Mathematical Modeling seminars

This seminar investigates Anderson localization in subwavelength elastic periodic systems with random perturbations. For the unperturbed structure, layer-potential methods recast the eigenvalue problem as boundary integral equations, yielding asymptotic expressions for subwavelength eigenvalues and establishing a band gap above the subwavelength band. A Floquet transform then brings the perturbed problem into a periodic formulation and produces equations for resonant frequencies under general perturbations. Numerical experiments on monomer and dimer structures test agreement with the analysis. Increasing the strength and number of random perturbations demonstrates localization, providing a mathematical basis for elastic metamaterial design.

The Ca2+ modulated pulsatile secretions of glucagon and insulin by pancreatic α and β cells play a key role in glucose metabolism and homeostasis. However, how different types of cells in the islet couple and coordinate to give rise to various Ca2+ oscillation patterns and how these patterns are being tuned by paracrine regulation are still elusive. Here we developed a microfluidic device to facilitate long-term recording of islet Ca2+ activity at single cell level and found that islets show heterogeneous but intrinsic oscillation patterns. The α and β cells in an islet oscillate in antiphase and are globally phase locked to display a variety of oscillation modes. A mathematical model of islet oscillation maps out the dependence of the oscillation modes on the paracrine interactions between α and β cells. Our study reveals the origin of the islet oscillation patterns and highlights the role of paracrine regulation in tuning them.