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Research

Current Research Students

Sam Bielawa Applied Math Major class of 2025
Danielle Vogt Applied Math Major class of 2026.

Grants & Consulting

ROGER WILLIAMS LAW SCHOOL - Consultant on the Project titled: Determinants of passing the �鶹ɫ��Ƭ RI/MA bar exams on the first attempt. 2009-2011

MICROSOFT (Baptist Group) consultant on the Project titled: Does Microsoft Sales folks help drive the client to upgrade their operating system (windows)? – Seattle, WA. 2011

NASA Project PI for RI Space Grant: The Numerical Solution of the Helmholtz Equation for the Superellipsoid via the Gallerkin Method. Summer 2012

NASA Project PI for RI Space Grant: The Numerical Solution of the Helmholtz Equation for the Superellipsoid via the Modified Gallerkin Method for the Impedance Boundary Value Condition. Summer 2014

EPSCoR - RID Grant: The Numerical Solution of the Helmholtz Equation for the Biconcave Disk (Blood Cell) for the Dirichlet Boundary Condition: Mars Project. Spring 2015

NASA Project PI for RI Space Grant (Experimental Program to Stimulate Competitive Research): Numerical Solutions of the Radiosity Equation for the Spherical Quatrefoil and Spherical Rhombus on Mars. Summer 2016-17

NASA Project PI for RI Space Grant (Experimental Program to Stimulate Competitive Research): Numerical Solutions of the Radiosity Equation for the Spherical Quatrefoil on Mars. Summer 2017

Other Synergistic Activities

Reviewer for Journal of Student Research and International Journal of Industrial Mathematics

Co-Facilitator for QTLC – Quantitative Teaching and Learning Community.

Team teaching Biostatistics with a colleague in Environmental Science. The course is designed as an interdisciplinary course that bridges statistics and biological phenomenon from the theoretical perspective with the aid of JMP (SAS for Windows) software.

Council of Undergraduate Research (CUR) – National Councilor for Mathematics and Computer Science Division - 2011 - 2014.

Professional Associations

American Mathematical Society
Mathematical Association of America

Courses Taught

Biostatistics
Differential Equations
Linear Programming
Engineering Mathematics
Numerical Analysis

2020 Joint Mathematics Meeting

Successful Alumni

Katie Gilbert 

BS Biochemistry/BS Applied Mathematics '22

Kate is pursuing a PhD in Chemstry and a MS in Data Science at Brown University.

Joshua Abston

BS Applied Mathematics, 2021

Joshua has been accepted into the Masters program in Computational Applied Mathematics at the University of Edinburgh, Scotland.

Gibson Leavitt

BS Environmental Science / Applied Mathematics, 2020

Gibson has been accepted into a Ph.D. program at the University of North Carolina, Chapel Hill where he will be studying with Alberto Scotti working on turbulent boundary layer modeling. Since graduating in May, he has been working for the US Geological Survey in Woods Hole, Massachusetts, modeling nearshore hydrodynamics and sediment transport.

Cole Foster

BS Engineering / Applied Mathematics, 2020

Cole has not traveled too far as he started his Ph.D. journey at Brown University in the Department of Electrical and Computer Engineering. He has been awarded a full fellowship and will be working in the Laboratory for Engineering Man/Machine Systems  (LEMS) in the area of computer vision. Elizabeth Gilchrist - BS Biology / Applied Mathematics, 2019

Elizabeth Gilchrist

BS Biology / Applied Mathematics, 2019

Elizabeth was awarded a full fellowship to Washington State University, where she is enrolled in the Ph.D. program in Applied Mathematics. She is currently a TA for undergraduate math courses and will begin her research.

