Abstract: Circadian clocks govern daily behaviors of organisms in all kingdoms of life. In mammals, the master clock resides in the suprachiasmatic nucleus (SCN) of the hypothalamus. It is composed of thousands of neurons, each of which contains a sloppy oscillator - a molecular clock governed by a transcriptional feedback network. Via intercellular signaling, the cell population synchronizes spontaneously, forming a coherent oscillation. This multi-oscillator is then entrained to its environment by the daily light/dark cycle. Both at the cellular and tissular levels, the most important feature of the clock is its ability not simply to keep time, but to adjust its time, or phase, to signals. We present the parametric impulse phase response curve (pIPRC), an analytical analog to the phase response curve (PRC) used experimentally. We use the pIPRC to understand both the consequences of intercellular signaling and the light entrainment process. Further, we determine which model components determine the phase response behavior of a single oscillator by using a novel model reduction technique. We reduce the number of model components while preserving the pIPRC and then incorporate the resultant model into a couple SCN tissue model. Emergent properties, including the ability of the population to synchronize spontaneously are preserved in the reduction. Finally, we present some mathematical tools for the study of synchronization in a network of coupled, noisy oscillators. --------------------------------------------------------------------- Professor Linda Petzold, Australian Mathematical Sciences Institute Lecturer 2007-2008: see http://www.amsi.org.au/ Dr. Linda Petzold is currently Professor in the Department of Computer Science (Chair 2003-2007) and the Department of Mechanical Engineering, and Director of the Computational Science and Engineering Program at the University of California Santa Barbara. She received her Ph.D. in Computer Science in 1978 from the University of Illinois. From 1978-1985 she was a member of the Applied Mathematics Group at Sandia National Laboratories in Livermore, California, from 1985-1991 she was Group Leader of the Numerical Mathematics Group at Lawrence Livermore National Laboratory, and from 1991-1997 she was Professor in the Department of Computer Science at the University of Minnesota. Dr. Petzold is a member of the US National Academy of Engineering. She is a Fellow of the ASME and of the AAAS. She was awarded the Wilkinson Prize for Numerical Software in 1991, the Dahlquist Prize in 1999, and the AWM/SIAM Sonia Kovalevski Prize in 2003. She served as SIAM (Society for Industrial and Applied Mathematics) Vice President at Large from 2000-2001, as SIAM Vice President for Publications from 1993-1998, and as Editor in Chief of the SIAM Journal on Scientific Computing from 1989-1993.