Accurate circadian (daily) timekeeping is essential to the survival of almost all organisms. A better understanding of circadian timekeeping is particularly important for the millions of Americans who travel overseas or work on irregular schedules. Recent research on circadian rhythms has discovered the complex biological mechanisms that generate timekeeping in each of the thousands of cells within the suprachiasmatic nuclei (SCN), the region of the brain that acts as the central circadian clock in mammals. Here, detailed mathematical models of neurons within the SCN are developed based on data from many experimental groups. These models are mathematically analyzed and simplified to determine the key properties that govern timekeeping. New mathematical and numerical methods are developed to allow for the study of complex neuronal networks and determine how timekeeping emerges from the collective behavior of coupled oscillators. These results should be applicable to many biological systems.
Circadian clocks are studied as a model system in cellular biology and electrophysiology to determine design principles that can be applied to other physiological systems, particularly those consisting of coupled oscillators. Mathematical models of electrophysiology will follow the Hodgkin-Huxley formalism. Mathematical models of cellular biology will follow the mass action formalism. Recent experimental research on the mechanisms of circadian timekeeping in individual cells will be incorporated into these mathematical models. Understanding the dynamics controlling circadian timekeeping will be extremely helpful for the field of circadian rhythms, which seeks to determine how a large number of proteins, ion channels and neurons work together to form the body's central clock. The general mathematical work in this proposal includes studying the attractors of biochemical feedback loops through iterative maps, a population density method for efficiently simulating large dimensional neuronal oscillators, and a new ansatz that can reduce a model of a large number of coupled oscillators to a two dimensional model. These mathematical approaches will be tested using our models of circadian timekeeping.
