Transition and pattern formation in physical and physiological systems

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The application of mathematical ideas to open problems in physics and biology is an ever growing area of research. The catalyst of this growth is scientific computing, which allows us to attack problems of a complexity that was, until recently, out of reach. In the current proposal, the mathematical theory of dynamical systems is used to study human brain dynamics and fluid turbulence. Although apparently unrelated, these applications have many traits in common. For instance, they are both known to be strongly nonlinear, and they both give rise to signals that can be coherent over a range of spatial scales and time scales. One common measurement of brain dynamics, the electroencephalograph, often shows oscillations at preferred frequencies, like the 8-13Hz alpha frequency, and also shows "brain waves", synchronized activity that spreads out over the cortex. In fluid turbulence we see vortical structures form, interact, and break down. The work proposed here aims to explain the dynamics and origin of such coherent behaviour.Over the last decades, the application of computational dynamical systems theory has unveiled part of the explanation. In fluid dynamics, some questions on how turbulent motion emerges from smooth flow have been answered in terms of buidling blocks like equilibrium and time-periodic flows. In brain modelling, bifurcation analysis has shed some light on transitions from normal to pathological states like seizures. However, there is still a lot of work to be done to bring this type of analysis closer to real-world problems. I propose several projects that attempt to close the gap.One project concerns the analysis of a physiologically plausible model of electrical signals in the human cortex. We will try to discern robust behaviour of the model as we vary its many parameters, linked to physiological processes, and classify it using bifurcation theory. In fluid turbulence, we will attempt to compute solutions that are localized in space and time, as are flows often observed in experiments, and to compute novel time-periodic solutions that have a Kolmogorov similarity spectrum, the hallmark of turbulence.