Inferring Effective Forces in Collective Animal Motion Dynamics

1. Inferring effective forces in collective motion Yael Katz, Christos Ioannou, Kolbjørn Tunstrøm and Iain Couzin Dept. of Ecology & Evolutionary Biology Princeton University Cristián Huepe Unaffiliated NSF Grantee Cristian Huepe Labs Inc. - Chicago IL This work was supported by the National Science Foundation under Grants No. DMS-0507745 & PHY-0848755

2. 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

3. – Background • 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

4. – Challenges • Current efforts • Quantitative experiments • Distinguishing generic and specific behaviors • Challenges in modeling • Different models produce similar dynamics • We can beprejudiced by familiar interactions • The inverse problem: • Deducing the interaction rules from collective dynamics

5. Generic rules (from computer graphics) Intuitive flocking algorithm (Craig Reynolds – Sony) • 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

6. – The Vicsek model • 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)

7. – A more biological model The “zones” model Journal of Theoretical Biology (2002) 218, 1-11 I. D. Couzin, J. Krause, R. James, G. D. Ruxton & N. R. Franks - “Insect-like” swarm: - Torus, “milling”: - Migration, flocking:

8. - Challenge: The inverse problem • 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?

9. 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

10. Experimental System Work with: Prof Iain Couzin, Dr Yael Katz, Dr Kolbjørn Tunstrøm Dr Christos Ioannou Other collaborators: Dr Andrey Sokolov Andrew Hartnett, Etc. Princeton University

11. 1000 fish dynamics

12. 1000 fish dynamics

13. The effective-force approach • 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

14. The two-fish system • 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

15. Position-dependent forces • Zero force  • high density • ||v||>0.5 BL/s • F||(y), F=(x)

16. Velocity-dependent forces • Higher speed  • larger forces & • preferred y-distance • Aligned  Higher F|| • Misaligned  Higher F

17. Temporal correlation Orientation information  Front to back Speed information  Both ways

18. The three-body problem

19. Intrinsic 3-body interaction Residual 3-body interaction: Residual 3-body interaction: “Non-negligible” “Negligible” Best match: Best match:

20. 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 orbehind • 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

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