New Directions in Non-linear Mathematical Asymptotics. Major challenges such as predicting epidemics or modelling cancer rely on our understanding of simple mathematical models with extremely complicated solutions. The first and only model in the literature to reproduce the three-phase cycle of immune response in HIV/AIDS was based on cellular automata. Its results are extremely sensitive to infinitesimally small changes in parameters. Yet, no technique exists to study such variation in cellular ....New Directions in Non-linear Mathematical Asymptotics. Major challenges such as predicting epidemics or modelling cancer rely on our understanding of simple mathematical models with extremely complicated solutions. The first and only model in the literature to reproduce the three-phase cycle of immune response in HIV/AIDS was based on cellular automata. Its results are extremely sensitive to infinitesimally small changes in parameters. Yet, no technique exists to study such variation in cellular automata. This research will provide new methods for prediction and analysis of such models. Read moreRead less
Determination of Conductivity Values for Anisotropic Tissue. Well established mathematical models governing the electrical potential in biological tissue can be combined with measurements of the electric potential on the surface of the tissue to provide insight into subsurface tissue damage. However, before such observations can be convincingly accepted, reliable values for the tissue conductivity must be obtained. The aim of this project is to develop mathematical techniques to calculate the co ....Determination of Conductivity Values for Anisotropic Tissue. Well established mathematical models governing the electrical potential in biological tissue can be combined with measurements of the electric potential on the surface of the tissue to provide insight into subsurface tissue damage. However, before such observations can be convincingly accepted, reliable values for the tissue conductivity must be obtained. The aim of this project is to develop mathematical techniques to calculate the conductivity values so that one can apply the equations to solve problems of potential distribution and proceed to accurately simulate electrical potential distributions in damaged tissue. More accurate and reliable conductivity values will allow a better understanding of the way electric current moves through the heart which, in turn, will result in more efficient defibrillators and better diagnosis of abnormal function.Read moreRead less
Multi-scale modelling of cell migration in developmental biology. Interpretative and predictive tools are needed for the comprehensive understanding of directed cell migration in the medical sciences. Mathematical models and modelling methodologies developed in this project will make a significant contribution to the investigation of cell migration and the testing and generation of hypotheses. Such models are needed to understand observed cellular patterns. This project will contribute to knowle ....Multi-scale modelling of cell migration in developmental biology. Interpretative and predictive tools are needed for the comprehensive understanding of directed cell migration in the medical sciences. Mathematical models and modelling methodologies developed in this project will make a significant contribution to the investigation of cell migration and the testing and generation of hypotheses. Such models are needed to understand observed cellular patterns. This project will contribute to knowledge of normal and abnormal developmental processes, especially in embryonic growth. Understanding these processes should lead to prediction and treatment of congenital disorders and contribute to a healthy start to life.Read moreRead less
A geometric theory for non-standard relaxation oscillators. This project aims to develop new geometric methods for the analysis of multi-scale models of biological rhythms, and design diagnostic tools to identify key parameters that cause and control these signals. Rhythms, such as breathing, neural and cardiac rhythms and pulsatile hormone secretion, are central for life. Many important biochemical cell signals exhibiting relaxation-type behaviour cannot be rigorously analysed with standard dy ....A geometric theory for non-standard relaxation oscillators. This project aims to develop new geometric methods for the analysis of multi-scale models of biological rhythms, and design diagnostic tools to identify key parameters that cause and control these signals. Rhythms, such as breathing, neural and cardiac rhythms and pulsatile hormone secretion, are central for life. Many important biochemical cell signals exhibiting relaxation-type behaviour cannot be rigorously analysed with standard dynamical systems tools due to an inherent non-uniform time-scale splitting in these models. This project aims to develop a unified mathematical theory that weaves together results from geometric singular perturbation theory and algebraic geometry to explain the genesis of complex rhythms and patterns in biological, non-standard, multi-scale systems, both at individual and network level.Read moreRead less
Geometric methods in mathematical physiology. This project will develop new geometric methods for the analysis of multiple-scales models of physiological rhythms and patterns, and will design diagnostic tools to identify key parameters that cause and control these signals. Thus, this project will deliver powerful mathematics for detecting and understanding fundamental issues of physiological systems.