My graduate studies focused on developing a physiologically accurate mathematical model of the human sleep-wake system. This involved using equations to represent the chemical and electrical mechanisms that govern sleep-wake cycle dynamics. Sleep occupies a large portion of the lives of all mammals - humans sleep for roughly one third of their entire lifetime. Despite encompassing such a large part of our lives, the physiological mechanisms and impacts of sleep are still not completely understood. Modeling sleep-wake dynamics can provide insight into the complex neural mechanisms that control sleep as well as provide a method of noninvasive experimentation. Noninvasive experimentation and predictive modeling are indispensable tools for investigating solutions of all biological problems, not just those related to sleep. My goal after graduate school was to continue my studies and computational modeling work as a postdoctoral research assistant. This would prepare me further for a career in applied mathematics research and academia. I received the NSF MSPRF, and took this opportunity to serve as postdoctoral researcher and instructor at NYU. I continued my work doing sleep-wake cycle modeling with the guidance of Dr. Charles Peskin. The three major goals of this project were to develop a more accurate, first-principles-derived model demonstrating the chemical and electrical mechanisms of the Human Sleep Wake System, to develop a deeper understanding of the physiology governing the human sleep wake system and to overall help bridge the gap between biology and applied mathematics through collaborative efforts. I used the intersection of mathematical theory and neurological and general physiological theory to develop a system of equations describing the chemical and electrical dynamics of communicating neural cells. This set of equations could then theoretically be entered into computing software, such as Python or MATLAB, to obtain solution trajectories that provide key information about an individual’s sleep cycle, such as optimal sleeping and waking times for a particular schedule. The model can also be used to test the effects of other chemicals, such as pharmaceuticals, on the system without using human or animal subjects. My research helped to bring awareness about the importance of sleep research to a part of the mathematical community. This research also helped demonstrate the merits of phenomenological modeling to the mathematical community. Most of the techniques used for computation and analysis are applicable to a wide variety of systems, not just biological systems. These techniques are also useful for educational purposes, giving students options for checking and solving problems, as well as showing them the physical motivation for using these techniques. The method of deriving and analyzing this model, as with any model, is based on equal parts mathematical prowess and scientific intuition. All of the derivation and analysis steps have both mathematical and physical meaning, and it is important to check the validity of both every step of the way. One of the most important things I developed while working on this research is the Algorithm for Constructing a Mathematical Model, which is a step-by-step guideline for solving modeling problems. This Algorithm can be used by all of the mathematical community, student and professional researcher alike, when solving complex problems. My research demonstrates the benefits of organization and attention to detail in modeling, as well as the benefits of using both first-principles and empirical data to derive models. I was able to relay my techniques and findings to students and other researchers in the community, hopefully inspiring them to take up similar methods. While participating in this project, I had ample opportunities to discuss and disseminate my research in interdisciplinary environments. My favorite arena to discuss research and techniques is in the classroom setting, where one is able to pass on knowledge and techniques to others. In the act of passing on knowledge to others, you become stronger as a mathematician and researcher yourself. This project helped me realize the importance of getting students inspired by and interested in mathematics at an early age, as well as teaching students through research and relatable applications. I realized that I did not want to stay in the world of college academia - rather, I wanted to relay the joys of mathematics and research to a larger audience, and a younger audience. I saw an opportunity to influence the course of education in the United States that also allows me to also continue to do research and develop models. Since the end of my tenure, I have been developing an academic tutoring and consulting company based on my teaching and research principles that I believe will help to positively influence American education. The NSF MSPRF not only helped me to successfully complete the goals I set out to achieve after graduate school, it also set me on path to a happy and prosperous future.
