| Abstract: |
| Homeostasis is a biological phenomenon in which a function of the state of the system remains approximately constant as an input parameter varies. Recently, Golubitsky and collaborators have developed a mathematical theory of homeostasis based on methods from singularity theory. In this theory, the state is a steady state, and the function is often a coordinate of the steady state. The weaker requirement that a function of the state remain approximately constant is replaced by the requirement that the derivative of the function with respect to the input parameter vanishes at an isolated point. This notion of a zero derivative has the name (steady-state) infinitesimal homeostasis, by which we mean that we are focusing on a function of a system steady state. Homeostasis can also apply to many oscillatory biological processes. For example, the period of circadian rhythms remains approximately constant as parameters such as ambient temperature and gene expression levels vary. In this talk, I will present results on infinitesimal homeostasis for periodic phenomena in models that undergo nondegenerate Hopf bifurcation. Our main result in this setup is that singularity theory can be used to find period homeostasis. This result is illustrated using numerical simulations of a simple model of the mammalian circadian clock. Our theoretical results are based on having two parameters (a bifurcation parameter and an input parameter) and the $S^1$ symmetry of Hopf normal form. |
|