Flocking starlings in 3D by Alessio Cimarelli

Flocking starlings in 3D Alessio Cimarelli

Flocking starlings in 3D The S TARFLAG project

Collective behaviour in biology A NIMAL GROUPS M ODELS

S TARFLAG project W HAT

Alessio Cimarelli Dept. of Physics, University of Rome “Sapienza”

W HERE / W HEN / W HO H OW

Structural analysis M ETRIC VS . TOPOLOGY

“Swarm Intelligence and Critical Behavior” Center for Interdisciplinary Research Bielefeld University March 22, 2011

D ENSITY GRADIENT

Velocity correlations Beyond starlings. . .

Contents

Flocking starlings in 3D Alessio Cimarelli

Collective behaviour in biology Animal groups Models S TARFLAG project What Where / When / Who How Structural analysis Metric vs. topology Density gradient Velocity correlations Beyond starlings. . .

Collective behaviour in biology A NIMAL GROUPS M ODELS

S TARFLAG project W HAT W HERE / W HEN / W HO H OW

Structural analysis M ETRIC VS . TOPOLOGY

D ENSITY GRADIENT

Velocity correlations Beyond starlings. . .

Animal groups â&#x2020;&#x2019; Self-organization From local rules to collective motion

Flocking starlings in 3D Alessio Cimarelli

Collective behaviour in biology A NIMAL GROUPS M ODELS

S TARFLAG project W HAT W HERE / W HEN / W HO H OW

Structural analysis M ETRIC VS . TOPOLOGY

D ENSITY GRADIENT

Velocity correlations Beyond starlings. . .

Animal groups â&#x2020;&#x2019; Self-organization From local rules to collective motion I

The group fulfils tasks beyond the abilities of the individuals Behavioral rules are selected to achieve collective performance What kind of interaction grants such efficient group behavior?

Flocking starlings in 3D Alessio Cimarelli

Collective behaviour in biology A NIMAL GROUPS M ODELS

S TARFLAG project W HAT W HERE / W HEN / W HO H OW

Structural analysis M ETRIC VS . TOPOLOGY

D ENSITY GRADIENT

Velocity correlations Beyond starlings. . .

Flocking starlings in 3D

Models Aoki, 1982 – Reynolds, 1987 – Huth & Wissel, 1992 – Warburton & Lazarus, 1991 – WL Romey, 1996 – Couzin et al., 2002 – Vicsek et al., 1995 – Toner & Tu, 1995 – Gregoire & Chate, 2004

Allelomimesis: imitation of neighbors I

alignment of velocities: go where the other go attraction to neighbors: stay with the group

short range repulsion: avoid collisions

constant velocity: self-propulsion

Alessio Cimarelli

Collective behaviour in biology A NIMAL GROUPS M ODELS

S TARFLAG project W HAT W HERE / W HEN / W HO H OW

Structural analysis M ETRIC VS . TOPOLOGY

D ENSITY GRADIENT

Velocity correlations nin nin X ~rij (t) 1 X ~ ~di (t + 1) = 1 wj dj (t) + + ~ηi (t) nin nin |~rij (t)| j=1

j=1

We need experimental data to control and verify numerical models!

Beyond starlings. . .

Contents

Flocking starlings in 3D Alessio Cimarelli

Collective behaviour in biology Animal groups Models S TARFLAG project What Where / When / Who How Structural analysis Metric vs. topology Density gradient Velocity correlations Beyond starlings. . .

Collective behaviour in biology A NIMAL GROUPS M ODELS

S TARFLAG project W HAT W HERE / W HEN / W HO H OW

Structural analysis M ETRIC VS . TOPOLOGY

D ENSITY GRADIENT

Velocity correlations Beyond starlings. . .

Flocking starlings in 3D

A case study: flocking

Alessio Cimarelli

Collective behaviour in biology A NIMAL GROUPS M ODELS

S TARFLAG project W HAT W HERE / W HEN / W HO H OW

Cohesion I

Coherence

robustness against perturbation large density fluctuations no fragmentation / few stragglers

Here it is our hero, the starling!

