Analysis of Sleep-Wake Transition Dynamics by Stochastic Mean Field Model and Metastable State

Sleep-wake dynamics change across postnatal development: wake bout durations of newborn rats are exponentially distributed but become power law distributed within three weeks, in contrast to sleep bout durations that exhibit exponential distributions at all ages. To gain insight into network processes that may underlie sleep-wake transitions in these animals, a stochastic model is constructed, consisting of two mutually-inhibitory Erdos-Renyi or scale-free random graphs in which each node represents a neuron that can be in one of three possible states (excited, basal and inhibited) with distinct firing rates. The two graphs alternate between high and low levels of activity with durations that can have either an exponential or a power law distribution depending on parameters. The dynamics are investigated by numerical and analytical studies of stochastic di↵erential equations related to the model, including the four-dimensional stochastic mean field equations. The full model is approximated by one dimensional reflected Brownian motion which has a power law distribution of bout durations on an intermediate regime, and the approximation is supported by large deviation theory and the penalization method. ii ACKNOWLEDGMENTS If I can only thank one person in this Acknowledgement, my advisor, Dr. Janet Best is the one. Her patient listening, many helpful suggestions and encouragement have been invaluable. She is a model for me as a researcher. I would like to thank Dr. Peter Kramer, a member of our sleep/wake research group, for providing sources of inspiration and fruitful suggestions. I would like to thank Dr. Fatih Olmez and Dr. Deena Schmidt for all the valuable discussions at the meeting. Especially, I thank Dr. Fatih Olmez for the numerical data from the nework model, and Dr. Deena Schmidt for proofreading this thesis. I would like to thank Dr. Ed Overman, and Dr. David Sivako↵ for serving on my dissertation committee. I would like to thank department of mathematics for its support during my graduate studies. Finally, special thanks to my husband Raeyong for providing emotional and physical support, and and my family members in South Korea. iii VITA 1998–2002 . BS in Mathematics Education, Korea University, Seoul, South Korea 2002–2006 . Graduate Study in Mathematics, Seoul National University, Seoul, South Korea 2007-Present . Graduate Teaching Associate, The Ohio State University PUBLICATIONS Y. -S. Chung, J. E. Kim and S. -Y. Chung Identification of multipoles via boundary measurements, European J. Appl. Math. 23 (2012), 289–313 J. E. Kim Mixed Mode Oscillations and Graded Persistent Activity Contribute to Memory Formation (in preparation) J. E. Kim, J. Best, D. Schmidt, F. Olmez, J. McSweeney and P. Kramer Analysis of Sleep-Wake Transition Dynamics by Stochastic Mean Field Model (in preparation) F. Olmez, J. E. Kim, J. Best, D. Schmidt and P. Kramer Improved Detection of Power-law Distributed Regions in Data Using Penalty Method (in preparation) iv FIELDS OF STUDY Major Field: Mathematics Specialization: Mathematical Biology v TABLE OF CONTENTS Abstract . ii Acknowledgments . iii Vita . iv List of Figures . viii List of Tables . xii CHAPTER PAGE