The simplicity of morphogenesis, manifested as collective shape changes, emerges from complex biophysical regulations within a multicellular embryo. Constructing a spatio-temporal atlas of mechanical stresses is central for understanding the emergence of this simplicity. Developing a new mathematical theory for the static mechanics of three-dimensional multicellular aggregates involving pressures and...
Motivated by rhythms in the brain, we investigate the synchronization of noisy and all-to-all pulse-coupled oscillators. We consider a case where the oscillatory excursions are of varying amplitude and where only sufficiently large excursions result in the output pulses that drive the interactions between the oscillators. In the regime of...
This dissertation is a review of three projects I worked on during my time in the Computational Photography Lab at Northwestern University. First, a source separation problem for the X-Ray Fluorescence images of painted works of art is addressed through the incorporation of Hyperspectral Reflectance data. Following this, a discussion...
Motivated by rhythms in the brain, we investigate the synchronization of noisy and all-to-all pulse-coupled oscillators. We consider a case where the oscillatory excursions are of varying amplitude and where only sufficiently large excursions result in the output pulses that drive the interactions between the oscillators. In the regime of...
This dissertation is a review of three projects I worked on during my time in the Computational Photography Lab at Northwestern University. First, a source separation problem for the X-Ray Fluorescence images of painted works of art is addressed through the incorporation of Hyperspectral Reflectance data. Following this, a discussion...
Each second, living organisms take in sensory input from an ever-changing environment and respond appropriately. Identifying and contextualizing stimuli is critical for survival, and it often necessitates distinguishing between sensory experiences that are similar to each other. Pattern separation characterizes the mechanisms by which neuronal networks extract and highlight differences...
Deterministic models are used to explain and predict the dynamics of ecosystems featuring cyclic competition schemes. The models are systems of reaction-diffusion partialdifferential equations that account for species mobility via Fickian diffusion and interspecies interactions according to the competition scheme. Length and temporal scales are
chosen to be appropriate for...
Dynamical modeling aims to capture the essential mechanics at work in real-world systems while remaining tractable enough to yield mathematical insights for predictions and interventions. The work presented here first takes this approach to the system of political ideology and influence, establishing a model for the continuous-time evolution of individual...
Originally motivated by the emergence of networked systems lacking central coordination such as multiprocessors, wireless sensor networks and smart grids, the study of distributed optimization algorithms has been an active field of research spanning multiple decades. More recently, the rapid growth in the availability of high-dimensional datasets has posed the...
For stochastic simulation optimization in a modern computing era, we introduce a new parallel framework for solving very large-scale problems using a ranking & selection (R&S) approach that simulates all systems or feasible solutions to provide a global statistical guarantee. We propose a parallel adaptive survivor selection (PASS) framework that...
Mixing by cutting-and-shuffling (like that for a deck of cards or a Rubik's cube) is a paradigm that has not been studied in detail even though it can be applied in a variety of situations including the mixing of granular materials. Mathematically, cutting- and-shuffling is described by piecewise isometries (PWIs),...
Circadian rhythms — physiological, behavioral, and metabolic oscillations with an approximate 24-h period — are controlled by an evolutionarily conserved set of core clock genes operating at the transcriptional and protein level. Entrainable by Zeitgebers (external environmental stimuli such as light, temperature, and food) that modulate time-of-day specific functions, the...
Modern data sets are increasingly vast, not only in the number of samples, but also in the number of measurements, or features, that they contain. This high-dimensionality poses a unique set of problems for data analysis due to a set of phenomena known as ``the curse of dimensionality.'' This thesis...
Recently, machine learning and deep learning, which have made many theoretical and empir- ical breakthroughs and is widely applied in various fields, attract a great number of researchers and practitioners. They have become one of the most popular research directions and plays a sig- nificant role in many fields, such...
Existing nonlinear optimization methods have proven reliable over the past few decades for a wide range of applications but have critically relied on accurate function and gradient evaluations. Modern nonlinear optimization problems arising from machine learning and scientific computing applications are increasingly complex and large scale, which make accurate evaluations...
Processing of sensory information in the brain is a pervasive and fundamental phenomenon across animal species and is involved in both "hard-wired" innate responses as well as learned and adaptive behaviors. Here, I show that the avoidance of hot temperature, a simple innate behavior, contains unexpected plasticity and complex processing...
The focus on this thesis is on the dynamics of colloidal particles in an applied electric field in a uniform bulk fluid and on a fluid-fluid interface. In a bulk fluid, the dynamics of an isolated particle, one pair, and a cluster of particles under an applied nonuniform electric field...
With neurons as its primary computational components, the brain operates at multiple timescales. In this thesis, we focus on two timescales: on a relatively slow timescale on the order of hours to days, the brain adapts to the environment it is exposed to and learns its circuitry by altering the...
We present a biophysical model of GCaMP6f calcium fluorescence in CA1 pyramidal neuron dendrites based upon results from imaging and electrophysiology experiments. This work was completed using experimental results from the laboratory of Professor Daniel Dombeck, Department of Neurobiology. Constraining the model to reproduce different objectives --- from in-vitro and...
