Current research projects
- Homeostasis as a control system
Collaborators: Yan Fossat, Adam Palanica (Klick Labs); Jacob Morra (Western).
Related publications: VPFM.
Key words: homeostasis, proportional-integral control, alternative biomarkers.
Abstract: Due to the proliferation of wearable and portable smart devices, an increasing number of data streams related to homeostasis of the human body are becoming available. Such quantities as core temperature, heart rate and blood glucose level can be measured quasi-continuously and the data can be read instantly on a smart phone or computer. In this research project, we explore the idea that from such data we can extract early warning signs of homeostatic dysfunction which, in turn, may be correlated to diseases like diabetes and migraine. If so, we can construct "alternative biomarkers", i.e. biomarkers that are much cheaper, faster and less invasive to obtain from a subject than the "gold standard" tests for specific conditions. Rather than to restrict the search for alternative biomarkers to data analysis, we try to do "forward modelling", i.e. the formulation of parsimonious dynamical models that predict the measured data.
- Unstable periodic orbits in homogeneous, isotropic turbulence
Collaborators: Shigeo Kida (retired, see Researchgate); Genta Kawahara (Osaka University, Japan); Tatsuya Yasuda (Nagoya Institute of Technology); Alberto Vela-Martín (Universidad Politécnica de Madrid).
Related publications: V3, VKK, KKV, VKY, VVKY, VVK.
Key words: turbulence, periodic orbit theory, scientific computing.
Abstract: In the first project, we computed unstable periodic solutions in Kida-Pelz flow and investigated their statistical and dynamical properties as well as their dependence on the Reynolds number. The ultimate goal is to compute periodic orbits which reproduce Kolmogorov statistics in the inertial range. Since the number of degrees of freedom this requires is too high for our current numerical methods and hardware, we resorted to LES modelling. In LES of box turbulence we identified a highly unstable periodic orbit that reproduces a small but significant Kolmogorov spectrum. We are currently investigating its Floquet spectrum.
- Scaling and dynamics in the Kuramoto-Sivashinsky equation
Collaborators: Kazumasa Takeuchi (Tokyo Institute of Technology, Japan).
Related publications: V4.
Key words: Yakhot's conjecture, scaling laws, scientific computing. Abstract: There is a long-standing conjecture due to Yakhot that the long-time, large-scale behaviour of the Kuramoto-Sivashinsky equation falls into the Kardar-Parisi-Zhang universality class, in which we find mostly stochastic models describing, for instance, ballistic deposition. There is no conclusive theoretical evidence in favour of this conjecture and numerical evidence is hard to produce. One must perform a great number of very accurate simulations on very large domains in order to obtain reliable statistics. In this project, we are trying to relate scaling exponents obtained from numerical experiments to those predicted by KPZ theory and dynamic renormalization, with the verification or rejection of the conjecture as the ultimate goal.
- Modelling collective motion of cells
Collaborators: Luciano Buono, Mitchell Kovacic, Eryn Frawley, Hendrick de Haan; Raluca Eftimie (University of Dundee, UK).
Related publications: bBEKV.
Key words: mathematical modelling, nonlocal PDEs, collective motion.
Abstract: Mitchell and Eryn studied models of animal aggregation in one and two spatial dimensions, respectively. These models are complicated because they contain nonlocal terms, modelling the processing of information by individuals in a finite spatial range aroud them. While these models were mostly meant to apply to large mammals, we now think that they might be particularly useful for decribing a certain type of cell motility in the high density limit. If we manage to tune a nonlocal PDE model to describe the collective motion of these cells, we could answer various questions about the self-organisation and dynamics without resorting to massive agent-based simulations.