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Inferring effective forces in collective motion
Yael Katz, Christos Ioannou, Kolbjørn Tunstrøm and Iain Couzin
Dept. of Ecology & Evolutionary Biology Princeton University
Cristián Huepe Unaffi liated NSF Grantee Cristian Huepe Labs Inc. - Chicago IL
This work was supported by the National Science Foundation under Grants No. DMS-0507745 & PHY-0848755
Outline
• Overview Background Some basic models of collective
motion Challenge: The inverse problem
• A detailed effective-force analysis Fish schooling: quasi 2D
experiments Model-free approach Effective-forces: results
Motivation
Collective motion is observed in diverse animal species, not only in bacteria.
Fish schools & bird flocks can involve from a few individuals to several thousands
Locust swarms can contain 109 individuals traveling thousands of kilometers
– Background
Current efforts Quantitative experiments Distinguishing generic and specific behaviors
Challenges in modeling Different models produce similar dynamics We can be prejudiced by familiar interactions
The inverse problem: Deducing the interaction rules from collective
dynamics
– Challenges
Intuitive flocking algorithm (Craig Reynolds – Sony)
Generic rules (from computer graphics)
– Flocks, Herds, and Schools: A Distributed Behavioral Model Computer Graphics, 21(4), pp. 25-34, 1987
– Defined Boids and simple interaction rules:
▪ Separation
▪ Alignment
▪ Cohesion
Motivation Non-equilibrium swarming dynamics
Emerging collective behavior
Statistical description
Complex behavior
The Vicsek model
Other models Agent-based algorithms
Discrete time Continuous time (ODEs)
Field-based descriptions (PDEs)
– The Vicsek model
The “zones” model
– A more biological model
Journal of Theoretical Biology (2002) 218, 1-11I. D. Couzin, J. Krause, R. James, G. D. Ruxton &N. R. Franks
- “Insect-like” swarm:
- Torus, “milling”:
- Migration, flocking:
Different algorithms yield similar collective motion
What interactions are animal swarms actually using?
Are we making underlying assumptions?
In other words:
Can we properly address the inverse problem?
- Challenge: The inverse problem
Outline
• Overview Background Some basic models of collective
motion Challenge: The inverse problem
• A detailed effective-force analysis Fish schooling: quasi 2D
experiments Model-free approach Effective-forces: results
Experimental System
Work with:
Prof Iain Couzin, Dr Yael Katz,
Dr Kolbjørn Tunstrøm
Dr Christos Ioannou
Other collaborators:Dr Andrey SokolovAndrew Hartnett,
Etc.
Princeton University
Method Measure mean effective forces on 2-fish & 3-fish systems Use large dataset: 14 experiments of 56 minutes each Use classical mechanics formalism (force-driven systems)
F=ma & trajectories given by (q,p) per degree of freedom
Goals “Model-free” approach on clear mathematical grounds Gain intuition over multiple possible dynamical
dependencies Study deviations from classical mechanics
Memory, higher-order interactions, etc.
Other methods Maximum entropy Bayesian inference
The effective-force approach
Space-like variables: Distance front-back Distance left-right
Velocity-like variables: Neighbor fish speed Focal fish speed Relative heading
Acceleration-like variables? Neighbor fish turning rate Neighbor fish speeding Focal fish turning rate Focal fish speeding
The two-fish system
Velocity-dependent forces
• Higher speed larger forces & preferred y-distance
• Aligned Higher F||
• Misaligned Higher F
Intrinsic 3-body interaction
Best match:
2neigbor 1neigbor 223 7.0 SS FFF 2neigbor 1neigbor 223 4.0 TT FFF
Residual 3-body interaction:
“Non-negligible” “Negligible”
Best match:
Residual 3-body interaction:
Conclusions
Using an effective-force approach we found that:
Within the interaction zone, speeding depends mainly on front-back distance, and turning on left-right distance
Trailing fish turn to follow fish in front but adjust speed to follow neighbors in front or behind
Alignment emerges from attraction/repulsion interactions: No evidence for explicit alignment
Tuning response is approximately averaged while speeding is between averaging and additive
Speeding response follows no linear superposition principle: Residual intrinsic three-body interaction
New models and simulations to analyze
New statistical/emergent properties to find … Fin