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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.


Simulation snapshots of the D’Orsogna model (11).Circles indicate positions of the N = 500 particles in an unbounded plane, line segments represent heading, and blue (red) agents are traveling (counter)clockwise. Over time, the group develops a hollow core and a double-mill structure in which a majority of agents travel clockwise, but a minority persists in the counterclockwise orientation. (A) Time t = 5. (B) Time t = 23. (C) Time t = 34. The other model parameters used in this simulation are α = 1.5, β = 0.5, Cr = 1, Lr = 0.5, Ca = 0.5, La = 2.
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pone.0126383.g011: Simulation snapshots of the D’Orsogna model (11).Circles indicate positions of the N = 500 particles in an unbounded plane, line segments represent heading, and blue (red) agents are traveling (counter)clockwise. Over time, the group develops a hollow core and a double-mill structure in which a majority of agents travel clockwise, but a minority persists in the counterclockwise orientation. (A) Time t = 5. (B) Time t = 23. (C) Time t = 34. The other model parameters used in this simulation are α = 1.5, β = 0.5, Cr = 1, Lr = 0.5, Ca = 0.5, La = 2.

Mentions: We conduct a simulation of (11) with N = 500 particles, and model parameters α = 1.5, β = 0.5, Ca = 0.5, Cr = 1, La = 2, Lr = 0.5. Fig 11 shows selected simulation snapshots. For convenience, we color blue all particles traveling clockwise with respect to the center of mass of the group; particles traveling counterclockwise are red. In panel (A), at t = 5, the particles occupy a disk-like region with somewhat disorganized velocities. At t = 23, we see that a hollow core has begun to form. At t = 45, the mill structure has a well defined core. The group favors the clockwise direction, though there is a minority group of particles traveling counterclockwise.


Topological data analysis of biological aggregation models.

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

Simulation snapshots of the D’Orsogna model (11).Circles indicate positions of the N = 500 particles in an unbounded plane, line segments represent heading, and blue (red) agents are traveling (counter)clockwise. Over time, the group develops a hollow core and a double-mill structure in which a majority of agents travel clockwise, but a minority persists in the counterclockwise orientation. (A) Time t = 5. (B) Time t = 23. (C) Time t = 34. The other model parameters used in this simulation are α = 1.5, β = 0.5, Cr = 1, Lr = 0.5, Ca = 0.5, La = 2.
© Copyright Policy
Related In: Results  -  Collection

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

pone.0126383.g011: Simulation snapshots of the D’Orsogna model (11).Circles indicate positions of the N = 500 particles in an unbounded plane, line segments represent heading, and blue (red) agents are traveling (counter)clockwise. Over time, the group develops a hollow core and a double-mill structure in which a majority of agents travel clockwise, but a minority persists in the counterclockwise orientation. (A) Time t = 5. (B) Time t = 23. (C) Time t = 34. The other model parameters used in this simulation are α = 1.5, β = 0.5, Cr = 1, Lr = 0.5, Ca = 0.5, La = 2.
Mentions: We conduct a simulation of (11) with N = 500 particles, and model parameters α = 1.5, β = 0.5, Ca = 0.5, Cr = 1, La = 2, Lr = 0.5. Fig 11 shows selected simulation snapshots. For convenience, we color blue all particles traveling clockwise with respect to the center of mass of the group; particles traveling counterclockwise are red. In panel (A), at t = 5, the particles occupy a disk-like region with somewhat disorganized velocities. At t = 23, we see that a hollow core has begun to form. At t = 45, the mill structure has a well defined core. The group favors the clockwise direction, though there is a minority group of particles traveling counterclockwise.

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.