Kaia Lindberg

BS Mathematics, 2019

I am working at Liberty Mutual Insurance in Boston, MA in the analytics development program. This one year rotational program includes trainings, 3 3-month rotations, and other projects to build technical, analytical, and strategic skills. I’m currently in my first rotation in Claims Analytics where I’m building an automated Power BI report to monitor productivity and customer satisfaction data for first notice of loss calls. I’m also working on a project analyzing loss frequency trends in Maryland, Kentucky, and West Virginia and making recommendations to improve profitability in these states.

Tyler Simmons

BS Marine Biology / Applied Mathematics, 2019

Tyler Simmons received his Ph.D from the University of Maryland Institute of Physical Science and Technology Biophysics in May 2024.

Andrew DelSanto

BS Engineering / Applied Mathematics, 2018

Andrew received a Masters degree in Civil Engineering from the University of Massachusetts, Amherst in January 2021. He continued in the Civil  Engineering Ph.D. program with a full fellowship to support his studies. He studied the effect of climate change (frequency and severity) on extreme hydrological events (floods and droughts) using machine learning techniques.

Andrew DelSanto received a PhD in Civil Engineering from the University of Massachusetts, Amherst in 2023. Dr. Andrew DelSanto is currently an assistant professor in the Department of Construction Management at Western New England University, MA.

Hien Ngo

BS Applied Mathematics / Biology, 2018

Hein received a full fellowship to Boston University, where she pursued a Masters degree in Mathematics. 

Kangi Chen

BS Biology / Applied Mathematics, 2018

Kangi is in his first year of the Masters degree program in Applied Mathematics at the University of Southern California.

Qiuyang Deng

BS Applied Mathematics, 2017

I am studying at the London School of Economics and Political Sciences, learning game theory and optimization. There are two terms for teaching courses and a summer term for writing a dissertation paper. Six courses are required and I chose: Game Theory I, Algorithms and Computation, Continuous Time Optimization, Games of Incomplete Information, Fundamentals of Operational Research, and Mathematical Optimization. My dissertation paper is focused on the game theory with incomplete information, the myopic equilibrium, which is a new equilibrium in Game Theory studied by my supervisor and my working area is primarily to research more applications for this equilibrium. I will graduate with a Masters degree in Operations Research (Linear Programming) in August, 2018.

Jill Resh

BS Applied Mathematics / Graphic Design Communications, 2016

Jill is currently working for the U.S. Census Bureau in Washington, DC.

Hy Dinh

BS Mathematics / BS Mechanical Engineering, 2015

Hy Dinh received a PhD in Mechanical Engineering from Tufts University in 2022. Dr. Hy Dinh is currently in Redwood Materials in Nevada.

Jane Pleskunas (Pellegren)

BS Mathematics, 2011

I graduated from Boston University School of Public Health in May 2015 with a concentration in Epidemiology and Biostatistics. Throughout my graduate school tenure, I had the opportunity to develop my research skills further across a wide variety of topics including examining chronic traumatic encephalopathy exposures in NFL players, exploring a link between latent tuberculosis infection and body mass index values in the United States, as well as publishing a collaborative manuscript in the American Journal of Public Health examining firearm homicide rates in the United States. In addition to academics, I had the opportunity to work at Novartis Vaccines in the department of Global Medical Affairs helping to manage the third party communications to ensure the medical and epidemiologic accuracy. I am a Senior Public Health Epidemiologist at the Rhode Island Department of Children, Youth & Families.

Karishma Silva

Transfer, May 2011

Since graduating from the Master's in Science in Economics program at the London School of Economics, I moved to Malawi to work as an Overseas Development Institute Fellow at the Ministry of Education, Science and Technology. I now work in education policy as a consultant for the World Bank. One of the key projects I'm working on is the Malawi Education Sector Improvement Project, which is a $44.9 million project funded by the Global Partnership for Education. My work mainly focuses on implementation, policy development and impact evaluation.