Structural analysis M ETRIC VS . TOPOLOGY

high coordination

collective response

Velocity correlations

quick information propagation

Beyond starlings. . .

D ENSITY GRADIENT

Our empirical observations From left to right: Fabio Stefanini, Raffaele Santagati, me

Flocking starlings in 3D Alessio Cimarelli

Collective behaviour in biology A NIMAL GROUPS M ODELS

S TARFLAG project W HAT W HERE / W HEN / W HO H OW

Structural analysis M ETRIC VS . TOPOLOGY

D ENSITY GRADIENT

Velocity correlations Beyond starlings. . .

Experimental seasons: 2006, 2007, 2008.

Starlings in action. . . Piazza dei Cinquecento, Roma

Flocking starlings in 3D Alessio Cimarelli

Collective behaviour in biology A NIMAL GROUPS M ODELS

S TARFLAG project W HAT W HERE / W HEN / W HO H OW

Structural analysis M ETRIC VS . TOPOLOGY

D ENSITY GRADIENT

Velocity correlations

Show video. . .

Beyond starlings. . .

Experimental approach: stereometry Cavagna et al., The STARFLAG handbook on collective animal behaviour, Animal Behaviour, vol. 76 (2008)

Flocking starlings in 3D Alessio Cimarelli

Collective behaviour in biology

d δs = xB − xA = f Z

A NIMAL GROUPS M ODELS

S TARFLAG project W HAT W HERE / W HEN / W HO H OW

Structural analysis M ETRIC VS . TOPOLOGY

D ENSITY GRADIENT

typical flocks distance: Z ∼ 100m

photographic resolution: δs ∼ 1pixel

required error on relative distance: δZ < 0.1m

Velocity correlations

z2 d> ∼ 23m Ωδz

Beyond starlings. . .

Experimental setting Cavagna et al., The STARFLAG handbook on collective animal behaviour, Animal Behaviour, vol. 76 (2008)

Flocking starlings in 3D Alessio Cimarelli

Collective behaviour in biology A NIMAL GROUPS M ODELS

S TARFLAG project W HAT W HERE / W HEN / W HO H OW

Structural analysis M ETRIC VS . TOPOLOGY

D ENSITY GRADIENT

Velocity correlations Beyond starlings. . .

The stereoscopic matching problem Cavagna et al., The STARFLAG handbook on collective animal behaviour, Animal Behaviour, vol. 76 (2008)

Flocking starlings in 3D Alessio Cimarelli

Collective behaviour in biology

Some previous 3D data: I

Cullen et al., 1965: 10 fishes Major & Dill, 1978: 45 birds Partridge et al., 1980: 30 fishes Pomeroy & Heppner, 1992: 16 birds

A NIMAL GROUPS M ODELS

S TARFLAG project W HAT W HERE / W HEN / W HO H OW

Structural analysis M ETRIC VS . TOPOLOGY

D ENSITY GRADIENT

Velocity correlations Beyond starlings. . .

Us: â&#x2C6;ź 4000 birds

Flocking starlings in 3D

From photos to pc screen Cavagna et al., The STARFLAG handbook on collective animal behaviour, Animal Behaviour, vol. 76 (2008)

Alessio Cimarelli

Collective behaviour in biology A NIMAL GROUPS M ODELS

Left camera (C)

Right camera (A)

S TARFLAG project W HAT W HERE / W HEN / W HO H OW

Structural analysis M ETRIC VS . TOPOLOGY

D ENSITY GRADIENT

Velocity correlations Beyond starlings. . .

3D positions from the A camera point of view

The dynamical tracking problem Stefanini F., Ricostruzione di traiettorie tridimensionali e analisi dinamica di stormi nell’ambito del progetto StarFlag, Thesis (February 2009)

Mean speed: ∼ 10 m/s

Camera time resolution: 1 s 10

Average nearest neighbour distance: ∼ 1 m 3D reconstruction efficiency: ∼ 90%

Flocking starlings in 3D Alessio Cimarelli

Collective behaviour in biology A NIMAL GROUPS M ODELS

S TARFLAG project W HAT W HERE / W HEN / W HO H OW

Structural analysis

1. Two-times matching is very efficient 2. Multi-time matching: requires optimization I I I

missing birds high density low temporal resolution

M ETRIC VS . TOPOLOGY

D ENSITY GRADIENT

Velocity correlations Beyond starlings. . .

