abstract: In the honey bee antennal lobe, uniglomerular projection neurons (uPNs) transiently spike to odor sensory stimuli with odor-specific response latencies, i.e., delays to first spike after odor stimulation onset. Recent calcium imaging studies show that the spatio-temporal response profile of the activated uPNs are dynamic and changes as a result of associative conditioning, facilitating odor-detection of learned odors. Moreover, odor-representation in the antennal lobe undergo reward-mediated plasticity processes that increase response delay variations in the activated ensemble of uniglomerular projection neurons. Octopamine is necessarily involved in these plasticity processes. Yet, the cellular mechanisms are not well understood. I hypothesize that octopamine modulates cholinergic transmission to uPNs by triggering translation and upregulation of nicotinic receptors, which are more permeable to calcium. Consequently, this increased calcium-influx signals transcription factors that upregulate potassium channels in the dendritic cortex of glomeruli, similar to synaptic plasticity mechanisms recently shown in various insect species. A biophysical model of the antennal lobe circuit is developed in order to test the hypothesis that increased potassium channel expression in uPNs mediate response delays to first spike, dynamically tuning odor-representations to facilitate odor-detection of learned odors. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics for the Life and Social Sciences 2016

abstract: This dissertation will look at large scale collaboration through the lens of online communities to answer questions about what makes a collaboration persist. Results address how collaborations attract contributions, behaviors that could give rise to patterns seen in the data, and the properties of collaborations that drive those behaviors. It is understood that collaborations, online and otherwise, must retain users to remain productive. However, before users can be retained they must be recruited. In the first project, a few necessary properties of the ``attraction'' function are identified by constraining the dynamics of an ODE (Ordinary Differential Equation) model. Additionally, more than 100 communities of the Stack Exchange networks are parameterized and their distributions reported. Collaborations do not exist in a vacuum, they compete with and share users with other collaborations. To address this, the second project focuses on an agent-based model (ABM) of a community of online collaborations using a mechanistic approach. The ABM is compared to data obtained from the Stack Exchange network and produces similar distributional patterns. The third project is a thorough sensitivity analysis of the model created in the second project. A variance based sensitivity analysis is performed to evaluate the relative importance of 21 parameters of the model. Results indicate that population parameters impact many outcome metrics, though even those parameters that tend towards a low impact can be crucial for some outcomes. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics for the Life and Social Sciences 2017

abstract: Functional magnetic resonance imaging (fMRI) is one of the popular tools to study human brain functions. High-quality experimental designs are crucial to the success of fMRI experiments as they allow the collection of informative data for making precise and valid inference with minimum cost. The primary goal of this study is on identifying the best sequence of mental stimuli (i.e. fMRI design) with respect to some statistically meaningful optimality criteria. This work focuses on two related topics in this research field. The first topic is on finding optimal designs for fMRI when the design matrix is uncertain. This challenging design issue occurs in many modern fMRI experiments, in which the design matrix of the statistical model depends on both the selected design and the experimental subject's uncertain behavior during the experiment. As a result, the design matrix cannot be fully determined at the design stage that makes it difficult to select a good design. For the commonly used linear model with autoregressive errors, this study proposes a very efficient approach for obtaining high-quality fMRI designs for such experiments. The proposed approach is built upon an analytical result, and an efficient computer algorithm. It is shown through case studies that our proposed approach can outperform the existing method in terms of computing time, and the quality of the obtained designs. The second topic of the research is to find optimal designs for fMRI when a wavelet-based technique is considered in the fMRI data analysis. An efficient computer algorithm to search for optimal fMRI designs for such cases is developed. This algorithm is inspired by simulated annealing and a recently proposed algorithm by Saleh et al. (2017). As demonstrated in the case studies, the proposed approach makes it possible to efficiently obtain high-quality designs for fMRI studies, and is practically useful. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics 2017

