Two Problems: Nonlinear Wave Equations and Mathematical Neurobiology

该项目由两个完全独立的部分组成：一个是对特定非线性波问题的分析研究，另一个是对哺乳动物视觉皮层中的方向调整的计算和建模研究。 第一部分考虑一个特殊的非线性偏微分方程耦合系统，PI和M。雪莱已经通过计算证明了有限时间奇点的“箍缩型”。 该系统是一个模型的一对不混溶的无旋无粘流体在二维流动。 在奇点中，两个演化函数中的一个从严格的正值变为正值，除了在一个点处为零之外，奇点对应于一股流体被一分为二。 严格正解的存在性和正则性问题将被考虑，因为将能量方法来证明有限时间奇异性。 在第二部分，PI将与实验人员B合作。Shapley和D. Ringach在方向调整的各个方面。 提供建模和计算的经验，各种模型将与实验结果进行比较，计算结果被用来建议进一步的实验的可能性增加。

The project consists of two completely separate parts: a largely analytical study of a particular nonlinear wave problem and a computational and modelling study of orientation tuning in the mammalian visual cortex. The first part considers a particular coupled system of nonlinear PDEs which the PI and M. Shelley have already computationally shown to have finite time singularities of "pinching type". The system is a model for the flow of a pair of immiscible irrotational inviscid fluids in two dimensions. The singularity, in which one of the two evolving functions goes from being strictly positive to positive except for vanishing at a point, corresponds to a strand of fluid being broken in two. Questions of existence and regularity of strictly positive solutions will be considered, as will energy-methods to prove a finite-time singularity. In the second part, the PI will work with experimentalists B. Shapley and D. Ringach on various aspects of orientation tuning. Providing experience in modelling and computation, various models will be compared to experimental results, with the added possibility of the computational results being used to suggest further experiments.