From static shots to vectorial field Stefanini F., Ricostruzione di traiettorie tridimensionali e analisi dinamica di stormi nellâ&#x20AC;&#x2122;ambito del progetto StarFlag, Thesis (February 2009)

Flocking starlings in 3D Alessio Cimarelli

Collective behaviour in biology A NIMAL GROUPS M ODELS

S TARFLAG project W HAT

Right camera (A)

Next shot (A)

W HERE / W HEN / W HO H OW

Structural analysis M ETRIC VS . TOPOLOGY

D ENSITY GRADIENT

Velocity correlations Beyond starlings. . .

Speed vectors from the A camera point of view

Contents

Flocking starlings in 3D Alessio Cimarelli

Collective behaviour in biology Animal groups Models S TARFLAG project What Where / When / Who How Structural analysis Metric vs. topology Density gradient Velocity correlations Beyond starlings. . .

Collective behaviour in biology A NIMAL GROUPS M ODELS

S TARFLAG project W HAT W HERE / W HEN / W HO H OW

Structural analysis M ETRIC VS . TOPOLOGY

D ENSITY GRADIENT

Velocity correlations Beyond starlings. . .

Use the structure as proxy of the interaction Ballerini et al., Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, PNAS, vol. 105(4) (2008)

Anisotropy maps

Flocking starlings in 3D Alessio Cimarelli

Collective behaviour in biology A NIMAL GROUPS M ODELS

S TARFLAG project W HAT W HERE / W HEN / W HO H OW

First neighbor

Structural analysis M ETRIC VS . TOPOLOGY

D ENSITY GRADIENT

Structure is the foremost effect of interaction and, conversely, interaction is ciphered in the inter-individual spatial structure

Velocity correlations Beyond starlings. . .

Tenth neighbor

Quantifying the anisotropy using the neighbours projection operator: M(n) =

1 N

(n)

with ui

i=1

(n)

|ui ihui |

Flocking starlings in 3D Alessio Cimarelli

Collective behaviour in biology A NIMAL GROUPS M ODELS

S TARFLAG project W HAT

unit vectors.

W HERE / W HEN / W HO H OW

~ (n) relative to the smallest eigenvalue of The eigenvector W (n) M defines the direction of minimal density of the n-th nearest neighbour. ~ ·W ~ (n) ]2 with V ~ centre of mass ⇒ Anisotropy: γ(n) = [V velocity. I

Isotropic case: γ =

Flocks: γ(1) ∼ 0.72

1 3

Structural analysis M ETRIC VS . TOPOLOGY

D ENSITY GRADIENT

Velocity correlations Beyond starlings. . .

Topological range of interaction Ballerini et al., Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, PNAS, vol. 105(4) (2008)

Two ranges: I

nc is the topological range of interaction (units of birds), the average distance rc of the neighbour nc is given by 1/3 rc â&#x2C6;ź ( n%c ) â&#x2C6;ź r1 nc

Flocking starlings in 3D Alessio Cimarelli

Collective behaviour in biology

The strength of the interaction decays with the distance.

rc is the metric range of interaction (units of meters)

Topological and metric ranges can not be both constant, since the density % varies strongly. . .

A NIMAL GROUPS M ODELS

S TARFLAG project W HAT W HERE / W HEN / W HO H OW

Structural analysis M ETRIC VS . TOPOLOGY

D ENSITY GRADIENT

Velocity correlations Beyond starlings. . .

Flocking starlings in 3D Alessio Cimarelli

Collective behaviour in biology A NIMAL GROUPS M ODELS

S TARFLAG project W HAT W HERE / W HEN / W HO H OW

Structural analysis M ETRIC VS . TOPOLOGY

Interaction depends on the topological distance in bird flocks!