abstract: Head and neck squamous cell carcinoma (HNSCC), the sixth most common cancer type worldwide, accounts for more than 630,000 new cases and 350,000 deaths annually. Drug-resistance and tumor recurrence are the most challenging problems in head and neck cancer treatment. It is hypothesized that a very small fraction of stem-like cells within HNSCC tumor, called cancer stem cells (CSCs), is responsible for tumor initiation, progression, resistance and recurrence. It has also been shown that IL-6 secreted by head and neck tumor-associated endothelial cells (ECs) enhances the survival, self-renewal and tumorigenic potential of head and neck CSCs. In this study we will use a mathematical multi-scale model which operates at the intracellular, molecular, and tissue level to investigate the impacts of EC-secreted IL-6 signaling on the crosstalk between tumor cells and ECs during tumor growth. This model will be calibrated by using the experimental in vivo data. Eventually the model will be modified to explore the responses of head and neck cancer cells to combination therapy involving Tocilizumab (an anti-IL-6R antibody) and Cisplatin (the most frequently used chemotherapy for head and neck cancer). The model will be able to predict the final proportion of CSCs in response to endothelial cell-secreted IL-6 and drug therapies. The model will be validated by directly comparing the experimental treatment data and the model predictions. This could potentially provide a condition under which we could control enlargement of the head and neck CSC pool and tumor recurrence. It may also suggest the best bounds for Cisplatin and/or Tocilizumab dose and frequency to be tested in the clinical trial. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics for the Life and Social Sciences 2017

abstract: Foraging strategies in social animals are often shaped by change in an organism's natural surrounding. Foraging behavior can hence be highly plastic, time, and condition dependent. The motivation of my research is to explore the effects of dispersal behavior in predators or parasites on population dynamics in heterogeneous environments by developing varied models in different contexts through closely working with ecologists. My models include Ordinary Differential Equation (ODE)-type meta population models and Delay Differential Equation (DDE) models with validation through data. I applied dynamical theory and bifurcation theory with carefully designed numerical simulations to have a better understanding on the profitability and cost of an adaptive dispersal in organisms. My work on the prey-predator models provide important insights on how different dispersal strategies may have different impacts on the spatial patterns and also shows that the change of dispersal strategy in organisms may have stabilizing or destabilizing effects leading to extinction or coexistence of species. I also develop models for honeybee population dynamics and its interaction with the parasitic Varroa mite. At first, I investigate the effect of dispersal on honeybee colonies under infestation by the Varroa mites. I then provide another single patch model by considering a stage structure time delay system from brood to adult honeybee. Through a close collaboration with a biologist, a honeybee and mite population data was first used to validate my model and I estimated certain unknown parameters by utilizing least square Monte Carlo method. My analytical, bifurcations, sensitivity analysis, and numerical studies first reveal the dynamical outcomes of migration. In addition, the results point us in the direction of the most sensitive life history parameters affecting the population size of a colony. These results provide novel insights on the effects of foraging and Varroa mites on colony survival. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics for the Life and Social Sciences 2017

abstract: Olfaction is an important sensory modality for behavior since odors inform animals of the presence of food, potential mates, and predators. The fruit fly, Drosophila melanogaster, is a favorable model organism for the investigation of the biophysical mechanisms that contribute to olfaction because its olfactory system is anatomically similar to but simpler than that of vertebrates. In the Drosophila olfactory system, sensory transduction takes place in olfactory receptor neurons housed in the antennae and maxillary palps on the front of the head. The first stage of olfactory processing resides in the antennal lobe, where the structural unit is the glomerulus. There are at least three classes of neurons in the antennal lobe - excitatory projection neurons, excitatory local neurons, and inhibitory local neurons. The arborizations of the local neurons are confined to the antennal lobe, and output from the antennal lobe is carried by projection neurons to higher regions of the brain. Different views exist of how circuits of the Drosophila antennal lobe translate input from the olfactory receptor neurons into projection neuron output. We construct a conductance based neuronal network model of the Drosophila antennal lobe with the aim of understanding possible mechanisms within the antennal lobe that account for the variety of projection neuron activity observed in experimental data. We explore possible outputs obtained from olfactory receptor neuron input that mimic experimental recordings under different connectivity paradigms. First, we develop realistic minimal cell models for the excitatory local neurons, inhibitory local neurons, and projections neurons based on experimental data for Drosophila channel kinetics, and explore the firing characteristics and mathematical structure of these models. We then investigate possible interglomerular and intraglomerular connectivity patterns in the Drosophila antennal lobe, where olfactory receptor neuron input to the antennal lobe is modeled with Poisson spike trains, and synaptic connections within the antennal lobe are mediated by chemical synapses and gap junctions as described in the Drosophila antennal lobe literature. Our simulation results show that inhibitory local neurons spread inhibition among all glomeruli, where projection neuron responses are decreased relatively uniformly for connections of synaptic strengths that are homogeneous. Also, in the case of homogeneous excitatory synaptic connections, the excitatory local neuron network facilitates odor detection in the presence of weak stimuli. Excitatory local neurons can spread excitation from projection neurons that receive more input from olfactory receptor neurons to projection neurons that receive less input from olfactory receptor neurons. For the parameter values for the network models associated with these results, eLNs decrease the ability of the network to discriminate among single odors. / Dissertation/Thesis / Ph.D. Applied Mathematics for the Life and Social Sciences 2013