Elizabeth Morgan (Silva)

BS Mathematics, 2009

My current job title is a Research Scientist II within the NYS Department of Health. I’m working in a Bureau handling the development and oversight of the NYS All Payer Database. This database will contain health care claims data from insurance carriers, health plans, third-party administrators, pharmacy benefit managers, Medicaid and Medicare in order to support research, management, evaluation, and analysis of the NYS health care system (especially in the monitoring of systemic efforts to reduce health care costs, and improve care quality and population health). Because this is a new and developing bureau my role is ever changing. The main focus of my position currently is to perform research, evaluation and analysis of these raw datasets as I work with our team on building a data model synthesizing these data streams to be utilized by other DOH researchers. It also includes responsibility for developing useful reports to consistently provide feedback on the volume, content, and quality of claims data being received from insurers. As this model becomes more defined my role will shift towards researching, developing and testing health care quality measures and provider assessment guidelines for APD dashboard designs which will be maintained by a contractor. 

Student Presentations

2021

Joint Mathematics Meeting (JMM)

January 6 - 9  Virtual

• Joshua Abston (Applied Mathematics '21)
  "A Computational Investigation of Ionic Transport and Gating due to Electrical Stimulation Treatments"
  • The precise mechanisms of DBS on ion flow is poorly understood and proves difficult to assess experimentally
  • This is an ideal area to investigate with Mathematical Modeling and Computational Simulation
  • Research suggests that DBS has an impact on ionic flux
  • We hypothesize that Ca 2+ ionic flow is enhanced by electrical stimulation treatments such as DBS
• Emma Dehetre (Engineering/Applied Mathematics '22)
  "COVID-19 Pandemic Volterra Integral Equation Model"
  • The COVID-19 pandemic has affected many people throughout not just the U.S. but across the whole world. The objective of this research project is to find a numerical solution through the Galerkin Method for the Volterra Integral Equation Model. The non-homogenous Volterra Integral Equation of the second kind is used to capture a broader range of disease distributions. Volterra Integral equation models are used in the context of mathematics, public health, and evolutionary biology. The mathematical model of this integral equation will yield convergence results for the COVID-19 data for Italy. The modeling of this country will be done using the Galerkin method, a type of numerical approximation using the Gaussian Quadrature nodes. Inspired by the COVID-19 pandemic, the model will include the number of initially infected individuals, the rate of infection, contact rate, death rate, fraction of recovered individuals, and the mean time an individual remains infected.
• Kate Gilbert (Biochemistry/Applied Mathematics '22)
  "Infectious Disease Model- COVID-19-Volterra integral Equation"
  • Inspired by the COVID-19 pandemic, this research investigates the feasibility of obtaining good convergence results for a model of the Volterra integral equation over the surface (geographic location). The Gallerkin Method was used to numerically solve the exterior boundary value problem. This model accounts for the number of initially infected individuals, suscept individuals, removed individuals, number of contacts per person, the recovery rate, and the total population. This model specifically looks at COVID-19 in South Africa for the first 200 days of the pandemic. The model accounts for the geography of the countries and uses Green’s Theorem. The numerical results of this research are expected to find good convergence for this model as well as limitations of the model such as the assumption for the number of contacts
• Kristen Norray (Engineering Spec/Applied Mathematics '21)
  "A Mathematical Modeling Approach to Cardiovascular Health and Interventions"
  • The leading cause of death in the United States is heart disease.
  • Mitral Stenosis is the narrowing of the mitral valve, resisting blood flow through the left side of the heart.
  • Pharmaceuticals are often used to help reduce the resistance in blood vessels.
  • Exercise has been shown to be beneficial in reducing symptoms for various diseases
• Elizabeth Wexler (Business Management/Mathematics '22)
  "Simulations of Transcranial Electrical Stimulation with Variable Tissue Conductivities"

2020

Joint Mathematics Meeting (JMM)

