USING TOPOLOGY TO MEASURE DYNAM- ICS OF BIOLOGICAL AGGREGATIONS August 7, 2019 Lori Ziegelmeier, Macalester College ICERM Workshop: Applied Mathematical Modeling with Topological Techniques
INTRODUCTION
BIOLOGICAL AGGREGATIONS In many natural systems, particles, organisms, or agents interact locally according to rules that produce aggregate behavior. 3
CLASSIC WAY TO ANALYZE BIOLOGICAL AGGREGATIONS � � N � � � 1 � Alignment Order Parameter: ϕ ( t ) � v i ( t ) � � Nv 0 � � � � i � 1 7 0 4
CLASSIC WAY TO ANALYZE BIOLOGICAL AGGREGATIONS � � N � � � 1 � Alignment Order Parameter: ϕ ( t ) � v i ( t ) � � Nv 0 � � � � i � 1 7 0 4
CLASSIC WAY TO ANALYZE BIOLOGICAL AGGREGATIONS � � N � � � 1 � Alignment Order Parameter: ϕ ( t ) � v i ( t ) � � Nv 0 � � � � i � 1 7 0 4
TOPOLOGICAL DATA ANALYSIS 1. Envision data as a point cloud ◦ e.g. position-velocity for one snapshot in time 2. Create connections between proximate points ◦ build simplicial complex 3. Determine topological structure of complex ◦ compute homology (measure # holes) 4. Vary proximity parameter to assess different scales ◦ calculate persistent homology 5
COMPUTE PERSISTENT HOMOLOGY b 0 ε 6
COMPUTE PERSISTENT HOMOLOGY b 0 1 6 3 2 7 6
COMPUTE PERSISTENT HOMOLOGY 6
COMPUTE PERSISTENT HOMOLOGY 6
TOPOLOGICAL DATA ANALYSIS 1. Envision data as a point cloud ◦ e.g. position-velocity for one snapshot in time 2. Create connections between proximate points ◦ build simplicial complex 3. Determine topological structure of complex ◦ compute homology (measure # holes) 4. Vary proximity parameter to assess different scales ◦ calculate persistent homology 7
TOPOLOGICAL DATA ANALYSIS 1. Envision data as a point cloud ◦ e.g. position-velocity for one snapshot in time 2. Create connections between proximate points ◦ build simplicial complex 3. Determine topological structure of complex ◦ compute homology (measure # holes) 4. Vary proximity parameter to assess different scales ◦ calculate persistent homology 5. Measure topology as time evolves. ◦ Crocker plots 7
EVOLVE IN TIME Compute the k th Betti number b k ( ε, t ) , 8
EVOLVE IN TIME Compute the k th Betti number b k ( ε, t ) , CROCKER plot Contour Realization Of Computed K -dimensional hole Evolution in the Rips complex (CROCKER) (Topaz, Z., Halverson 2015) Topological Data Analysis of Biological Aggregation Models 8
CROCKER AS EXPLORATORY TOOL
D’ORSOGNA MODEL � Dynamical system describing motion of interacting point particles in an unbounded plane in continuous time. � Model written as: � � � x i v i i � 1 , . . . , N � v i | 2 � � m � � � α − β |� v i − ∇ i U i v i � � x j |/ ℓ r − C a e −|� C r e −|� x i −� x i −� x j |/ ℓ a U i � j � i � After nondimensionalization, we have 4 parameters: α, β, C � C r , ℓ � ℓ r C a ℓ a (D’Orsogna, et al 2006) Self-Propelled Particles with Soft-Core Interactions: Patterns, Stability, and Collapse 10
SNAPSHOTS IN TIME α � 1 . 5 , β � 0 . 5 , C � 2 , ℓ � 0 . 25 with N � 500 particles 11
ORDER PARAMETERS TO SUMMARIZE COLLECTIVE BEHAVIOR � � N � i � 1 v i ( t ) Polarization: P ( t ) � � � � N i � 1 | v i ( t )| � � 12
ORDER PARAMETERS TO SUMMARIZE COLLECTIVE BEHAVIOR � � N � i � 1 v i ( t ) Polarization: P ( t ) � � � � N i � 1 | v i ( t )| � � Are the particles moving in the same direction? 12
ORDER PARAMETERS TO SUMMARIZE COLLECTIVE BEHAVIOR � � N � i � 1 v i ( t ) Polarization: P ( t ) � � � � N i � 1 | v i ( t )| � � Angular Momentum: � � N � i � 1 r i ( t )× v i ( t ) M ang ( t ) � � � � N � i � 1 | r i ( t )|| v i ( t )| � 12
ORDER PARAMETERS TO SUMMARIZE COLLECTIVE BEHAVIOR � � N � i � 1 v i ( t ) Polarization: P ( t ) � � � � N i � 1 | v i ( t )| � � Angular Momentum: Are the particles rotating (in the same � � N � i � 1 r i ( t )× v i ( t ) M ang ( t ) � � � direction)? � N � i � 1 | r i ( t )|| v i ( t )| � 12
