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2008 Summer Program in Mathematical Bioengineering Each summer the MBI hosts a three-week education program. The first week is spent in a tutorial, which combines morning lectures with active learning laboratories in the afternoon. The following two weeks are spent working on guided team projects and participating in a miniconference to share project results. The program is meant primarily for graduate students; college instructors and qualified undergraduates will also be considered. The 2008 Summer Program dates are July 7 - 25 July 7-11, 2008 In these lectures Bertram will discuss examples of how mathematical modeling is used in the areas of neuroscience and physiology. Topics include the dynamics of electrically excitable cells, calcium dynamics and waves, fast and slow time scales, bursting oscillations, phase oscillators, circadian gene oscillations, and synchronization of oscillators. A basic familiarity with ordinary and partial differential equations is assumed. Techniques for the analysis of nonlinear ordinary differential equations using phase plane and bifurcation diagrams will be discussed throughout the series of lectures.
Team Projects Project 1: Dynamical properties of biochemical reaction networks Modern biological research provides countless examples of biological interaction networks at very different scales: molecular, cellular, tissue, organism, and population. At the intracellular level, the nodes of these interaction networks could be signaling molecules, genes, and gene products. At the ecosystem level, the nodes could be the various species and energy resources. In order to understand the role played by some of these interactions (for example the role of a signaling pathway in a cell, or the effect of introducing a foreign species in an ecosystem) one often faces great difficulties in trying to interpret the effect of positive and negative feedbacks, nonlinear interactions, and other complex signaling between the nodes of the network. These difficulties are due to the inherent complexity of the dynamics of nonlinear systems, which can give rise to such phenomena as multistability, oscillations, and chaotic behavior. Our aim will be to analyze the connections between reaction network structure and the capacity for complex dynamic behavior. These connections will play an important role in facilitating the understanding of very diverse sets of experimental data, such as the intracellular dynamics of genes and proteins, the effect of a drug in a cell or a tissue, the dynamics of an infectious disease, and the population dynamics of species in an ecosystem. Project 2: Mathematical modeling in immunity Every day our bodies are bombarded by foreign microbes that we inhale or ingest; and every day our immune system works to eliminate them, orchestrating an amazing response consisting of numerous cell types and molecules. Sometimes this response is not even noticeable to the host and other times not only are the effects (e.g. fever, headaches, elevated heart rate) clearly felt, it is possible that the immune response is not capable of effectively resolving the conflict. (e.g. think of the mortality associated with the black plague, circa 1340!) Consequently, in order for therapies and vaccines to be effectual, it is necessary to have a good understanding of the way in which insults such as disease causing agents (pathogens) or tumor cells interact with a host's defense system. In this project, we will not only examine such interactions biologically, but mathematically as well, exploring ordinary differential equation (ODE) models describing these processes. The goal of this project is to delve into the fascinating world of immunity and 1) understand the underlying biology and questions involved, 2) understand the modeling techniques used and evaluate the methods/assumptions, and (3) depending on time and interest, apply engineering control methods (using Matlab/Simulink) to test the effectiveness of possible therapies. Project 3: Microarray Data Analysis Recent advances in high-throughput technologies such as microarrays have brought a revolution in our understanding of normal and abnormal molecular processes. Microarrays are a great tool for identifying differentially expressed genes, testing hypotheses, studying developmental processes, and performing clinical classification. Therefore, a large amount of microarray data has been generated over the last several years. This wealth of data has proposed both great opportunities and challenges for statisticians and computational scientists because of the complexity of the data, the large data size, the small number of biological replicates, and so on. In this project, we will conduct microarray data analysis on a publicly available data set. The specific goals of this project are 1) understand the rationale and details behind the microarray experiment; 2) learn how to preprocess the data by examining the noise, correcting for background effects, and normalizing the data to reduce systematic bias; 3) identify differentially expressed genes and interpret their meaning; and 4) depending upon time and interest, integrate genetic variations (e.g., differential expression patterns) with epigenetic changes (e.g., DNA methylation or histone modifications). Project 4: Development of the primary visual cortex: ocular dominance, competition for neurotrophins, and the cortical laminae The primary visual cortex (V1) is the first cortical area to receive visual input generated from the left and right eyes. The information is said to be mapped onto V1, which physically is akin to a sheet with shallow depth. The most studied cortical map is that of ocular dominance (OD), characterized by a significant influence of, say, left-eye over right-eye activation determining a neuron's response properties, that is, the feedforward connections originating from left-eye thalamic regions are stronger than the connections from the right-eye thalamic regions. The segregation of ocular streams may play a crucial role in stereopsis, i.e., depth perception. OD maps appear not to be predetermined and depend critically upon the driving activity during early development. As such, the OD maps for different species take on a variety of patterns from a stripe-like pattern in macaque monkey and humans where the left and right eye drives are believed to be approximately balanced to a patchy pattern in cats where the contralateral eye more powerfully drives V1 during early development. OD maps from animals that have been monocularly deprived at young age, either via strabismus, suturing, or enucleation, are qualitatively and quantitatively different from OD maps from normally raised animals. This dependence on activity in the system motivates a class of models for the development of cortical maps, i.e., activity--driven developmental models (see the reviews of Swindale 1996 and vanOoyen 2001). In this project, we will examine and simulate a model proposed by Ermentrout and Harris that takes into account neurotrophins which the axonal projections must compete for in order to synapse on a specific point. We hope to extend this model in a manner that takes into account the laminar structure of cortex, which is often neglect. In particular, after the OD map is set, plasticity first occurs in the upper laminae and through feedback appears to sculpt the thalamic input which occurs in the lower laminae, a counter intuitive finding. Project 5: An ODE/PDE model for tumor growth |
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