Spatiotemporal pattern formation in some nonlocal and local dynamical systems

这一建议旨在理解围绕一个分支组织的一些空间扩展系统的时空模式。更具体地说，这项研究将研究与视觉皮层活动和对流模式有关的问题的自发和自组织的模式。模型系统是定义在无界空间域上的非局部方程和局部方程。一种方法是利用神经场设置（非局部）来研究视觉皮层的时空动态，利用Swift-Hohenberg族（局部）来分析对流模式中域边界的形成。建议集中于三个主要项目。第一个项目的重点是对视幻觉的时空动态和在接近不同类型分岔的六边形对称存在下正在进行的皮层活动提供一些新的见解。第二个项目是提供平面行波、同心行波和螺旋波等不同类型二维皮质波的存在性和稳定性结果。最后一个项目是将晶界分析扩展到三个方向：1)研究小角度晶界；2)证明不对称晶界的存在；3)研究六边形晶界。本文从现象学的角度，系统地探讨了模型中不同参数的作用。所开发的方法在神经科学、流体和晶体动力学中具有广泛的适用性。视幻觉、二维皮层波和对流模式大多是围绕时空相干结构组织的。对它们的结构、稳定性和分岔进行详细的理论和概念分析是必要的。

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.