Most people in my age group are simply not getting adequate sleep to function. This is a problem I really wanted to solve and wanted to delve deeper into it. As a mathematician, I enjoy solving complex problems, whether it be Reimann's work in approximations, Kleeper's discovery and derivations of planetary laws, Weyl's work with groups and isomorphism, or Schrodinger's works in quantum mechanics and relevant systems, I started research on ways mathematics, the language that permeates and reverberates so thoroughly throughout the known universe, could help in solving the problem. As a research intern at Arcascope,

I got an opportunity to delve deeper into the research of applied mathematics on circadian rhythms. Arcascope is a startup that builds products centered on sleep and the effects of light. It took me about two years to complete this work - over the course of these years, I worked indefatigably to create an efficient MVP which utilized advanced APIs and geolocation & metadata. I started this project in 2021 and just wrapped it up in January of 2024. The 2 most important applications I have developed are –

Cos.R - It is a natural harmonic oscillator simulator, and it shows oscillations in the body’s circadian rhythms. It is based on the research of differential equations of sleep modeling, like Chronauer – Juwett model. This application helps to visualize the results of the ODEs and creates a framework in which natural oscillations ingrained in the body can be reproduced. This helps turn theoretical literature and research into practical results via helping to see what body functions are in terms of natural oscillations and what they look like.

I consulted trial-based academic research on circadian rhythms. I worked on the code/software in modules, dealing with each component separately and breaking down and designing the functions and methods I would be using so that they didn't give me trouble in future endeavors. In addition to this, I used layers of reviews and PRs (pull requests) from my advisors to update and revise my code.

Sleepstructure.R - My sleep structure application is made in R language, Shiny studio. The sleep structure asks information to form the user “UI” in order to generate optimal sleep and wake time for adolescent/students. The application MVP (minimal viable product) is made to take in a location and “the user city” and uses that input and geo spatial information in R dependencies. In addition to this the applications also uses circadian rhythms modeling based of an earlier software repository.

The aim of the Sleepstructure.R application is to create an API (application

programming interface) to recommend optimal fall asleep and wake times. So the students can enjoy a more fulfilling sleep schedule better suited for their everyday life because of it’s adaptive UI, this application allows for local data to be transferred into real results for students who maybe wanting a a better sleep schedule.

An example from the Sleep Structure app.

The development of these apps had a lot of scientific, research-inspired thought behind them. Much of the research that went into the work for these apps were based on the studies of circadian rhythms – it would be a good idea to use some examples of papers that studied the effects of variables like light and circadian rhythms on sleep patterns, especially in the younger age groups.

The apps were crafted with a conceptual framework in mind: sleep structures are often disrupted by certain parameters like jetlag or social constraints. For example, in “The effects of self-selected light-dark cycles and social constraints on human sleep and circadian timing: a modeling approach, ” we see oscillations, sometimes consistent or inconsistent, for sleep. Social constraints like school start times (which the app deals with through an input box for school start times) further disrupt cycles that are vital for human sleep.

This figure from the aforementioned study illustrates a relationship that shows the internal pressure from your rhythms that encourages you to go to bed at a certain time and the days of the week, a discrete variable.

In “A simple model of the human circadian pacemaker” by Forger, Jewett, & Kronauer, I used an instance of mathematical modelling to help me with creating cos.R’s ODE solving algorithm. This particular study focued on differential equations modeling natural light and its effects.

Examples of equations that incentivized my cos.R app making.

The results of the mathematics in this work were focused on oscillations and nonlinearity which yielded helpful information for how to compute certain variables from givens. A snapshot from cos.R is shown below.