Homogenization in cardiac electrophysiology and blow-up in bacterial chemotaxis

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.