Grant: Discovery Grants Program - Individual

The proposed research concerns complex behaviour which occurs in spatially extended systems. One important example of such a system is fluid turbulence. Although turbulence has been on the menu of physicists, mathematicians and engineers alike over the past century, many fundamental problems remain unsolved. One important reason for this tantalizing lack of understanding is the fact that the governing equations, due to Navier and Stokes, are used to simulate turbulence in numerical experiments, but not usually to analyse the results. This is commonly done by statistical means, which blur the intantaneous structure of the flow and discard important dynamical detail. Recently, however, increased computer power and a string of fresh theoretical ideas have enabled us to study solutions to the governing equations with the tools of dynamical systems theory. This kind of analysis gives us an unprecedented insight in the role of coherent structures, such as vortex tubes and sheets, in establishing the observed statistics of turbulence. We will address several open questions in isotropic and shear turbulence, using novel computational methods in conjunction with a detailed analysis of the elementary physical processes. Another area in which dynamical systems theory can fill the gap between phenomenology and the underlying processes is neuroscience. In the second component of the proposed research program we will attempt to explain some well-known, yet ill-understood features of the human cortex. Among a host of proposed models of human brain function, we will focus on a mean-field model of cortical dynamics. The output of this model is directly related to the signal measured with the electroencephalogram. This offers the exiting prospect of direct comparison of model analyis with experiment. In doing this we will attempt to find out which dynamical regime of the underlying mean field model can give rise to the observed spectral properties of the measured EEG, in particular to the well-known alpha and beta rythms.