Research Spotlights

Abstract

We have two Research Spotlights papers in the lineup this issue. The first of these, “Persistent Homology of Geospatial Data: A Case Study with Voting," written by Michelle Feng and Mason Porter, is quite timely. The article begins with a gentle introduction into the what and why of persistent homology (PH): after converting the point cloud of data into a series of simplicial complexes at different scales, the homology groups are computed at each scale, with homological features tracked across scales. It is generally assumed that persistence of features can then be used to distinguish signal (important patterns) from noise. Of concern is the method of converting the data into the simplicial complexes. The authors argue that PH using the simplicial complexes computed via the distance-based approach can lead to erroneous assumptions or missed characteristics about the voting data they consider. They propose to counter these deficiencies by developing two new methods, one based on adjacency networks and one that utilizes level sets to capture the manifold nature of the data, for building a filtered simplicial complex. The dramatic differences achieved by varying the complexes on the resulting PH are colorfully illustrated and interpreted for real voter data. Readers may appreciate not only the tools used and results obtained, but also the broad range of problems---transportation networks, spatial demography, granular materials, biological structures---identified by the authors for which the new approaches may prove relevant. An animal's ability to localize odor sources and follow trails of odor is key to many aspects of its survival. Authors Nour Riman, Jonathan D. Victor, Sebastian D. Boie, and Bard Ermentrout focus their article, “The Dynamics of Bilateral Olfactory Search and Navigation," on providing a comprehensive mathematical analysis of the dynamics of the differential equations that model the way that animals use bilateral information to navigate odor sources. When odor sensors are located at two different positions of the body, it enables comparison of odor concentrations and thus allows the animal to determine whether to move toward or away from the stimulus. Tropotaxis in the present context refers to the movement of the animal in response to an odor directly toward or away from the source of the odor. The sensor angles, which the animal can control, play an important role in enabling the animal to find and follow a trail. The other model parameters include sensor length and sensitivity to concentration change. The authors investigate tropotaxis as functions of the model parameters under varying conditions for the odor sources and trails. In some cases, the authors are able to reduce the dynamics to a planar system. Graphical illustrations compliment the discussion of the behavior. The paper concludes with a discussion of future research, particularly with respect to exploring the effects of noise on the models. It is the hope that this new “understanding [of] the underlying dynamics of the bilateral model will help in building models that use bilateral information together with other strategies."