<p> This thesis is concerned with modeling and simulation of the mitral valve, one of the four valves in the human heart. The valve is composed of leaflets attached to a ring, the free edges of which are supported by a system of chordae, which themselves are anchored to muscles inside the heart. First, we examine valve anatomy and show the results of original dissections. These display the gross anatomy and information on fiber structure of the mitral valve. Next, we build a model valve following a design-based approach to elasticity. We incorporate information from the dissections to specify the fiber topology of this model. We assume the valve achieves mechanical equilibrium while supporting a static pressure load. The solution to the resulting differential equations determines the pressurized configuration of the valve model. To complete the model we then specify a constitutive law based on experimental stress-strain relations from the literature. Finally, using the immersed boundary method, we simulate the model valve in fluid in a computer test chamber. The aim of this work is to determine the basic principles and mechanisms underlying the anatomy and function of the mitral valve.</p><p>

Mathematical modelling and simulations were carried to study diblock copolymer system confined in circular annular pores, cylindrical pores and spherical pores using Cell Dynamics simulation (CDS) method employed in physically motivated discretization. The lamella, cylindrical and spherical forming systems were studied in the neutral surfaces and the wetting surfaces. To employ CDS method in polar, cylindrical and spherical coordinates, the Laplacian operators were discretized and isotropised in polar, cylindrical and spherical coordinate systems.

In this thesis we summarize the classical cases of one-dimensional oligomers and two-dimensional plaquettes, respecting the parity-time ([special characters omitted]) symmetry. We examine different types of solutions of such configurations with linear and nonlinear gain or loss profiles. For each configuration, we develop a dynamical model and examine its [special characters omitted] symmetry. The corresponding nonlinear modes are analyzed starting from the Hamiltonian limit, with zero value of the gain-loss coefficient. Once the relevant waveforms have been identified (analytically or numerically), their stability as well as those of the ghost states in certain regimes is examined by means of linearization in the vicinity of stationary points. This reveals diverse and, occasionally, fairly complex bifurcations. The evolution of unstable modes is explored by means of direct simulations.

Birdsong is a model system for the production of learned, serially ordered motor movements, such as playing a musical instrument or riding a bicycle. To this end, the neural mechanisms underlying birdsong have been studied in great depth, and many tools have been developed for analyzing the spectral and temporal structure of song. In this dissertation, I develop mathematical neural network models to explain how the nuclei in the song system interact to produce song. These models are constrained by the structural connectivity of the song system, the signaling properties of individual neurons and circuits, and circuit-breaking behavioral studies. Chapter 1 provides an overview of songbirds as a model system for speech production, outlines the structure of song, and describes the structure and function of the song system. Chapter 2 describes the neurophysiology and mathematical models of a premotor nucleus, called HVC (proper name), that is essential for song learning and production. In Chapter 3, I develop the neural network model for song production and use it to explain the effects of partial lesions of HVC on song. Furthermore, I use the model to make predictions about the behavioral effects of these lesions and reanalyze the data, validating those predictions. Finally, in Chapter 4 I develop a simplified version of the model that sacrifices spiking dynamics of neurons while maintaining the essential higher-level features of the model. I use this model to study interhemispheric synchronization and the effects of unilateral perturbations of HVC on song. The model captures the effects of these perturbations, particularly unilateral temperature manipulation and electrical stimulation of HVC, and makes predictions about the circuit-level effects of these perturbations. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Spring Semester 2018. / February 26, 2018. / Birdsong, Bursting, Computational Modeling, Neural Networks, Zebra Finch / Includes bibliographical references. / Richard Bertram, Professor Directing Dissertation; Paul Q. Trombley, University Representative; Nicholas G. Cogan, Committee Member; Richard L. Hyson, Committee Member; Frank Johnson, Committee Member; Theodore Vo, Committee Member.