flocking starlings in 3d

Flocking starlingsin 3D

Alessio Cimarelli

Collectivebehaviour inbiologyANIMAL GROUPS

MODELS

STARFLAGprojectWHAT

WHERE / WHEN /WHO

HOW

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


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STARFLAGprojectWHAT

WHERE / WHEN /WHO

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DENSITY GRADIENT



Contents

Collective behaviour in biologyAnimal groupsModels

STARFLAG projectWhatWhere / When / WhoHow

Structural analysisMetric vs. topologyDensity gradient

Velocity correlations

Beyond starlings. . .


Alessio Cimarelli


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Animal groups→ Self-organizationFrom local rules to collective motion


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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


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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

nin∑j=1

wj~dj (t) +

1nin

nin∑j=1

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

+ ~ηi (t)

We need experimental data to control and verify numericalmodels!


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Contents







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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


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

Experimental seasons: 2006, 2007, 2008.


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Starlings in action. . .Piazza dei Cinquecento, Roma

Show video. . .


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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

d >z2

Ωδz∼ 23m

δs = xB − xA = fdZ


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


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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


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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:

110 s

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


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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


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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


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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)

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


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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. . .


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Interaction depends on the topological distance in bird flocks!

〈nc〉 = 7


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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


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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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Contents







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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.


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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


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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


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STARFLAGprojectWHAT

WHERE / WHEN /WHO

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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 )

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

C(r ; L) = 1rγ f ( r

L )⇒ C∞(r) ∼ 1

rγ

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


Alessio Cimarelli


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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 )

dd( r

ξ )C( r

ξ )|r/ξ=1 =

1ξγ g′(1) ∼ − 1

ξγ ∼ −1

Lγ

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


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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


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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


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Contents







Alessio Cimarelli


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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


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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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Alessio Cimarelli


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Why a topological rather than a metric interaction?

Topological flocks are morerobust against externalperturbations!


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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


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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


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STARFLAGprojectWHAT

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HOW


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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)

(a

r + a

)νexp

[−(

rξ

)λ]

flocking starlings in 3d

Documents

contentscollective behaviour

collective motioni

groupi short range repulsion

critical behaviorcenter

case study

efficient groupbehavior

kind of interactiongrants

gregoire chate