Topological Data Analysis of Biological Aggregation Models

  title={Topological Data Analysis of Biological Aggregation Models},
  author={Chad M. Topaz and Lori Ziegelmeier and Tom Halverson},
  journal={PLoS ONE},
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… 
Modelling Topological Features of Swarm Behaviour in Space and Time With Persistence Landscapes
It is demonstrated that the proposed model may be used to perform retrieval and clustering of swarm behavior in terms of topological features, and it is discovered that clustering returns clusters corresponding to the swarm behaviors of flock, torus, and disordered.
A topological approach to selecting models of biological experiments
Analysis of the a priori order parameters indicates that the interactive model better describes the experimental data than the control model does, suggesting the utility of the topological approach in the absence of specific knowledge of mechanisms underlying the data.
Analyzing collective motion with machine learning and topology
This work uses topological data analysis and machine learning to study a seminal model of collective motion in biology that describes agents interacting nonlinearly via attractive-repulsive social forces and gives rise to collective behaviors such as flocking and milling.
Spatiotemporal Persistent Homology for Dynamic Metric Spaces
This paper extends the Rips filtration stability result for (static) metric spaces to the setting of DMSs and proposes to utilize a certain metric d for comparing these invariants, including the rank invariant or the dimension function of the multidimensional persistence module that is derived from a DMS.
Geometrical and topological approaches to Big Data
Towards the prediction of critical transitions in spatially extended populations with cubical homology
En route to a global extinction event, it is found that the Betti number time series of a population exhibits characteristic changes, which is likely to aide in the prediction of critical transitions in spatially extended systems.
A Short Survey of Topological Data Analysis in Time Series and Systems Analysis
This paper will review recent developments and contributions where topological data analysis especially persistent homology has been applied to time series analysis, dynamical systems and signal processing and cover problem statements such as stability determination, risk analysis, systems behaviour, and predicting critical transitions in financial markets.
Analyzing Collective Motion Using Graph Fourier Analysis
This work shows how the field of graph signal processing can be used to fuse the two approaches to analysis of swarms by collectively analyzing swarm properties using graph Fourier harmonics that respect the topological structure of the swarm.
Tracking collective cell motion by topological data analysis
Time series of data generated by numerical simulations are used to automatically group, track and classify the advancing interfaces of cellular aggregates by means of bottleneck or Wasserstein distances of persistent homologies, which are relevant for tissue engineering, biological effects of materials, tissue and organ regeneration.
Extracting Topological Features from Big Data Using Persistent Density Entropy
This paper defines an entropy called persistent density entropy, which gives the uncertainty of each simplicial complex approximating the underlying space of the data set and can be used to detect outliers to a certain extent.


Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study
It is argued that a topological interaction is indispensable to maintain a flock's cohesion against the large density changes caused by external perturbations, typically predation, and supported by numerical simulations.
Nonlocal Aggregation Models: A Primer of Swarm Equilibria
This work derives a nonlocal partial differential equation describing the evolving population density of a continuum aggregation and finds exact analytical expressions for the equilibria.
Inferring individual rules from collective behavior
It is found that important features of observed flocking surf scoters can be accounted for by zonal models with specific, well-defined rules of interaction, including strong short-range repulsion, intermediate-range alignment, and longer-range attraction.
Topological estimation using witness complexes
This paper tackles the problem of computing topological invariants of geometric objects in a robust manner, using only point cloud data sampled from the object, and produces a nested family of simplicial complexes, which represent the data at different feature scales, suitable for calculating persistent homology.
Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis
It is shown that maximum persistence at the point-cloud level can be used to quantify periodicity at the signal level, prove structural and convergence theorems for the resulting persistence diagrams, and derive estimates for their dependency on window size and embedding dimension.
A Primer of Swarm Equilibria
The model can reproduce a model of locust swarms, which in nature are observed to consist of a concentrated population on the ground separated from an airborne group, and quasi-two-dimensionality of the model plays a critical role.
Nonlocal aggregation equations: A primer of swarm equilibria
Biological aggregations (swarms) exhibit morphologies governed by social interactions and responses to environment. Starting from a particle model we derive a nonlocal PDE, known as the aggregation
A Nonlocal Continuum Model for Biological Aggregation
A continuum model for biological aggregations in which individuals experience long-range social attraction and short-range dispersal is constructed, and energy arguments are used to understand the nonlinear selection of clump solutions, and to predict the internal density in the large population limit.
Detecting coherent structures using braids