January 15 - 18  Denver, CO

• Matthew D'Amico (Applied Mathematics '20)
  "Modified Gakerkin Method for the Wiener-Hopf Integral Equation for a Semi-Inverted Cassini"
• Emma Dehetre (Engineering/Applied Mathematics '22)
  "Social Determinants of Health: Case Study of 10 U.S. States"
  • The Behavioral Rick Factor Surveillance System (BRFSS) is an annual survey conducted by the Centers for Disease Control and Prevention (CDC). This survey is given to all fifty states, as well as other U.S. territories such as Guam and Puerto Rico. We used the survey to conduct a social determinant model using a multiple logistic regression test and a principal component analysis model. By using these two statistical methods, it was determined how the variables impact one’s safety and general health which can be used to create recommendations for better public health outcomes in 10 U.S. states: Iowa, Massachusetts, Minnesota, Mississippi, New Hampshire, Pennsylvania, Utah, West Virginia, Wisconsin, and Wyoming. For the multiple logistic regression model, we used general health as the dependent variable with general health being defined as excellent, very good, and good as a positive outcome. Through the principal component analysis, 98.5% of the variation was explained by looking at one’s notion of safety relative to access to food.
• Lance Dengelegi (Applied Mathematics '20)
  "Characterizations of string stability with applications to automobile platoons"
  • The notion of String stability characterizes the longitudinal safety margins of automobile platoons. In this work, we provide two characterizations of string stability. One characterization offers a physical interpretation of string stability. The second characterization provides a simplified mathematical framework to analyze the safety margins of automobile platoons. We illustrate the practical and theoretical significance of each characterization. Additionally, we provide how these ideas support the design of Adaptive Cruise Control (ACC) systems and Cooperative Adaptive Cruise Control (CACC) systems
• Cole Foster (Engineering/Applied Mathematics '20)
  "Detecting Underwater Objects through Scattering Theory; the Wiener-Hopf Integral Equation"
  • The Galerkin method is used to numerically solve the exterior boundary value problem for the WienerHopf Integral Equation over the boundary of a Spherical Biconcave Disk. The Wiener-Hopf Integral Equation is a mathematical model representing the radiative transfer over the half plane of the Spherical Biconcave Disk. The equation is used in many diffusion problems, and can be used detect objects underwater through scattering theory. The scattering process forces some forms of wave radiation to deviate from a straight trajectory and includes deviation of reflected radiation from an angle predicted by the law of reflection. We used the Green’s theorem to solve the integral equation on the boundary of the surface for the Dirichlet problem. The Exterior Boundary problem will be solved using the Gaussian Quadrature Method, where rotations of the coordinates would be used to minimize the inherent singularity that is present in the fundamental solution of the equation.
• Jess Messina (Applied Mathematics '20)
  "Developing Stocks Volatility Prediction Model Using Neural Networks"
  • We implement machine learning techniques to develop stocks volatility prediction models. This is a typical time series problem: a study of the past evolution of the phenomenon with respect to time in order to predict the future trend. Generally, statistical models such as the Auto Regressive Moving Average (ARMA) model can be used to perform time series analysis. However, because of the availability of Big Data and easy access to computational resources, data scientists have recently been moving towards machine learning techniques to analyze these kinds of problems. Our model is based on Neural Networks, particularly a Long Short-Term Memory (LSTM), which is a type of Recurrent Neural Network (RNN). We execute this model in the Python programming environment using TensorFlow and Keras APIs. The model produced impressive fitting results when we implemented it in a few common stocks listed in NYSE and NASDAQ. Various model selection strategies, including cross-validation, are used to test the effectiveness and resilience of our model.
• Abigail Small (Applied Mathematics/Computer Science '20)
  "A Mathematical Approach to Assessing tDCS Efficacy for Post-traumatic Stress Disorder"
  • Post-traumatic stress disorder (PTSD) is a neurological condition which results from a traumatic experience caused by physiological shock or physical harm. Clinical results show success in combating the symptoms of PTSD with tran-scranial direct current stimulation (tDCS). Though effective, the underlying mechanisms of the treatment are not fully understood. To help elucidate reasons for its efficacy, a partial differential equation (PDE) based mathematical model of tDCS for PTSD has been implemented. Using the finite element method, numerical simulations generate results that predict and quantify the electrical energy distribution during tDCS sessions. Simulations utilize real-world electrode montages and treatment parameters for PTSD and a three-dimensional, MRI-derived cranial cavity with biologically-based tissue conductivities. Regions of the brain thought to be targeted by tDCS treatments are confirmed with in silico experiments, thus validating the model and approaches. Finally, a novel, PDE based multiscale mathematical model of tDCS is presented, which adds the ability to quantify neural tissue response via tDCS-induced transmembrane voltage polarization.

