Date Published: March 15, 2019
Publisher: Public Library of Science
Author(s): M. Ulmer, Lori Ziegelmeier, Chad M. Topaz, Bard G. Ermentrout.
We use topological data analysis as a tool to analyze the fit of mathematical models to experimental data. This study is built on data obtained from motion tracking groups of aphids in [Nilsen et al., PLOS One, 2013] and two random walk models that were proposed to describe the data. One model incorporates social interactions between the insects via a functional dependence on an aphid’s distance to its nearest neighbor. The second model is a control model that ignores this dependence. We compare data from each model to data from experiment by performing statistical tests based on three different sets of measures. First, we use time series of order parameters commonly used in collective motion studies. These order parameters measure the overall polarization and angular momentum of the group, and do not rely on a priori knowledge of the models that produced the data. Second, we use order parameter time series that do rely on a priori knowledge, namely average distance to nearest neighbor and percentage of aphids moving. Third, we use computational persistent homology to calculate topological signatures of the data. Analysis of the a priori order parameters indicates that the interactive model better describes the experimental data than the control model does. The topological approach performs as well as these a priori order parameters and better than the other order parameters, suggesting the utility of the topological approach in the absence of specific knowledge of mechanisms underlying the data.
Given data potentially described by various mathematical models, how might one choose between those models? In the context of experiments on social insects, we use topological data analysis (TDA) to inform this choice, and compare the topological approach to more traditional methods. Our investigation complements the rich literature on biological model selection [1, 2].
We reprise some details of the experiments and modeling in . During nine experimental trials, groups of 7 to 33 aphids were filmed for 45 minutes walking in a circular arena 40 cm in diameter. Fig 1 shows the initial state of the aphids in each trial. Fig 2(A) shows trajectories of the aphids in Experiment 9, obtained via motion-tracking software.
Later, we will analyze our data with tools from topological data analysis (TDA). For now, we explain these methods. We provide a brief and nontechnical review of the main ideas followed by a slightly more technical explanation for the mathematically-inclined reader.
We simulate aphid movement using both the interactive and control models, initialized with the same configuration as each of the nine experiments. We run 100 trials for each model-initial condition pair, for a total of 1800 trials. Each simulation has the same number of frames as the experiment from which its initial conditions were taken, ranging in number between 4605 and 5883. We compare the fitness of the two models of aphid motion with the experimental data using two different approaches: first, via several order parameters, and then, via topological data in the form of crockers.
We have assessed the goodness-of-fit of two models of aphid motion: an interactive model that describes social behavior of the insects and a control model that omits this effect. To compare the model results to experimental data, we performed statistical tests on three different sets of measures: order parameters that do not use a priori knowledge of the models, order parameters that do use this information, and topological measures in the form of crocker plots. Order parameters that rely on a priori knowledge give a consistent message, namely that the interactive model better describes the experimental data. The topological measures match the performance of the a priori order parameters and outperform the other order parameters. Of course, the topological measures do not rely on knowledge of the models. Thus, we find that adopting a topological lens may be a useful approach for characterizing and comparing motion of biological groups when one has little information about what model mechanisms might be important.