From individual to collective
information processing in fish schools

Guy Theraulaz
Centre de Recherches sur la Cognition Animale
CNRS, UMR 5169, Toulouse, France

Crowd Dynamics
International Conference onMathematical Modeling and Applications
Meiji University, Tokyo, January 11, 2015

© Alex Mustard

From individual to collective
information processing in fish schools
 To coordinate their actions fish
have to exchange some
information and perform somespecific actions (controllinglinear and turning velocities) in
response to this information

Scomber scombrus© Christopher Swann1

Collective motion in biological systems© Eshel Ben Jacob

Paenibacillus vortex

Paenibacillus vortex

Sphyraena barracuda

Sphyraena barracuda

Collective motion in bird flocks

Sturnus vulgaris ( © C. Carrere )Ballerini, M. et al., PNAS (2008); Cavagna, A. et al., PNAS (2010); Bialek, W. et al., PNAS (2012)2

Collective motion in fish schools
Océans, (2010)
Directed byJacques Perrin,
Pathé Distribution

Sphyraena barracuda

Collective motion in fish schools
Schooling manoeuvres in response to predator attacks

3Collective motion in fish schools
Collective patterns arising from interactions between fish

What are the interactions rules and behavioral mechanisms
involved in the coordination of collective motion?

Determining interactions rules
What kind of information is collected?

4Determining interactions rules
What kind of information is collected?

Determining interactions rules
What kind of information is collected?

ii
jposition
velocityheading?

5Determining interactions rules
What are the neighboring partners involved in the interaction?

Determining interactions rules
What are the neighboring partners involved in the interaction?

6

Determining interactions rules
What are the neighboring partners involved in the interaction?

Determining interactions rules
What are the neighboring partners involved in the interaction?

7

Determining interactions rules
How is the information combined?

iCavagna, A. et al., Proc. R. Soc. B (2013)

A methodology to build reliable models
Individual level Quantification of behaviours atdifferent scales and in groupsof different sizes Connecting micro-level tomacro-level: how do complex
patterns emerge out of local
interactions among individuals?

Trajectory analysis

 Trajectory analysis is used tobuild a model of spontaneous
movement
Weitz, S. et al., Plos One (2012)
Gautrais, J. et al., Plos ComputationalBiology (2012)8A methodology to build reliable models
Pair interactions
 Interactions between pairs ofindividuals are characterized Computational models aredeveloped to test assumptions
about the combination of
information sources at theindividual level and theresulting behavioural decision

S1

Behavioral
response The already-determinedparameters are kept
unchanged and the stimulusresponse function and thecorresponding parameters aredetermined from data

S2

Response functions

Weitz, S. et al., Plos One (2012)
Gautrais, J. et al., Plos ComputationalBiology (2012)

S3

ComputationalmodelstimulusBehavioral
response

A methodology to build reliable models
Collective level: combining manyinteractions
 To how many neighbors do
individuals respond, and what is
the relative weight given to each?

 Quantitative predictions: is themodel able to accurately predictthe collective dynamics and theoutcome of novel situations?

Dynamics of interaction network

 The incremental analysis allows abetter understanding of the role
played by each behavioral
component in the observed
collective pattern

Weitz, S. et al., Plos One (2012)
Gautrais, J. et al., Plos ComputationalBiology (2012)9Empirical investigation of fish schooling
Study siteSaint-Leu

La Réunion

Coral Farm

Empirical investigation of fish schooling
Study species: the barred flagtail

10 cm

Khulia mugil

10Empirical investigation of fish schoolingExperimental setup and video tracking
 Single fish and groups ofincreasing size (N = 2, 5,10, 15, 30) have beenrecorded in a 4-m wide and1-m depth tank

5 Fish4m

 Two minutes wereextracted from digital video
recordings and the positionof the individual's head wastracked every 1/12 s (1440
data-points per trajectory)

30 FishEmpirical investigation of fish schoolingExperimental setup and video tracking

Group polarizationbAlignment

5 Fish30 FishTime (s)11Modeling single fish behavior and wall avoidanceSpontaneous fish movement
 The swimming speed ofeach fish is constantS (t)Curvilinear distance

Swimming speed(m/s)

Time (s)Gautrais, J. et al., J. Math. Biol., (2009)

Modeling single fish behavior and wall avoidanceSpontaneous fish movement
 The swimming speed ofeach fish is constant The observed motion
results from a control of theturning speed
 There is a significant
autocorrelation of theturning speed

Swimming speed(m/s)

Turning speed(rad/s) There is a spontaneoustendency of the fish to
randomly change theturning speed

Gautrais, J. et al., J. Math. Biol., (2009)12

Modeling single fish behavior and wall avoidance
A persistent turning walker model of fish movement

Spontaneous swimming: characteristic time of the (exponentially decaying)autocorrelation function ofWall
Gautrais, J. et al., J. Math. Biol., (2009)

Modeling single fish behavior and wall avoidance
A persistent turning walker model of fish movement

