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.
