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Topological data analysis of biological aggregation models.

Topaz CM, Ziegelmeier L, Halverson T - PLoS ONE (2015)

Bottom Line: To interpret our results, we introduce a visualization that displays Betti numbers over simulation time and topological persistence scale.We compare our topological results to order parameters typically used to quantify the global behavior of aggregations, such as polarization and angular momentum.The topological calculations reveal events and structure not captured by the order parameters.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics, Statistics, and Computer Science, Macalester College, Saint Paul, Minnesota, United States of America.

ABSTRACT
We apply tools from topological data analysis to two mathematical models inspired by biological aggregations such as bird flocks, fish schools, and insect swarms. Our data consists of numerical simulation output from the models of Vicsek and D'Orsogna. These models are dynamical systems describing the movement of agents who interact via alignment, attraction, and/or repulsion. Each simulation time frame is a point cloud in position-velocity space. We analyze the topological structure of these point clouds, interpreting the persistent homology by calculating the first few Betti numbers. These Betti numbers count connected components, topological circles, and trapped volumes present in the data. To interpret our results, we introduce a visualization that displays Betti numbers over simulation time and topological persistence scale. We compare our topological results to order parameters typically used to quantify the global behavior of aggregations, such as polarization and angular momentum. The topological calculations reveal events and structure not captured by the order parameters.

No MeSH data available.


Related in: MedlinePlus

Aggregate behavior of the Vicsek model, simulation #3.A typical snapshot is shown in Fig 7(C). (A) Normalized average velocity order parameter φ(t). (B) Contour plot of Betti number . (C) Contour plot of Betti number . The topological analysis shows essentially no cluster formation; the narrow region in which  arises from an isolated agent. The two persistent topological circles are consistent with highly aligned particles covering both dimensions of the periodic spatial domain. Topological features become fairly stagnant once the entire group forms a large, aligned cluster traveling as a rigid body early in the simulation. See text for a more comprehensive analysis.
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pone.0126383.g010: Aggregate behavior of the Vicsek model, simulation #3.A typical snapshot is shown in Fig 7(C). (A) Normalized average velocity order parameter φ(t). (B) Contour plot of Betti number . (C) Contour plot of Betti number . The topological analysis shows essentially no cluster formation; the narrow region in which arises from an isolated agent. The two persistent topological circles are consistent with highly aligned particles covering both dimensions of the periodic spatial domain. Topological features become fairly stagnant once the entire group forms a large, aligned cluster traveling as a rigid body early in the simulation. See text for a more comprehensive analysis.

Mentions: Fig 10 shows results from the same simulation as the snapshot in Fig 7(C). Here, N = 300, ℓ = 5, and η = 0.1. In panel (A), the order parameter φ(t) rises steeply to one with a small intermediate step. The order parameter here and in Fig 6(A) share some features, and yet we will see that the topological calculations capture the differences in the dynamics already apparent in the snapshots of Fig 7.


Topological data analysis of biological aggregation models.

Topaz CM, Ziegelmeier L, Halverson T - PLoS ONE (2015)

Aggregate behavior of the Vicsek model, simulation #3.A typical snapshot is shown in Fig 7(C). (A) Normalized average velocity order parameter φ(t). (B) Contour plot of Betti number . (C) Contour plot of Betti number . The topological analysis shows essentially no cluster formation; the narrow region in which  arises from an isolated agent. The two persistent topological circles are consistent with highly aligned particles covering both dimensions of the periodic spatial domain. Topological features become fairly stagnant once the entire group forms a large, aligned cluster traveling as a rigid body early in the simulation. See text for a more comprehensive analysis.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4430537&req=5

pone.0126383.g010: Aggregate behavior of the Vicsek model, simulation #3.A typical snapshot is shown in Fig 7(C). (A) Normalized average velocity order parameter φ(t). (B) Contour plot of Betti number . (C) Contour plot of Betti number . The topological analysis shows essentially no cluster formation; the narrow region in which arises from an isolated agent. The two persistent topological circles are consistent with highly aligned particles covering both dimensions of the periodic spatial domain. Topological features become fairly stagnant once the entire group forms a large, aligned cluster traveling as a rigid body early in the simulation. See text for a more comprehensive analysis.
Mentions: Fig 10 shows results from the same simulation as the snapshot in Fig 7(C). Here, N = 300, ℓ = 5, and η = 0.1. In panel (A), the order parameter φ(t) rises steeply to one with a small intermediate step. The order parameter here and in Fig 6(A) share some features, and yet we will see that the topological calculations capture the differences in the dynamics already apparent in the snapshots of Fig 7.

Bottom Line: To interpret our results, we introduce a visualization that displays Betti numbers over simulation time and topological persistence scale.We compare our topological results to order parameters typically used to quantify the global behavior of aggregations, such as polarization and angular momentum.The topological calculations reveal events and structure not captured by the order parameters.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics, Statistics, and Computer Science, Macalester College, Saint Paul, Minnesota, United States of America.

ABSTRACT
We apply tools from topological data analysis to two mathematical models inspired by biological aggregations such as bird flocks, fish schools, and insect swarms. Our data consists of numerical simulation output from the models of Vicsek and D'Orsogna. These models are dynamical systems describing the movement of agents who interact via alignment, attraction, and/or repulsion. Each simulation time frame is a point cloud in position-velocity space. We analyze the topological structure of these point clouds, interpreting the persistent homology by calculating the first few Betti numbers. These Betti numbers count connected components, topological circles, and trapped volumes present in the data. To interpret our results, we introduce a visualization that displays Betti numbers over simulation time and topological persistence scale. We compare our topological results to order parameters typically used to quantify the global behavior of aggregations, such as polarization and angular momentum. The topological calculations reveal events and structure not captured by the order parameters.

No MeSH data available.


Related in: MedlinePlus