My research is focused on applications of dynamical systems theory to areas outside of mathematics, in particular to understand various oscillatory phenomena in cell biology, biochemistry, and neuroscience. Dynamical systems theory provides a powerful framework for understanding temporal patterns. Pertinent mathematical questions of biological interest are: existence and stability of equilibria, periodic oscillations, switching phenomena, and bifurcations, i.e., changes of these dynamical properties as key parameter vary.
Slow-fast dynamical systems in cell biology
One important aim of my work is to extend and develop mathematical methods for the mathematical analysis of ODE models arising in cell biology. I focus on models of intracellular processes of great biomedical importance, e.g., cell division cycle.
Realistic models arising in applications are typically too large for theoretical analysis. However, there is evidence from simulations that often only a small or moderate number of components or parts of these large systems play essential roles, while large parts of the system have a more passive or even negligible role. The mathematical reason for this behavior is the occurence of dependent variables and parameters with very different orders of magnitude. Hence, some variables may vary little and can thus be treated as constants. Some parameters may have almost no effect and can be neglected. Some variables may rapidly approach a (quasi)equilibrium and can thus be slaved to other variables. In mathematical terms this allows the systematic use of regular or singular perturbation methods. In particular slow-fast dynamical systems, i.e., systems with solutions varying on very different timescales are abundant in biology in general and in cellular biology in particular.
The approach in my project relies strongly on novel dynamical systems methods for systems with multiple time scale dynamics, known as geometric singular perturbation theory (GSPT). Interestingly, the models under investigation do not have the standard form of slow-fast systems and are not covered by the existing theory and the analysis must be based on identifying and using hierarchies of local approximations based on various – sometimes hidden – forms of scale separation.
Relaxation oscillations in slow-fast systems beyond the standard form
Relaxation oscillations are highly non-linear oscillations, which appear to feature many important biological phenomena such as heartbeat, neuronal activity, and population cycles of predator-prey type. They are characterized by repeated switching of slow and fast motions and occur naturally in singularly perturbed ordinary differential equations, which exhibit dynamics on different time scales. Traditionally, slow-fast systems and the related oscillatory phenomena – such as relaxation oscillations – have been studied by the method of the matched asymptotic expansions, techniques from non-standard analysis, and recently a more qualitative approach known as geometric singular perturbation theory.
It turns out that relaxation oscillations can be found in a more general setting; in particular, in slow-fast systems, which are not written in the standard form. Systems in which separation into slow and fast variables is not given a priori, arise frequently in applications. Many of these systems include additionally various parameters of different orders of magnitude and complicated (non-polynomial) non- linearities. This poses several mathematical challenges, since the application of singular perturbation arguments is not at all straightforward. For that reason most of such systems have been studied only numerically guided by phase-space analysis arguments or analyzed in a rather non-rigorous way. It turns out that the main idea of singular perturbation approach can also be applied in such non-standard cases.
This thesis is concerned with the application of concepts from geometric singular perturbation theory and geometric desingularization based on the blow-up method to the study of relaxation oscillations in slow-fast systems beyond the standard form. A detailed geometric analysis of oscillatory mechanisms in three mathematical models describing biochemical processes is presented. In all the three cases the aim is to detect the presence of an isolated periodic movement represented by a limit cycle. By using geometric arguments from the perspective of dynamical systems theory and geometric desingularization based on the blow-up method analytic proofs of the existence of limit cycles in the models are provided. This work shows – in the context of non-trivial applications – that the geometric approach, in particular the blow-up method, is valuable for the understanding of the dynamics of systems with no explicit splitting into slow and fast variables, and for systems depending singularly on several parameters.