flocking starlings in 3d

Flocking starlingsin 3D

Alessio Cimarelli

Collectivebehaviour inbiologyANIMAL GROUPS

MODELS

STARFLAGprojectWHAT

WHERE / WHEN /WHO

StructuralanalysisMETRIC VS.TOPOLOGY

DENSITY GRADIENT

Velocitycorrelations

Beyondstarlings. . .

Flocking starlings in 3DThe STARFLAG project

Alessio CimarelliDept. of Physics, University of Rome “Sapienza”

“Swarm Intelligence and Critical Behavior”Center for Interdisciplinary Research

Bielefeld UniversityMarch 22, 2011

Alessio Cimarelli

MODELS

STARFLAGprojectWHAT

WHERE / WHEN /WHO

DENSITY GRADIENT

Contents

Collective behaviour in biologyAnimal groupsModels

STARFLAG projectWhatWhere / When / WhoHow

Structural analysisMetric vs. topologyDensity gradient

Velocity correlations

Beyond starlings. . .

Alessio Cimarelli

MODELS

STARFLAGprojectWHAT

WHERE / WHEN /WHO

DENSITY GRADIENT

Animal groups→ Self-organizationFrom local rules to collective motion

Alessio Cimarelli

MODELS

STARFLAGprojectWHAT

WHERE / WHEN /WHO

DENSITY GRADIENT

Animal groups→ Self-organizationFrom local rules to collective motion

I The group fulfils tasksbeyond the abilities of theindividuals

I Behavioral rules are selectedto achieve collectiveperformance

I What kind of interactiongrants such efficient groupbehavior?

Alessio Cimarelli

MODELS

STARFLAGprojectWHAT

WHERE / WHEN /WHO

DENSITY GRADIENT

ModelsAoki, 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 ofneighbors

I alignment of velocities: gowhere the other go

I attraction to neighbors: staywith the group

I short range repulsion: avoidcollisions

I constant velocity:self-propulsion

~di (t + 1) =1

nin∑j=1

wj~dj (t) +

nin∑j=1

~rij (t)|~rij (t)|

+ ~ηi (t)

We need experimental data to control and verify numericalmodels!

Alessio Cimarelli

MODELS

STARFLAGprojectWHAT

WHERE / WHEN /WHO

DENSITY GRADIENT

Contents

Alessio Cimarelli

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A case study: flocking

CohesionI robustness

againstperturbation

I large densityfluctuations

I no fragmentation/ few stragglers

Here it is ourhero, thestarling!

CoherenceI high coordinationI collective

responseI quick information

propagation

Alessio Cimarelli

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DENSITY GRADIENT

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

Experimental seasons: 2006, 2007, 2008.

Alessio Cimarelli

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Starlings in action. . .Piazza dei Cinquecento, Roma

Show video. . .

Alessio Cimarelli

MODELS

STARFLAGprojectWHAT

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DENSITY GRADIENT

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

I typical flocks distance:Z ∼ 100m

I photographic resolution:δs ∼ 1pixel

I required error on relativedistance: δZ < 0.1m

Ωδz∼ 23m

δs = xB − xA = fdZ

Alessio Cimarelli

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STARFLAGprojectWHAT

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

Alessio Cimarelli

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

Some previous 3Ddata:

I Cullen et al., 1965:10 fishes

I Major & Dill, 1978:45 birds

I Partridge et al.,1980: 30 fishes

I Pomeroy &Heppner, 1992: 16birds

Us: ∼ 4000 birds

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

Left camera (C) Right camera (A)

3D positions from the A camera point of view

Alessio Cimarelli

MODELS

STARFLAGprojectWHAT

WHERE / WHEN /WHO

DENSITY GRADIENT

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

I Mean speed: ∼ 10 m/sI Camera time resolution:

I Average nearestneighbour distance: ∼ 1 m

I 3D reconstructionefficiency: ∼ 90%

1. Two-times matching isvery efficient

2. Multi-time matching:requires optimization

I missing birdsI high densityI low temporal

resolution

Alessio Cimarelli

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From static shots to vectorial fieldStefanini F., Ricostruzione di traiettorie tridimensionali e analisi dinamica di stormi nell’ambito del progetto StarFlag, Thesis(February 2009)

Right camera (A) Next shot (A)

Speed vectors from the A camera point of view

Alessio Cimarelli

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STARFLAGprojectWHAT

WHERE / WHEN /WHO

DENSITY GRADIENT

Contents

Alessio Cimarelli

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

Structure is the foremosteffect of interaction and,conversely, interaction is

ciphered in theinter-individual spatial

structure

Anisotropy maps

First neighbor

Tenth neighbor

Alessio Cimarelli

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WHERE / WHEN /WHO

DENSITY GRADIENT

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

Quantifying the anisotropy using the neighbours projectionoperator:

M(n) = 1N

∑Ni=1 |u

(n)i 〉〈u

(n)i |

with u(n)i unit vectors.

The eigenvector ~W (n) relative to the smallest eigenvalue ofM(n) defines the direction of minimal density of the n-thnearest neighbour.

⇒ Anisotropy: γ(n) = [~V · ~W (n)]2 with ~V centre of massvelocity.

I Isotropic case: γ = 13

I Flocks: γ(1) ∼ 0.72

Alessio Cimarelli

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

Two ranges:I nc is the topological

range of interaction(units of birds), theaverage distance rc ofthe neighbour nc isgiven byrc ∼ ( nc

% ) ∼ r1n1/3c

I rc is the metric rangeof interaction (units ofmeters)

The strength of theinteraction decays with thedistance.

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

Alessio Cimarelli

MODELS

STARFLAGprojectWHAT

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DENSITY GRADIENT

Interaction depends on the topological distance in bird flocks!

〈nc〉 = 7

Alessio Cimarelli

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DENSITY GRADIENT

Internal density gradientBallerini 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 notalways well defined. . .

The average nearestneighbour distancegrows from the edgeto the center, theedge is denser thanthe core of flocks.

I Surface tensionI Confusion effect

Alessio Cimarelli

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STARFLAGprojectWHAT

WHERE / WHEN /WHO

DENSITY GRADIENT

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

Integrated conditional density: Γ(r) = 1nc

∑nci=1

Ni (r)4/3πr3

Scale of homogeneity is equalto the average nearest

neighbour distance.

There is no scale ofhomogeneity because of a

strong density gradient.

Alessio Cimarelli

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STARFLAGprojectWHAT

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DENSITY GRADIENT

Contents

Alessio Cimarelli

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DENSITY GRADIENT

Using velocity correlations to investigatecoordination and response

Correlation measures how the behavioural changes of oneanimal influences that of other animals across the group:

I interaction local (few individuals)I correlation effective perception range

Behavioural correlations are therefore ultimately responsibilefor the group’s ability to respond collectively to its environment:correlation←→ response

Response, unlike order, is the real signature ofself-organization.

Alessio Cimarelli

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DENSITY GRADIENT

A look at the velocity fields

Absolute values Fluctuations

~VCM = 1N

∑i ~vi ∼ 11 m/s

ϕ = | 1N∑

i~vi|~vi || ∼ 0.95

Fast polarized groups

δ~vi = ~vi − ~VCM ⇒∑

i δ~vi = 0

Large correlated region

Alessio Cimarelli

MODELS

STARFLAGprojectWHAT

WHERE / WHEN /WHO

DENSITY GRADIENT

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

How a change in orientationof one individual influencesthe change in orientation ofanother individual atdistance r :

C(r) = 1c0

∑ij δ~vi · δ~vj δ(r−rij )∑

ij δ(r−rij )

ξ = correlation lengthI zero of the correlation

functionI extension of the

correlated domainsI much larger than the

interaction range!