We present two novel, computational models of biofilm growth within an experimental flow cell. First, we use asymptotic approximations to develop a reduced model that captures the large-scale dynamics within an entire flow cell. The reduced model's predicted growth and nutrient distribution are close to the values predicted by previous...
Cells are often precisely organized into patterns within developing tissues. This precision must emerge from biochemical processes within, and between cells, that are inherently stochastic. I investigated the impact of stochastic gene expression on self-organized pattern formation, focusing on Senseless (Sens), a key target of Wnt and Notch signaling during...
Soft matter is the field of science concerning soft and deformable materials: such as liquids, gels, and foams. Active matter is a sub-field of soft matter that considers systems that contain active agents or particles that consume energy for self-propulsion or to exert mechanical stress on the surrounding system. In...
Perhaps because of the influence of the central limit theorem, it is common for scientists to assume distributions in the real world are singly peaked and unimodal. However, many quantities in nature are actually better represented by multimodal distributions. One must provide an explanation for this disconnect between the central...
There is a rich history on the study of the interplay between symmetry and synchronization in networks. At the most fundamental level, many synchronization patterns are induced by underlying network symmetries. However, when stability is taken into account, the relation between symmetry and synchronization is far from monotonic. In this...
A series of theories and models are developed and used to investigate the growth of protective oxide films on metal and alloy surfaces for cases in which Wagner's classical model of oxidation does not hold. First, irreversible thermodynamics is applied to formulate a model for the outward growth of rocksalt...
Histone methylation plays an important role as an epigenetic regulator, capable of driving stable, persistent changes in gene expression without changing a cell's genetic code. Previous work has used stable isotope labeling (SILAC) in combination with mass spectrometry to observe the relationship between the methylation of two neighboring lysine sites...
Optical fibers utilize nonlinear effects to help transmit soliton or near soliton pulses in a variety of contexts including optical communication systems and fiber lasers. Fiber lasers produce ultra-short pulses, down to a few femtoseconds in duration, via a process called mode-locking where modes of the optical cavity are synchronized...
While optimization has received much attention in the machine learning community, most of them consider unconstrained supervised learning models such as neural networks and support vector machine. In this dissertation, we introduce a new class of optimization problems called scale invariant problems that include interesting unsupervised learning models such as...
This thesis focuses on applications of recurrent neural networks (RNNs) for three aspects of sequential classification. In the first chapter, a novel method to generate synthetic minority data generation to improve imbalanced classification is discussed. Generative Adversarial Networks (GANs) have been used in many different applications to generate realistic synthetic...
The ever growing desire for accurate estimation and efficient learning necessitates the efforts to quantitatively characterize uncertainties for models. In this thesis, four problems pertaining to uncertainty quantification are discussed: A sequential stopping framework of constructing fixed-precision confidence regions is proposed for a class of multivariate simulation problems where variance...
Forecasting U.S. elections has been a field of interest for many researchers, with various statistical and mathematical models being proposed. In my research, I analyzed a prior election model, the SIS election model. In this model, a system of differential equations traditionally used in epidemiology to study disease transmission (but...
Forecasting the outcomes of U.S. elections is a relevant and complex task that has been approached in many ways, most commonly incorporating statistics or proprietary methods that include some degree of subjectivity. Our approach differs from this convention in that we use multidisciplinary methods from applied mathematics. Specifically, we use...
To survive, animals, including human beings, have developed an amazing ability to learn the constantly changing environment. Specifically, detecting specific odorants in a noisy, variable background is crucial for finding food and water, mating, and avoiding potential dangers. For this purpose, rodents have developed an olfactory system that is powerful...
This dissertation considers a periodically-forced 1-D Langevin equation that possesses two stable periodic solutions in the absence of noise. We aim at answering the question: is there a most likely noise-induced transition path between these periodic solutions that allows one to identify a preferred phase of the forcing when the...
Many industrial fluid flow problems involve the interaction between heavy, rigid objects and one or more fluid phases. For several decades, there has been a vested interest in simulating these fluid-structure interaction (FSI) problems in order to improve engineering design processes. However, numerical simulations of these problems can be challenging...
This dissertation presents two projects with the goal of understanding how to quantitatively describe biological data, particularly data that is highly dynamic. The first study presents an improved quantitative tool for the analysis of particulate trajectories. Particulate trajectory data appears in several different biological contexts, and the majority of analyses...
Gene regulatory networks (GRNs) are important abstractions of the complex regulatory interplays between genes, proteins, metabolites, and other molecular-level entities. Comprehensive GRNs provide high-level overviews of the topology of gene-gene interactions and their purposes, thereby enabling a comprehensive understanding of their role in phenotypic variation, disease mechanisms, and other biological...
This thesis focuses on the application of neural networks to three types of classification tasks, each work with its own chapter. The objective of the first work is to take advantage of deep neural networks in order to make next day crime count predictions in a fine-grain city partition. We...