D ENSITY GRADIENT

Velocity correlations

hnc i = 7

Beyond starlings. . .

Internal density gradient Ballerini et al., Empirical investigation of starling flocks: a benchmark study in collective animal behaviour, Animal Behaviour, vol. 76 (2008)

Not all the flocks are homogeneous, so main density is not always well defined. . .

Flocking starlings in 3D Alessio Cimarelli

Collective behaviour in biology A NIMAL GROUPS M ODELS

S TARFLAG project

The average nearest neighbour distance grows from the edge to the center, the edge is denser than the core of flocks. I

Surface tension

Confusion effect

W HAT W HERE / W HEN / W HO H OW

Structural analysis M ETRIC VS . TOPOLOGY

D ENSITY GRADIENT

Velocity correlations Beyond starlings. . .

Flocking starlings in 3D

Conditional density Cavagna et al., New statistical tools for analyzing the structure of animal groups, Mathematical Biosciences, vol. 214 (2008)

Integrated conditional density:

Γ(r ) =

1 nc

Ni (r ) i=1 4/3πr 3

Pnc

Alessio Cimarelli

Collective behaviour in biology A NIMAL GROUPS M ODELS

S TARFLAG project W HAT W HERE / W HEN / W HO H OW

Structural analysis M ETRIC VS . TOPOLOGY

D ENSITY GRADIENT

Velocity correlations

Scale of homogeneity is equal to the average nearest neighbour distance.

There is no scale of homogeneity because of a strong density gradient.

Beyond starlings. . .

Contents

Flocking starlings in 3D Alessio Cimarelli

Collective behaviour in biology Animal groups Models S TARFLAG project What Where / When / Who How Structural analysis Metric vs. topology Density gradient Velocity correlations Beyond starlings. . .

Collective behaviour in biology A NIMAL GROUPS M ODELS

S TARFLAG project W HAT W HERE / W HEN / W HO H OW

Structural analysis M ETRIC VS . TOPOLOGY

D ENSITY GRADIENT

Velocity correlations Beyond starlings. . .

Using velocity correlations to investigate coordination and response

Flocking starlings in 3D Alessio Cimarelli

Collective behaviour in biology A NIMAL GROUPS

Correlation measures how the behavioural changes of one animal influences that of other animals across the group: I

interaction local (few individuals)

correlation effective perception range

Behavioural correlations are therefore ultimately responsibile for the group’s ability to respond collectively to its environment: correlation ←→ response Response, unlike order, is the real signature of self-organization.

M ODELS

S TARFLAG project W HAT W HERE / W HEN / W HO H OW

Structural analysis M ETRIC VS . TOPOLOGY

D ENSITY GRADIENT

Velocity correlations Beyond starlings. . .

Flocking starlings in 3D

A look at the velocity fields

Alessio Cimarelli

Collective behaviour in biology A NIMAL GROUPS M ODELS

S TARFLAG project W HAT W HERE / W HEN / W HO H OW

Structural analysis M ETRIC VS . TOPOLOGY

D ENSITY GRADIENT

Absolute values

~ CM = V

1 N

ϕ = | N1

~vi ∼ 11 m/s

~ vi | i |~ vi |

∼ 0.95

Fast polarized groups

Fluctuations

~ CM ⇒ δ~vi = ~vi − V

Velocity correlations

δ~vi = 0

Large correlated region

Beyond starlings. . .

Velocity-velocity correlation function Cavagna et al., Scale-free correlations in starling flocks, PNAS, vol. 107(26) (2010)

How a change in orientation of one individual influences the change in orientation of another individual at distance r :

C(r ) =

1 c0

δ~vi · δ~vj δ(r −rij ) P ij δ(r −rij )

Flocking starlings in 3D Alessio Cimarelli

Collective behaviour in biology A NIMAL GROUPS M ODELS

S TARFLAG project W HAT W HERE / W HEN / W HO H OW

Structural analysis

ξ = correlation length I

zero of the correlation function

extension of the correlated domains

much larger than the interaction range!