2019

International Engineering in Medicine and Biology Society Conference 

July 23 - 27  Berlin, Germany

• Abigail Small (Applied Mathematics/Biology '20)
  "Mathematical Modeling of Neurostimulation for Post-Traumatic Stress Disorder: A Migration towards Multiscale Modeling to Assess Neural Response to Transcranial Direct Current Stimulation Treatments"

Engineering in Medicine and Biology Society Conference on Neural Engineering (NER) 

March 20 - 23, San Francisco, CA

•  Kaia Lindberg (BS Mathematics '19)
  

Joint Mathematics Meeting (JMM)

January 18 - 21 Baltimore, MD

• Cole Foster (Engineering/Applied Mathematics '20)
  "Wiener-Hopf Integral Equation Model: Underwater Applications"
  • The objective of this research project is to find a n umerical s olution t o t he N eumann b oundary value p roblem for the Wiener-Hopf Integral Equation Model. The Wiener-Hopf Integral Equation has applications in radiative transfer, electromagnetics, and optical oceanography. The mathematical model of this integral equation will yield convergence results of incoming external waves over the surface of spherical shapes that satisfy Greens Theorem. The modeling of these shapes will be done using the Galerkin method, a type of numerical approximation using the Gaussian Quadrature nodes. Inspired by the loss of Malaysian Airline flight 370, this study into scattering theory and optical oceanography would help provide the theoretical framework for understanding light propagation in the ocean, which can aid in the search of submerged objects.
• Elizabeth Gilchrist (Biology/Applied Mathematics '19)
  "A Computational Approach for Constructing an Intracellular Signaling Pathway Mathematical Model with Applications to Parkinson's Disease"
  • Parkinsons disease (PD) is the second most common neurodegenerative disorder. Despite this, there is no cure and the cellular level pathogenesis remains elusive. In an attempt to gain new insights, we created a mathematical model of the intracellular signaling pathway of a dopaminergic neuron cell with application to PD. A comprehensive literature search was conducted to construct a wiring diagram, which was used to generate a system of ordinary differential equations using the law of mass action and the Michaelis-Menten equation. Many of the kinetics are presently unknown, so a novel computationally-based reverse engineering method was used to identify them; this approach uses expected system behavior and the Metropolis Algorithm to numerically determine appropriate values. Suitable rates were ranked based on performance in a phenotype-based computational assessment, and then robustly screened using a k-means clustering assessment, sensitivity analyses, and an eigen-analysis. The result is a mathematical model that efficiently emulates the signaling network of a dopaminergic neuron model; it showcases the intracellular processes of both a healthy and PD-like dopaminergic neuron.
• Kaia Lindberg (Mathematics '19)
  "Investigating Cellular-Level Effects of Neurostimulation Therapies with a Partial Differential Equation Based Mathematical Model"
  • Neurostimulation therapies have demonstrated success in mitigating symptoms of neurodegenerative diseases, but the cellular-level impacts of these treatments remain elusive. We have implemented a mathematical model that integrates the Poisson-Nernst-Planck system of PDEs and Hodgkin-Huxley based ODEs to model the effects of this neurotherapy on transmembrane voltage, ion channel gating, and ionic mobility. The PDEs are solved using the finite element method on a biologically inspired discretized domain. Our results suggest two possible mechanisms by which neurostimulation achieves therapeutic