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Homogenization in cardiac electrophysiology and blow-up in bacterial chemotaxis.

机译：心脏电生理的均质化和细菌趋化性的爆发。

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摘要

In the first part of this dissertation, we investigate three different issues involving homogenization in cardiac electrophysiology.;We present a modification for how heart tissue is typically modeled in order to derive values for intracellular and extracellular conductivities needed for bidomain simulations. In our model, cardiac myocytes are rectangular prisms and gap junctions appear in a distributed manner as flux boundary conditions for Laplace's equation. In other models, gap junctions tend to be explicit geometrical entities. Using directly measurable microproperties such as cellular dimensions and end-to-end and side-to-side gap junction coupling strengths, we inexpensively obtain effective conductivities close to those given by simulations with a detailed cyto-architecture. This model provides a convenient framework for studying the effect on conductivities of aligned vs. brick-like arrangements of cells and the effect of different distributions of gap junctions between the sides and ends of myocytes.;We further illustrate this framework by investigating the effect on conductivity of non-uniform distributions of gap junctions within the ends of cells. We show that uniform distributions are local maximizers of conductivity through analytical perturbation arguments.;We also derive a homogenized description of an ephaptic communication mechanism along a single strand of cells. We perform numerical simulations of the full model and its homogenization. We observe that the two descriptions agree when gap junctional coupling is at physiologically normal levels. When gap junctional coupling is low, the homogenized description does not capture the behavior that the ephaptic mechanism can speed up action potential propagation.;In the second part of this dissertation, we investigate finite-time blow-up and stability of the Keller-Segel model for bacterial chemotaxis. We use a second moment calculation to establish finite-time blow-up for the Keller-Segel system on a disk with Dirichlet boundary conditions and a supercritical mass.;We numerically investigate the evolution and stability of the Keller-Segel system in order to provide a conjecture about the generality of boundary blow-up for supercritical mass under the Jager-Luckhaus boundary conditions.;Finally, we use the free energy of solutions to Keller-Segel equations to derive a functional inequality that may be helpful for analyzing the stability of steady states.
机译：在本论文的第一部分中，我们研究了涉及心脏电生理学均质化的三个不同问题。我们提出了对通常如何对心脏组织进行建模的修改，以得出双域模拟所需的细胞内和细胞外电导率值。在我们的模型中，心肌细胞是矩形棱柱，间隙连接作为拉普拉斯方程的通量边界条件以分布方式出现。在其他模型中，间隙连接往往是明确的几何实体。使用直接可测量的微观性质，例如细胞尺寸以及端对端和侧向间隙连接偶联强度，我们可以廉价地获得接近电导率的有效电导率，该电导率与详细的细胞结构模拟所给出的电导率接近。该模型提供了一个方便的框架，用于研究对齐的或砖状排列的细胞对电导率的影响以及心肌细胞的侧面和末端之间的间隙连接的不同分布的影响。细胞末端间隙连接的不均匀分布的电导率。我们通过分析扰动论证表明均匀分布是电导率的局部最大化。;我们还沿着细胞的单链得出了一种关于ephaptic通讯机制的均质描述。我们对完整模型及其均质化进行数值模拟。我们观察到，当间隙连接偶联处于生理正常水平时，这两个描述是一致的。当缝隙连接耦合度低时，均质描述不能捕捉到麻药机制可以加速动作电位传播的行为。在本论文的第二部分，我们研究了Keller-Segel的有限时间爆破和稳定性。细菌趋化性模型。我们使用第二矩计算来建立具有Dirichlet边界条件和超临界质量的圆盘上的Keller-Segel系统的有限时间爆炸。;我们通过数值研究了Keller-Segel系统的演化和稳定性，以提供关于在Jager-Luckhaus边界条件下超临界质量边界爆破的一般性的一个猜想。稳定状态。

著录项

  • 作者

    Hand, Paul Earl.;

  • 作者单位

    New York University.;

  • 授予单位 New York University.;
  • 学科 Mathematics.
  • 学位 Ph.D.
  • 年度 2009
  • 页码 172 p.
  • 总页数 172
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类 数学;
  • 关键词

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