Alessio Cimarelli

MODELS

STARFLAGprojectWHAT

WHERE / WHEN /WHO

DENSITY GRADIENT

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

ξ scales with the flock’s size

⇒ ξ(bL) = bξ(L)⇒ correlations arescale-free

C(r) = 1ξγ g( r

ξ(bL) = bξ(L)⇒ C(r ; L) = bγC(br ; bL)

C(r ; L) = 1rγ f ( r

L )⇒ C∞(r) ∼ 1

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

Alessio Cimarelli

MODELS

STARFLAGprojectWHAT

WHERE / WHEN /WHO

DENSITY GRADIENT

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

. . . a very slow decaying power law!

C(r) = 1ξγ g( r

ξ )C( r

ξ )|r/ξ=1 =

1ξγ g′(1) ∼ − 1

ξγ ∼ −1

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

Alessio Cimarelli

MODELS

STARFLAGprojectWHAT

WHERE / WHEN /WHO

DENSITY GRADIENT

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

Csp(r) = 1c0

∑ij ϕiϕjδ(r−rij )∑

ij δ(r−rij )with ϕi = |~vi | − 〈|~v |〉

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

Alessio Cimarelli

MODELS

STARFLAGprojectWHAT

WHERE / WHEN /WHO

DENSITY GRADIENT

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

All modes are scale-free correlated:

Orientation ⇒ continuous symmetry breakingGoldstone’s theoremtransverse soft modes

Speed ⇒ no symmetry explanationsome kind of criticality may actuallybe present in this system. . .

Correlations are unusually long-range: interactions must bemore complex than simple alignment. . .The complete dynamical state of one individual is stronglycorrelated to that of all other individuals within the group.Information can be transferred to all animals no matter theirdistance, enhancing the collective response of the group.But correlations are unusually long-range. . . maximalresponse? Flocks as critical systems?

Alessio Cimarelli

MODELS

STARFLAGprojectWHAT

WHERE / WHEN /WHO

DENSITY GRADIENT

Contents

Alessio Cimarelli

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DENSITY GRADIENT

Art Swarm on IIT

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

Alessio Cimarelli

MODELS

STARFLAGprojectWHAT

WHERE / WHEN /WHO

DENSITY GRADIENT

Thanks to. . .

The STARFLAG group is at the SMC Centre ISC-CNR, Dept. ofPhysics, University of Rome “Sapienza”

http://www.smc.infm.it/biological-systems/flocking.html

Giorgio Parisi(coordinator of theEU project)

Andrea Cavagna(coordinator of theINFM-CNR node)

Irene Giardina(vice-coordinatorof the INFM-CNRnode)

Alessio Cimarelli

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STARFLAGprojectWHAT

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DENSITY GRADIENT

Alessio Cimarelli

MODELS

STARFLAGprojectWHAT

WHERE / WHEN /WHO

DENSITY GRADIENT

Why a topological rather than a metric interaction?

Topological flocks are morerobust against externalperturbations!

Alessio Cimarelli

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Why this range?

Antipredatory optimum

Not noisy, but short ranged Long ranged, but too noisy

Constraints on the cortical elaboration of the visual input

In many species 6− 7seems to be the upperthreshold of subitizing(object tracking), we foundnc = 7

Alessio Cimarelli

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STARFLAGprojectWHAT

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DENSITY GRADIENT

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

Velocities ~v Fluctuations δ~v = ~v − ~V

Randomϕ ∼ 0.9

57-03ϕ ∼ 0.99

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

Alessio Cimarelli

MODELS

STARFLAGprojectWHAT

WHERE / WHEN /WHO

DENSITY GRADIENT

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

Synthetic random vector fields generated with correlationlength ξ and localized on real space distribution of birds showthe link between the zero of the correlation function and thecorrelation length.

G(r) = G(0)

)νexp

flocking starlings in 3d

contentscollective behaviour

collective motioni

groupi short range repulsion

critical behaviorcenter

case study

efficient groupbehavior

kind of interactiongrants

gregoire chate

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