success. First, neurostimulation polarizes the cell membrane, elevating resting membrane potential to facilitate action potential firing. Second, a neurostimulation-induced calcium influx alters cytosolic calcium concentrations, which is essential for proper neurotransmitter secretion and its dyshomeostatis is a known associate of neurodegenerative diseases. We also compare the effects of two different types of neurostimulation (transcranial electrical stimulation and deep brain stimulation) showcasing cellular-level differences resulting from these distinct forms of electrical therapy.
• Matthew Rose (Engineering/Applied Mathematics '19)
  "Characterizations of string stability of interconnected automobile systems"
  • String stability plays an important role in modeling self-driving automobile systems and automated smart traffic flow systems. This plays an important role specially in designing Adaptive Cruise Control (ACC) systems and Cooperative Adaptive Cruise Control (CACC) systems. In this work we present a few variants of string stability conditions and we analyze these variants. We provide characterizations for certain types of string stability. We analyze theoretical significance of these stability notions together with numerical validations.
• Tyler Simmons (Applied Mathematics/Marine Biology '19)
  "How Healthy is Rhode Island: Life Satisfaction and Risk Prevention Models"
  • Multiple logistic regression allows several different components of an experiment to be observed in how they affect an overall outcome. Using the Behavioral Risk Factor Surveillance System survey data provided by the Rhode Island Department of Health, life satisfaction and preventative health models were constructed through a public health perspective. With these models it is possible to identify which aspects of peoples everyday lives have the greatest effect or impact on an individuals overall health and life satisfaction. To determine which variables had the largest impact, or how they compared to other variables, several different methods were used including the log odds and odds ratios. As these were predictive models, they can be used for future years to give the Rhode Island Department of Health recommendations for better public health outcomes. The statistical analysis used for this project was performed on JMP, the windows version of SAS.
• Abigail Small (Applied Mathematics/Biology '20)
  "A Mathematical Approach for Assessing tDCS Efficacy for Post-Traumatic Stress Disorder"
  • Post-Traumatic Stress Disorder (PTSD) is a neurological condition caused by distressing or traumatic events. It has been recently found that symptoms of PTSD can be combated using forms of neurostimulation, in particular, transcranial direct current stimulation (tDCS). While it is known that the electrical energy delivered by this treatment to targeted areas of the brain is effective in treating PTSD, the optimal positioning of tDCS electrodes and treatment parameters for achieving the greatest efficacy is unknown. We have implemented a partial differential equation based mathematical model of tDCS with application to PTDS, and have generated numerous numerical simulations using the finite element method, all using distinct electrode montages, treatment parameters known to mitigate PTSD symptoms, and a three-dimensional MRI-derived cranial cavity with biologically-based tissue conductivities. The model predicts not only voltage and electrical current density within the head cavity, but also the sensitivity of the brain tissue to fire an action potential during treatments. We present our current results and findings that begin to shed light on ideal tDCS settings for treating PTSD.

2018

Joint Mathematics Meeting (JMM)

