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

1000 fish dynamics

1000 fish dynamics

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

Position-dependent forces

• Zero force high density

• ||v||>0.5 BL/s

• F||(y), F=(x)

Velocity-dependent forces

• Higher speed larger forces & preferred y-distance

• Aligned Higher F||

• Misaligned Higher F

Temporal correlation

Orientation information Front to back

Speed information Both ways

The three-body problem

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

… Fin