Spontaneous swimming

1m

Wall avoidance
distance before collision

WallGautrais, J. et al., J. Math. Biol., (2009)

13Modeling single fish behavior and wall avoidance
A persistent turning walker model of fish movement
Data

Model

Mean squaredisplacement ( m 2 s -1 )Data

Wall avoidance

ModelTime (s)Gautrais, J. et al., J. Math. Biol., (2009)

Empirical investigation of fish schooling
Deciphering interactions between pairs of fish

Tracking the movement
patterns in groups of 2 fish

Mean speed

Fish n°2
Fish n°1

E1

E2

E3E4E5E6Experiments

Gautrais, J. et al., Plos Computational Biology (2012)14

Empirical investigation of fish schooling
Influence of swimming speed on coordination and alignment_Polarization

 Synchronization increases
with swimming speed

Speed Directional information is arequired component toachieve synchronization
 Positional information is
likely to play a key role inthe cohesion of the school+Time

Gautrais, J. et al., Plos Computational Biology (2012)

Empirical investigation of fish schooling
Influence of swimming speed on coordination and alignment

Proportion of time
Spent schooling The degree ofsynchronization increases
as a result of an increaseof swimming speed Directional information is arequired component toachieve synchronization
LightDark

 Positional information is
likely to play a key role inthe cohesion of the school

Astyanax mexicanus
Kowalko, J.E. et al., Current Biology (2013)15

Modeling pair interactions
Characterizing interactions between pairs of fish in the model Positional effect (attraction): turn towards the neighbor
Short inter-distanceFish jFish iGautrais, J. et al., Plos Computational Biology (2012)

Modeling pair interactions
Characterizing interactions between pairs of fish in the model Positional effect (attraction): turn towards the neighbor
Large inter-distance

Gautrais, J. et al., Plos Computational Biology (2012)16

Modeling pair interactions
Characterizing interactions between pairs of fish in the model Directional information (alignment): turn to align with the neighbor
Low swimming speed

Gautrais, J. et al., Plos Computational Biology (2012)

Modeling pair interactions
Characterizing interactions between pairs of fish in the model Directional information (alignment): turn to align with the neighborHigh swimming speed

Gautrais, J. et al., Plos Computational Biology (2012)17

Data-driven fish school model
Modeling pair interactions
Parameter estimation

WalleffectPositional
effectDirectional
effectGautrais, J. et al., Plos Computational Biology (2012)

Data-driven fish school model
Modeling pair interactions
DataModel
2 FishWalleffectPositional
effectMean interindividual distance

Polarization

Data
Model

Directional
effect0.024 m0.94

28.9 m-1 s-1/2
0.41 m-1 s-1

Speed
Speed
2.7 m-1
Gautrais, J. et al., Plos Computational Biology (2012)18Data-driven fish school model
Determining the number and position of influential neighbors
Metric interactions
 To how many neighbors do
individuals respond, andwhat is the relative weight
given to each?

rc

 In starling flocks each birdinteracts with a fixed
number of neighbors (6-7)
irrespective of theirdistance

Ballerini, M. et al., PNAS (2008)
Raynaud, F. 3D Collective Behavior Models.
Ph.D. thesis, Université Paris 7 (2009)

Data-driven fish school model
Determining the number and position of influential neighbors
Topological interactions
 To how many neighbors do
individuals respond, andwhat is the relative weight
given to each?
 In starling flocks each birdinteracts with a fixed
number of neighbors (6-7)
irrespective of theirdistance

Ballerini, M. et al., PNAS (2008)
Raynaud, F. 3D Collective Behavior Models.
Ph.D. thesis, Université Paris 7 (2009)19

Data-driven fish school model
Determining the number and position of influential neighbors
 K Nearest Neighbors?
KNN = 7

Data-driven fish school model
Determining the number and position of influential neighbors
 Approximating "visible" neighbors with a greedy triangulation

Gautrais, J. et al., Plos Computational Biology (2012)20

Data-driven fish school model
Weighting all influential neighbors
DataModel Simplest assumption:
averaging the influence ofall "visible" neighbors

5 FishMean interindividual distance

Polarization

Data
Model

0.024 m28.9 m-1 s-1/2
0.41 m-1 s-1

0.94

Speed
Speed
2.7 m-1
Gautrais, J. et al., Plos Computational Biology (2012)

Data-driven fish school model
Comparison between model predictions and experimental data
Polarization

Mean interindividual distanceData

2 FishModel
No interaction

5 Fish10 Fish15 Fish30 Fish21

Data-driven fish school model
Quantification of group-size effectGregarious locusts

Serotoninlevel

Solitarious locustGroup size

Anstey et coll.,Science (2009)

Data-driven fish school model
Comparison between model predictions and experimental data
Polarization

Mean interindividual distanceData

2 FishModel
No interaction

5 Fish10 Fish15 Fish30 Fish22

Data-driven fish school model
Comparison between model predictions and experimental data
Data

Model
30 FishPolarization

Mean interindividual distance
Data
Model
No interaction

Gautrais, J. et al., PLoS Computational Biology (2012)

Model properties
Speed-induced transition to schooling
100 Fish What are the the model
predictions for larger groups inunbounded space?