January 10 - 13 San Diego, CA

• Kaia Lindberg (Mathematics '19)
  "Computational Simulation of a Partial Differential Equation Based Model of Electrostatic Forces on Neuronal Electrodynamics"
  • Neurostimulation therapies demonstrate success as a medical intervention for individuals with neurodegenerative diseases. Despite promising results from these treatments, the influence of an electric current on ion concentrations and subsequent transmembrane voltage is unclear. This project focuses on developing a unique cellular-level mathematical model of neurostimulation to better understand its effects on neuronal electrodynamics. The Poisson-Nernst Planck system of PDEs is used to model electric potential, transmembrane voltage, and ion concentrations. This system is decoupled using the Gauss Siedal method and then the equations are solved using the finite element method on a biologically-inspired discretized domain. Using FEniCS we have conducted numerous numerical experiments on several two-dimensional neuronal geometries involving action potential generation and external current application. Preliminary results demonstrate the influence of applied external currents on membrane voltage. Future work will include extending these computational simulations to three-dimensional neuron domains and integrating an ODE based intracellular signaling pathway model. Hopefully this work will ultimately help elucidate the principles by which neurostimulation alleviates disease symptoms. (Received September 25, 2017)
• Kangji Chen (Biology/Applied Mathematics '18)
  "Evolution of microRNA diversity and regulation: Statistical Modeling"
  • MicroRNAs (miRNAs) are small molecules found in all plants and animals. Despite their relatively small size and limited number, they regulate many cellular processes in healthy and disease states. In the c. 600-million-year history of animal evolution, the number of miRNA-encoding genes has grown from 8 in sponges, to over 1,400 in humans. Despite this overall trend, the number of miRNA genes among species is highly variable, and has no obvious relationship to the age of the species. In our project, we are developing statistical models to evaluate the strength of the association between miRNA number and the following species-specific parameters besides evolutionary age: organism mass, surface area, tissue complexity, genome size and genome complexity. In order to investigate these relationships, we will consider both parametric and non-parametric statistical techniques (implemented using JMP software).
• Andrew DelSanto (Engineering/Applied Mathematics '18)
  "Time-Independent Solution of the Poisson-Nerst-Planck System of Equations for Neurological Applications"
  • Electromagnetic brain stimulation has been shown to benefit medical patients with a variety of brain diseases. The molecular mechanisms by which these therapies operate are still largely unknown; nevertheless, these treatments are electromagnetic in nature, and so we hypothesize that they must have a direct influence on the distribution of ionic species adjacent to neural cells and their transfer between extracellular and intracellular spaces in neural tissue. We implement a finite element solution to the steady-state Poisson-Nerst-Planck system of partial differential equations to investigate these questions. To solve this system, we employ a variety of numerical methods including a Gauss-Siedal decoupling method and a weighted iterative Gummel scheme to facilitate numerical convergence. Implementation is performed using FEniCS on a biologically-inspired domain constructed in GMSH. We present our numerical simulation results which show the influence of an electric field on ionic distributions, and in addition, present numerical method convergence metrics which gauge simulation efficiency and accuracy
• Madison Guitard (Engineering/Applied Mathematics '20)
  "An Interdisciplinary Approach to Computational Neurostimulation"
  • Mathematical models of electrotherapies are proving to be a valuable accompaniment to these medical treatments; in particular, modeling the transcranial direct current stimulation (tDCS) mode provides a predictive component to assess and quantify voltage and electrical current distributions in the head cavity, which in turn gives a direct measure of treatment efficacy. One fundamental drawback of these simulations, however, is the fact that precise material conductivity values are unknown for a particular patient, and in addition, are highly variable within a tissue type. As these values greatly impact simulation results, simulation utility is highly dependent on them. To address this issue, we have imple-mented a mathematical model of tDCS using a stochastic partial differential equation coupled with mixed boundary conditions. The finite element method is used to solve our governing system, and Monte Carlo experiments over scalp, skull, brain fluid, and brain tissue variabilities are performed. Numerical simulations are performed on an idealized two dimensional mesh, and then extended to an MRI-derived three dimensional head geometry. Our preliminary results showcase the importance of incorporating and quantifying this conductivity uncertainty within mathematical models of tDCS.
• Hien Ngo (Applied Mathematics/Biology '18)
  "Radiosity Equation Model for an Interior Space Illumination Design; Mars Project"
  • This research project is focused on finding the true solution of the exterior Dirichlet problem for the Radiosity equation to determine the convergence of a Spherical Quatrefoil in three dimensions at its boundaries, using the Galerkin Method. A mathematical model, based on the Radiosity equation will be utilized to investigate the role of incoming light waves for different surfaces with different emissivity and reflectivity functions. Theoretical and computational details of the method will provide sufficient information for designing proper lighting of an interior space inside a spacecraft that can ultimately be used for future endeavors of Mars exploration. H. Voderberg constructed a tile with the property that two copies of the tile can enclose one or two other copies. This tile’s design can be extended to a general form which has the property that any number of copies can be enclosed within just two tiles. This property is known as the n-enclosing property, where n is the number of tiles enclosed.
• Abigail Small (Applied Mathematics/Biology '20)
  "Quantifying Electromagnetic Properties of Applied Electric Fields on Neural Tissues"

