Research Spotlights

Abstract

This issue's Research Spotlights section contains three papers. First, “Canards, Folded Nodes, and Mixed-Mode Oscillations in Piecewise-Linear Slow-Fast Systems" extends the theory of canards---a method commonly used for smooth systems with multiple time scales---to the case of piecewise linear systems. In two dimensions, a classical canard is a solution to a smooth system of differential equations such that one variable is moving much faster than the other. The result is a periodic solution with a graph that appears almost discontinuous, with the shape resembling a cartoon duck (canard being French for both duck and a small joke). In this paper, the theory is extended to include systems that are only piecewise smooth, such as for Filippov vector fields. In addition to new results, this paper presents a self-contained introduction to recent approaches in multiple time scale systems. It is worth a read for those interested in systems with multiple time scales, particularly those with an interest in neuronal modeling. The second article, “Communicability Angle and the Spatial Efficiency of Networks," considers communication processes in a connected network, with the goal of understanding whether the graphs are spatially efficient. The paper develops a new measure of spatial efficiency, called the communicability angle, relating this angle to more traditional graph measures, such as the level of clustering and the existence of holes. The methods are applied to a series of 120 real-world data networks, including networks of urban streets, the power grid, the Internet, and the brain. This compelling new idea would be worthwhile for anyone with interest in theoretical, computational, or data networks. The third paper in the section uses a stochastic birth-death population model for competitive growth between a population of healthy cells and a population of cancer cells. “An Evolutionary Model of Tumor Cell Kinetics and the Emergence of Molecular Heterogeneity Driving Gompertzian Growth" explores the formation of tumors under this model and investigates different treatment methods within the subclinical, clinical, and lethal regimes of tumor growth. This paper is an excellent example of the power of mathematical modeling: The model borrows ideas from stochastic population models which were developed in a completely different applied context. It allows for new insights on tumor control, as it is possible to systematize the finding of optimized strategies, and it is also possible to develop therapies for subclinical tumors, which with current methods would not be clinically detected.