ORDER PARAMETERS TO SUMMARIZE COLLECTIVE BEHAVIOR � � N � i � 1 v i ( t ) Polarization: P ( t ) � � � � N i � 1 | v i ( t )| � � Angular Momentum: � � N � i � 1 r i ( t )× v i ( t ) M ang ( t ) � � � � N � i � 1 | r i ( t )|| v i ( t )| � Absolute Angular Momentum: � � N � i � 1 | r i ( t )× v i ( t )| M abs ( t ) � � � � N � i � 1 | r i ( t )|| v i ( t )| � 12
ORDER PARAMETERS TO SUMMARIZE COLLECTIVE BEHAVIOR � � N � i � 1 v i ( t ) Polarization: P ( t ) � � � � N i � 1 | v i ( t )| � � Angular Momentum: Are the particles rotating? � � N � i � 1 r i ( t )× v i ( t ) M ang ( t ) � � � � N � i � 1 | r i ( t )|| v i ( t )| � Absolute Angular Momentum: � � N � i � 1 | r i ( t )× v i ( t )| M abs ( t ) � � � � N � i � 1 | r i ( t )|| v i ( t )| � 12
ORDER PARAMETERS TO SUMMARIZE COLLECTIVE BEHAVIOR � � N � i � 1 v i ( t ) Polarization: P ( t ) � � � � N i � 1 | v i ( t )| � � Angular Momentum: � � N � i � 1 r i ( t )× v i ( t ) M ang ( t ) � � � � N � i � 1 | r i ( t )|| v i ( t )| � Absolute Angular Momentum: � � N � i � 1 | r i ( t )× v i ( t )| M abs ( t ) � � � � N � i � 1 | r i ( t )|| v i ( t )| � Average Nearest Neighbor Distance: N � D nn ( t ) � 1 1 ≤ j ≤ N | x i ( t ) − x j ( t )| min N i � 1 12
ORDER PARAMETERS TO SUMMARIZE COLLECTIVE BEHAVIOR � � N � i � 1 v i ( t ) Polarization: P ( t ) � � � � N i � 1 | v i ( t )| � � Angular Momentum: How close are the particles? � � N � i � 1 r i ( t )× v i ( t ) M ang ( t ) � � � � N � i � 1 | r i ( t )|| v i ( t )| � Absolute Angular Momentum: � � N � i � 1 | r i ( t )× v i ( t )| M abs ( t ) � � � � N � i � 1 | r i ( t )|| v i ( t )| � Average Nearest Neighbor Distance: N � D nn ( t ) � 1 1 ≤ j ≤ N | x i ( t ) − x j ( t )| min N i � 1 12
D’ORSOGNA SIMULATION ANALYSIS 1.00 (A) 0.6 (A) Order Parameters 0.75 Order Parameter 0.50 P M 0.25 t = 5 M abs −0.6 −0.6 0.6 0.00 0 10 20 30 40 50 0.6 (B) Simulation Time Proximity Parameter ε (B) level 1.5 b 0 = 1 1 2 1.0 t = 23 −0.6 3 −0.6 0.6 0.5 4 b 0 ≥ 5 0.6 5 (C) 0.0 0 10 20 30 40 50 Simulation Time t Proximity Parameter ε (C) t = 34 level −0.6 1.5 b 1 = 0 b 1 = 1 1 −0.6 0.6 2 1.0 3 0.5 b 1 ≥ 5 4 b 1 = 2 5 0.0 0 10 20 30 40 50 Simulation Time t 13
D’ORSOGNA SIMULATION ANALYSIS 1.00 (A) 0.6 (A) Order Parameters 0.75 Order Parameter 0.50 P M 0.25 t = 5 M abs −0.6 −0.6 0.6 0.00 0 10 20 30 40 50 0.6 (B) Simulation Time Proximity Parameter ε (B) level 1.5 b 0 = 1 1 2 1.0 t = 23 −0.6 3 −0.6 0.6 0.5 4 b 0 ≥ 5 0.6 5 (C) 0.0 0 10 20 30 40 50 Simulation Time t Proximity Parameter ε (C) t = 34 level −0.6 1.5 b 1 = 0 b 1 = 1 1 −0.6 0.6 2 1.0 3 0.5 b 1 ≥ 5 4 b 1 = 2 5 0.0 0 10 20 30 40 50 Simulation Time t 13
PARAMETER IDENTIFICATION
QUESTION OF INTEREST Forward Problem: 15
QUESTION OF INTEREST Forward Problem: Inverse Problem: Research Question What is an informative way to summarize collective behavior for machine learning techniques? 15
PHENOTYPES IN THE D’ORSOGNA MODEL Different patterns emerge by altering parameters C and ℓ 16
PIPELINE Simulate Swarm Behavior Design Construct Order Parameters Topology Problem dependent Problem independent ε /2 Order Parameter based Topology based Description Description 25 parameter sets x 100 realizations = 2500 simulations Construct Feature Vectors value scale time Machine Learning time Predict Parameters & Patterns 17
PHENOTYPES AND FEATURE VECTORS Single Mill: Double Ring: Escape: 18
SUPPORT VECTOR MACHINE � Supervised machine learning algorithm � 5-fold cross validation, with each training simulation labeled with ( C , ℓ ) , 20% of data used for each test � Accuracy � out-of-sample simulations with ( C ,ℓ ) correct out-of-sample simulations 19
CLASSIFICATION RESULTS Summary Feature Dimension Accuracy P ( t ) 87 57.7% M ang ( t ) Order 87 34.4% M abs ( t ) Parameters 87 68.0% D NN ( t ) 87 91.1% All 4*87 89.2% TDA b 0 200*86 99.6% (time-delayed b 1 200*86 99.3% position) b 0 & b 1 2*200*86 99.1% 20
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