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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.


Aggregate behavior of the D’Orsogna model.Snapshot of the time evolution are shown in Fig 11. (A) Three order parameters: polarization P (red), angular momentum M (green), and absolute angular momentum Mabs (blue). (B) Contour plot of Betti number . (C) Contour plot of Betti number . At times below t ≈ 20, there is little topological structure. For t > 20, we have one or—intermittently—two connected components of data points. There are two discernible topological circles for smaller ɛ and one circle for larger ɛ. These circles survive for long periods of simulation time. The topological signature of the first two Betti numbers, , is consistent with a double mill structure. See text for a more comprehensive analysis.
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pone.0126383.g012: Aggregate behavior of the D’Orsogna model.Snapshot of the time evolution are shown in Fig 11. (A) Three order parameters: polarization P (red), angular momentum M (green), and absolute angular momentum Mabs (blue). (B) Contour plot of Betti number . (C) Contour plot of Betti number . At times below t ≈ 20, there is little topological structure. For t > 20, we have one or—intermittently—two connected components of data points. There are two discernible topological circles for smaller ɛ and one circle for larger ɛ. These circles survive for long periods of simulation time. The topological signature of the first two Betti numbers, , is consistent with a double mill structure. See text for a more comprehensive analysis.

Mentions: Fig 12(A) shows time series of three order parameter metrics used in [12] to characterize the global behavior of the system. Polarization P, defined in (Eq 1), is similar to φ(t) for the Vicsek model; it measures the degree to which agents are aligned. Because the group is traveling around a circle, P (red curve) remains low for the duration of the simulation. Angular momentum M, also defined in (Eq 1), helps quantify the rotation of the group. For a perfect mill structure, M = 1. For our simulation, we see an evolution from M ≈ 0 early in time to M ≈ 0.93 for later times (green curve). However, as mentioned previously, the metric M cannot distinguish between single and double mills, and so [12] introduces the absolute angular momentum Mabs, defined in (Eq 2). The fact that Mabs approaches unity signals that the asymptotic behavior of the group is rotational. The fact that M approaches a number close to but less than unity signals that a small minority of the group members are rotating counter to the majority.


Topological data analysis of biological aggregation models.

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

Aggregate behavior of the D’Orsogna model.Snapshot of the time evolution are shown in Fig 11. (A) Three order parameters: polarization P (red), angular momentum M (green), and absolute angular momentum Mabs (blue). (B) Contour plot of Betti number . (C) Contour plot of Betti number . At times below t ≈ 20, there is little topological structure. For t > 20, we have one or—intermittently—two connected components of data points. There are two discernible topological circles for smaller ɛ and one circle for larger ɛ. These circles survive for long periods of simulation time. The topological signature of the first two Betti numbers, , is consistent with a double mill structure. 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.g012: Aggregate behavior of the D’Orsogna model.Snapshot of the time evolution are shown in Fig 11. (A) Three order parameters: polarization P (red), angular momentum M (green), and absolute angular momentum Mabs (blue). (B) Contour plot of Betti number . (C) Contour plot of Betti number . At times below t ≈ 20, there is little topological structure. For t > 20, we have one or—intermittently—two connected components of data points. There are two discernible topological circles for smaller ɛ and one circle for larger ɛ. These circles survive for long periods of simulation time. The topological signature of the first two Betti numbers, , is consistent with a double mill structure. See text for a more comprehensive analysis.
Mentions: Fig 12(A) shows time series of three order parameter metrics used in [12] to characterize the global behavior of the system. Polarization P, defined in (Eq 1), is similar to φ(t) for the Vicsek model; it measures the degree to which agents are aligned. Because the group is traveling around a circle, P (red curve) remains low for the duration of the simulation. Angular momentum M, also defined in (Eq 1), helps quantify the rotation of the group. For a perfect mill structure, M = 1. For our simulation, we see an evolution from M ≈ 0 early in time to M ≈ 0.93 for later times (green curve). However, as mentioned previously, the metric M cannot distinguish between single and double mills, and so [12] introduces the absolute angular momentum Mabs, defined in (Eq 2). The fact that Mabs approaches unity signals that the asymptotic behavior of the group is rotational. The fact that M approaches a number close to but less than unity signals that a small minority of the group members are rotating counter to the majority.

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.