M ETRIC VS . TOPOLOGY

D ENSITY GRADIENT

Velocity correlations Beyond starlings. . .

Flocking starlings in 3D

Scale-free correlations

Alessio Cimarelli

Cavagna et al., Scale-free correlations in starling flocks, PNAS, vol. 107(26) (2010)

Collective behaviour in biology A NIMAL GROUPS

ξ scales with the flock’s size

M ODELS

S TARFLAG project

⇒ ξ(bL) = bξ(L) ⇒ correlations are scale-free

W HAT W HERE / W HEN / W HO H OW

Structural analysis M ETRIC VS . TOPOLOGY

C(r ) = ξ1γ g( ξr ) ξ(bL) = bξ(L)

⇒

C(r ; L) = bγ C(br ; bL) ⇒ C(r ; L) = r1γ f ( Lr )

C∞ (r ) ∼

1 rγ

D ENSITY GRADIENT

Velocity correlations Beyond starlings. . .

The asymptotic correlation in starling flocks is a scale-free power law. . .

Flocking starlings in 3D

Scale-free correlations

Alessio Cimarelli

Cavagna et al., Scale-free correlations in starling flocks, PNAS, vol. 107(26) (2010)

Collective behaviour in biology

. . . a very slow decaying power law!

A NIMAL GROUPS M ODELS

C(r ) =

S TARFLAG project

1 r ξ γ g( ξ )

r d d( ξr ) C( ξ )|r /ξ=1 1 1 0 ξ γ g (1) ∼ − ξ γ

W HAT W HERE / W HEN / W HO

= ∼

H OW

− L1γ

Structural analysis M ETRIC VS . TOPOLOGY

D ENSITY GRADIENT

No apparent flattening of the derivative for larger sizes. γ ∼ 0.2, but data are equally compatible with a log decay and even with a constant. . . The asymptotic correlation function is practically not decaying with the distance, flocks display very long-ranged scale-free correlations.

Velocity correlations Beyond starlings. . .

Speed correlations Cavagna et al., Scale-free correlations in starling flocks, PNAS, vol. 107(26) (2010)

Flocking starlings in 3D Alessio Cimarelli

Collective behaviour in biology A NIMAL GROUPS M ODELS

S TARFLAG project W HAT W HERE / W HEN / W HO H OW

Structural analysis M ETRIC VS . TOPOLOGY

Csp (r ) =

P ij ϕi ϕj δ(r −rij ) 1 P with ϕi = |~ vi | − h|~ v |i c0 ij δ(r −rij )

Also speed correlations are scale-free and long-range!

D ENSITY GRADIENT

Velocity correlations Beyond starlings. . .

Flocking starlings in 3D

Results Cavagna et al., Scale-free correlations in starling flocks, PNAS, vol. 107(26) (2010)

All modes are scale-free correlated: Orientation

Speed

⇒

continuous symmetry breaking Goldstone’s theorem transverse soft modes no symmetry explanation some kind of criticality may actually be present in this system. . .

Correlations are unusually long-range: interactions must be more complex than simple alignment. . .

Alessio Cimarelli

Collective behaviour in biology A NIMAL GROUPS M ODELS

S TARFLAG project W HAT W HERE / W HEN / W HO H OW

Structural analysis M ETRIC VS . TOPOLOGY

D ENSITY GRADIENT

The complete dynamical state of one individual is strongly correlated to that of all other individuals within the group.

Velocity correlations

Information can be transferred to all animals no matter their distance, enhancing the collective response of the group.

Beyond starlings. . .

But correlations are unusually long-range. . . maximal response? Flocks as critical systems?

Contents

Flocking starlings in 3D Alessio Cimarelli

Collective behaviour in biology Animal groups Models S TARFLAG project What Where / When / Who How Structural analysis Metric vs. topology Density gradient Velocity correlations Beyond starlings. . .

Collective behaviour in biology A NIMAL GROUPS M ODELS

S TARFLAG project W HAT W HERE / W HEN / W HO H OW

Structural analysis M ETRIC VS . TOPOLOGY

D ENSITY GRADIENT

Velocity correlations Beyond starlings. . .