2017

Northeast Regional IDeA Conference (NERIC)

August 16 – 18  Burlington, VT

• Kaia Lindberg (Mathematics ‘19) and Abigail Small (Applied Mathematics ’20)
  “A Mathematical Model of the Effects of Neurostimulation Treatment on Neuronal Electrodynamics”
  • Post-Traumatic Stress Disorder (PTSD) is a neurological condition caused by distressing or traumatic events. It has been recently found that symptoms of PTSD can be combated using forms of neurostimulation, in particular, transcranial direct current stimulation (tDCS). While it is known that the electrical energy delivered by this treatment to targeted areas of the brain is effective in treating PTSD, the optimal positioning of tDCS electrodes and treatment parameters for achieving the greatest efficacy is unknown. We have implemented a partial differential equation based mathematical model of tDCS with application to PTDS, and have generated numerous numerical simulations using the finite element method, all using distinct electrode montages, treatment parameters known to mitigate PTSD symptoms, and a three-dimensional MRI-derived cranial cavity with biologically-based tissue conductivities. The model predicts not only voltage and electrical current density within the head cavity, but also the sensitivity of the brain tissue to fire an action potential during treatments. We present our current results and findings that begin to shed light on ideal tDCS settings for treating PTSD.
• Elizabeth Gilchrist (Biology/Applied Mathematics ’19)
  “A Computational Approach to Modeling Dopaminergic Neurons with Application to Parkinson’s Disease” 

Summer Undergraduate Research Fellowship Conference (SURF)

July 28 – URI, Rhode Island

Kaia Lindberg (Mathematics ‘19) and Abigail Small (Applied Mathematics/Biology ’20)
“A Mathematical Model of the Effects of Neurostimulation Treatment on Neuronal Electrodynamics”

Elizabeth Gilchrist (Biology/Applied Mathematics ’19)
“A Computational Approach to Modeling Dopaminergic Neurons with Application to Parkinson’s Disease”

Biology and Medicine Through Mathematics Conference (BAMM)

May 18 – 20  VCU, Virginia

Madison Guitard (Engineering/Applied Mathematics ’20)
“An Interdisciplinary Approach to Computational Neurostimulation”

Women’s Intellectual Network Research Symposium

March 4  Brown University, Rhode Island

Madison Guitard (Engineering/Applied Mathematics ’20)
“An Interdisciplinary Approach to Computational Neurostimulation”

Amanda Becotte (Mathematics ’17), Ashley Crane (Mathematics ’17), Alexandra Halligan (Mathematics ’17)
“A Mathematical Model for the Effects of Fertilization on Nitrogen Concentrations in Unsaturated Soil on Blueberry Farms”

Joint Mathematics Meeting (JMM)

January 4 – 7  Atlanta, Georgia

Hien Ngo (Applied Mathematics/Biology ’18)
“Numerical Solutions of the Radiosity Equation (Brightness) via the Galerkin Method: Mars Project”

Qiuyang Deng (Applied Mathematics ’17)
“Numerical Solutions of the Radiosity Equation by the Galerkin Method for the Spherical Pyramid (Mars Project)”

2016

Joint Mathematics Meeting (JMM)

January 6 – 9  Seattle, Washington

Jill Resh (Applied Mathematics/Graphic Design Communications ’16)
“The Numerical Solutions of the Helmholtz Equation for the Bloodcell Shape: Mars Project”