Speed = 0.2 m/s

Swarming

Polarization

Speed = 0.9 m/s

Schooling
Speed
Calovi, D. et al., New Journal of Physics (2014)23

Model properties
Speed-induced transition to schoolingGroup polarization
increases with velocity

 What are the the model
predictions for large groups inunbounded space?

Speed

Polarization

Time

Speed

Empirical investigation of fish schooling
Neighbors’ influence depends on their position
Neighbor ahead

Attraction

Alignment

Neighbor behind24

Empirical investigation of fish schooling
Neighbors’ influence depends on their position
 In a confined environment
the influence of the spatial
modulation of interactions
on collective motion
patterns cannot bedetected (reaction to thewall and to the density offish)

Attraction

Alignment

Neighborahead

NeighborbehindModel properties
Angular weighting of fish interactionsji

ji

Calovi, D. et al., New Journal of Physics (2014)25

Exploring the parameter space
Parameter space effect on polarization (schooling) Polarisation (Schooling) : Orderparameter that tends to unity as fish
are aligned, and zero when swarming

Alignment ( β )Polarization

Attraction ( γ ) Transition to schooling is
independent of the attraction

Attraction ( γ ) Transition to schoolingdepends on the attraction= 0.8 m/s
(N = 100 fish)

Exploring the parameter space
Parameter space effect on angular momentum (Milling) Angular momentum (Milling) : Orderparameter that tends to unity as fish
are swimming in a perfect circle

Alignment ( β )Milling

Attraction ( γ ) No milling region detected

Attraction ( γ ) A large milling region isobserved= 0.8 m/s
(N = 100 fish)26

Exploring the parameter space
Region I: Schooling

PolarizationMilling
Polarization
Nearestneighbor distance

Alignment ( β )I

Polarization & Milling

MillingTime ( s)Attraction ( γ )Calovi, D. et al., New Journal of Physics (2014)

Exploring the parameter space
Region II: Milling

Polarization

Milling
Polarization

Alignment ( β )Nearestneighbor distance

Polarization & Milling

MillingTime ( s)II

Attraction ( γ )Calovi, D. et al., New Journal of Physics (2014)27

Exploring the parameter space
Region III: Winding zone (line configuration)
Milling

Polarization

Polarization

Alignment ( β )Nearestneighbor distance

Polarization & Milling

MillingTime ( s)IIIAttraction ( γ )Calovi, D. et al., New Journal of Physics (2014)

Exploring the parameter space
Region III: Winding zone (line configuration)
Milling
Polarization

Alignment ( β )Nearestneighbor distance

School of Atlantic herring(Clupea harengus)© P. Brehmer, IRD

III
Attraction ( γ )Calovi, D. et al., New Journal of Physics (2014)28

Exploring the parameter space
Region I-II: Transition zone

Polarization

Milling
Polarization

Alignment ( β )Nearestneighbor distance

Polarization & Milling

MillingTime ( s)I-IIAttraction ( γ )Tunstrøm, K. et al., PLoS Computational Biology (2013)

Model properties

Alignment ( β )Enhanced collective response to perturbationsin the transition zone

Attraction ( γ )Calovi, D. et al., Journal of the Royal Society Interface (2015)29

Model properties
Enhanced collective response to perturbationsin the transition zone close to criticality
N = 100fish The
maximum
4.102 simulation runs
responseof1039
the s
Average overschool toperturbations isreached In thetransition zone
betweenschooling andmilling

Alignment ( β )II-IIIIIII
Attraction ( γ )Difference in averagepolarization
Calovi, D. et al., Journal of the Royal Society Interface (2015)

Emergent coordination in fish schools
Wrap-up: main findings
 An incremental methodology
was used to build a fish
behavior model completely
based on interactions with thephysical environment andneighboring fish There is a continuous balancing
between attraction andalignment behavior as afunction of the distance
between fish

Khulia mugil

30

Emergent coordination in fish schools
Conclusions
 The modulation of strength of thealignment and attraction behaviors
plays a key role in the kind ofcollective motion pattern thatemerge at the school level By providing a high responsivenessto perturbations the transition region
between milling and schooling is ahighly desired state that optimizes
the ability of the fish to react
collectively (e.g. to a predatorattack) thus increasing the survivalof fish

© Octavio Aburto© Octavio Aburto
Caranx sexfasciatum

Acknowledgements

J. Gautrais

D.S. Calovi

M. Soria

H. ChatéF. Ginelli

S. Blanco

R. FournierCentre de Recherches sur la Cognition
Animale, CNRS, Université Paul Sabatier
Toulouse, France
UMR EME, Institut de Recherche
pour le Développement (IRD)
La Réunion, France
Service de Physique de l’Etat
Condensé, CEA
Gif-sur-Yvette, France
Laboratoire Plasma et Conversion
d’Energie, CNRS, Université Paul
Sabatier, Toulouse, France31