Art Swarm on IIT

Flocking starlings in 3D Alessio Cimarelli

Collective behaviour in biology A NIMAL GROUPS M ODELS

S TARFLAG project W HAT

Now we are working to adapt our experimental setting to study flies swarm and try to compare previous results with a totally different species. . . from birds to insects, stay tuned! :)

W HERE / W HEN / W HO H OW

Structural analysis M ETRIC VS . TOPOLOGY

D ENSITY GRADIENT

Velocity correlations Beyond starlings. . .

Flocking starlings in 3D

Thanks to. . .

Alessio Cimarelli

The S TARFLAG group is at the SMC Centre ISC-CNR, Dept. of Physics, University of Rome “Sapienza”

Collective behaviour in biology A NIMAL GROUPS M ODELS

S TARFLAG project W HAT W HERE / W HEN / W HO H OW

Structural analysis M ETRIC VS . TOPOLOGY

http://www.smc.infm.it/biological-systems/ flocking.html Giorgio Parisi (coordinator of the EU project)

Andrea Cavagna (coordinator of the INFM-CNR node)

Irene Giardina (vice-coordinator of the INFM-CNR node)

D ENSITY GRADIENT

Velocity correlations Beyond starlings. . .

Flocking starlings in 3D Alessio Cimarelli

Collective behaviour in biology A NIMAL GROUPS M ODELS

S TARFLAG project W HAT W HERE / W HEN / W HO H OW

Structural analysis M ETRIC VS . TOPOLOGY

D ENSITY GRADIENT

Velocity correlations Beyond starlings. . .

Why a topological rather than a metric interaction?

Flocking starlings in 3D Alessio Cimarelli

Collective behaviour in biology A NIMAL GROUPS M ODELS

S TARFLAG project W HAT W HERE / W HEN / W HO H OW

Structural analysis M ETRIC VS . TOPOLOGY

D ENSITY GRADIENT

Velocity correlations

Topological flocks are more robust against external perturbations!

Beyond starlings. . .

Flocking starlings in 3D Alessio Cimarelli

Collective behaviour in biology

Why this range? Antipredatory optimum

A NIMAL GROUPS M ODELS

S TARFLAG project W HAT W HERE / W HEN / W HO H OW

Structural analysis M ETRIC VS .

Not noisy, but short ranged

Long ranged, but too noisy

Constraints on the cortical elaboration of the visual input In many species 6 â&#x2C6;&#x2019; 7 seems to be the upper threshold of subitizing (object tracking), we found nc = 7

TOPOLOGY

D ENSITY GRADIENT

Velocity correlations Beyond starlings. . .

Flocking starlings in 3D

Velocity fluctuations Cavagna et al., Scale-free correlations in starling flocks, PNAS, vol. 107(26) (2010)

Velocities {~v }

~ Fluctuations δ~v = ~v − V

Alessio Cimarelli

Collective behaviour in biology A NIMAL GROUPS

Random ϕ ∼ 0.9

M ODELS

S TARFLAG project W HAT W HERE / W HEN / W HO H OW

Structural analysis M ETRIC VS .

57-03 ϕ ∼ 0.99

TOPOLOGY

D ENSITY GRADIENT

Velocity correlations Beyond starlings. . .

It’s not necessary that a polarized vector field has fluctuations structured in large correlated regions.

Velocity fluctuations Cavagna et al., Scale-free correlations in starling flocks, PNAS, vol. 107(26) (2010)

Synthetic random vector fields generated with correlation length ξ and localized on real space distribution of birds show the link between the zero of the correlation function and the correlation length. " # ν λ a r G(r ) = G(0) exp − r +a ξ

Flocking starlings in 3D Alessio Cimarelli

Collective behaviour in biology A NIMAL GROUPS M ODELS

S TARFLAG project W HAT W HERE / W HEN / W HO H OW

Structural analysis M ETRIC VS . TOPOLOGY

D ENSITY GRADIENT

Velocity correlations Beyond starlings. . .