Spatiotemporal pattern formation in some nonlocal and local dynamical systems

University Of Minnesota-Twin Cities, Minneapolis MN

Investigators

Abstract

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.