分析对流模式中的域边界的形成。建议重点关注三个主要项目。第一个项目的重点是提供一些新的见解的时空动力学的幻视和正在进行的皮质活动中存在的六边形对称接近不同类型的分叉。第二个项目是提供不同类型的二维皮层波，如平面行波，同心行波和螺旋波的存在性和稳定性的结果。最后，最后一个项目是将晶界的分析扩展到三个方向：i）研究小角度晶界，ii）显示不对称晶界的存在和iii）研究六边形晶界。从唯象学的角度出发，系统地研究了模型中不同参数的作用，所发展的方法在神经科学、流体和晶体动力学等领域具有广泛的适用性。幻视、二维皮层波和对流模式大多数时候都是围绕时空相干结构组织的。对它们的结构、稳定性和分叉进行详细的理论和概念分析是必不可少的。