BCCS_seminar_Dec2014

Stochastic modelling of zebrafish locomotion :
collective motion from the bottom-up
Adam Zienkiewicz1 • David A. W. Barton1 • Maurizio Porfiri2 • Mario di Bernardo1,3
1Department of Engineering Mathematics, University of Bristol, UK
2Department of Mechanical and Aerospace Engineering,
New York University Polytechnic School of Engineering, USA
3Department of Information and Computer Engineering,
University of Naples Federico II, Italy
Adam Zienkiewicz Stochastic modelling of zebrafish locomotion
Complexity (BCCS) Seminar – 16 December 2014

Overview
(1 / 25)
Background and motivation Why study animal motion? Why (zebra)fish?
Individual model
…motion of (isolated) zebrafish
Multi-agent modeling
…modeling schools and collective motion
Emergence of leadership in groups
…towards methods of control
• requirements of a new model
• experiments, observations
• modeling and results
• observations and experiments
• model extension & results
• macro-level dynamics
Data-driven stochastic modeling of
zebrafish locomotion.
Zienkiewicz et al. , J. Math. Biology (2014)

Introduction
• Recent stochastic models of fish locomotion
- “Persistent Turning Walker” (PTW) model : Kulia mugil
displacement described in terms of turning speed and its autocorrelation
- later extended to multiple fish, modelling collective behaviour (same authors)
...motivation and background
Gautrais et al. (2009) J. Math. Biol.
58(3) and (2012) PLoS Comp. Biol. 8(9)
(2 / 25)
Berdahl et al. (2013). Science, 339(6119)
• Speed regulation as a primary response mechanism
- decomposition of interaction forces within groups of small, shoaling fish (golden shiners)
Katz et al. and Herbert-Read et al. (2011) PNAS 108(46)
- emergent sensing (distributed / decentralised) of environment via speed regulation
Berdahl et al. (2013) Science, 339(6119)
Katz et al. (2011) PNAS 108(46)

Model species ...zebrafish
Zebrafish (Danio rerio) - Increasingly predominant species for neurobiological,
developmental and behavioural laboratory studies
small ● fast reproductive cycle ● genetic homology
high stocking density ● strong social groups
≈ 3 cm
© Azul 2005
cf. Kulia Mugil ≈ 20-25 cm
No suitable model! - Can similar stochastic PTW type model be used?
(3 / 25)

Introduction ...objectives
Using data collected from video tracking of individual trajectories,
develop a data-driven model framework describing the motion
characteristics of individual zebrafish
• Adapt Gautrais’ PTW (constant speed) model of fish locomotion to capture
salient swimming features of small, shoaling fish with burst-and-coast
swimming mode
- augment stochastic differential model with additional dynamic speed process
- identify sufficient key metrics required to describe fish locomotion
- allow for and characterise interactions with environment (wall-avoidance)
(4 / 25)
• Model framework features to provide foundation for extension to multiple
fish dynamics : biologically realistic models of (zebrafish) collective motion
- speed regulation : dynamic speed interaction models
vs. (canonical) constant speed

Experiments
Zebrafish trajectory data -
...data capture : automated tracking
• 10 isolated individuals observed for 5 min. Each
- shallow (10 cm), square tank (120 x 120 x20 cm)
• overhead video capture - 30 fps  5 fps (Hz)
• automated visual tracking of centroid
(point) position of fish in tank
(Kalman filter)
(rate of change of heading angle of vt )
VIDEO: automated position
tracking of live zebrafish
Automated fish (particle) tracking
• compute velocity
• speed
• turning speed
Dynamical Systems Laboratory (NYU)
(5 / 25)

Observations ...zebrafish trajectories
Raw trajectory data: 10 zebrafish, 5 min observations Speed (ut)
Individual zebrafish exhibit a variety of locomotory patterns:
- smooth, fluid turning
- erratic (stop / start) with sharp or spiralling turning
- wall following (thigmotaxis)
BL.s-1
(6 / 25)

Observations ...segment trajectories
Swimming segment trajectories (coloured by parent fish ID F1...F10)
‘Swimming’ data isolated from individual zebrafish : 28 segments of equal duration
(7 / 25)

Trajectory analysis ...primary characteristics
Time series, distribution and autocorrelation (segment S9)
Time lag (s)
ut and ωt characterised by their distributions and autocorrelation (ACF)
- approximate with normal distributions and
Typically sharper
than Gaussian
(8 / 25)

Trajectory analysis ...primary characteristics
Speed / Turning speed cross correlation (log-frequency : all segments)
Strong correlation within ut and ωt joint distribution across all segments
(9 / 25)

Individual Model ...stochastic differential model
Capture salient characteristics of trajectory data using mean-reverting
stochastic processes : coupled stochastic differential equations
Wiener processes: dWt and dZt
Speed:
Ornstein-Uhlenbeck (type) processes:
Couple equations
with function:
Wall interactions
via bias function:
• mean speed: , mean turning speed:
• exp. decaying ACF with rates:
• process volatilities: and
6 variable parameters :
estimated from each
swimming segment
Turning-speed:
(10 / 25)

Individual Model (2) ...wall avoidance and SDE coupling
Boundary effect on turning speed (S27)Quantify effect of wall interactions
on trajectories:
- induced fwd. acceleration : inconclusive
- induced turning : trajectories bent away from
wall dependent on projected collision angle
Define a coupling function to restrict the volatility of Ωt process
as a function of Ut
• fc → σ0 as Ut → 0 (upper bounded)
• fc → 0 as Ut → ∞ (lower bounded)
• fc → σω /2 as Ut → μu (σω estimated from data)
estimate constants (A,B σ0 )
(11 / 25)

Results ...segment-wise calibration and simulation
Qualitative comparisons : S13 vs. RW13
Speed Turning speed
Dist.ACF
Calibrate SDE parameters [μ,σ,ϑ]u,ω from experimental speed / turning speed data
max. likelihood est.
assume standard
(Gaussian) O-U processes
dU, dΩ numerically integrated
(Euler-Maruyama)
 generate random walks
Trajectories : S13 vs. RW
(12 / 25)

Results ...process coupling
Coupling function fc recovers joint distribution of Ut and Ωt
Speed / Turning speed cross correlation (composite of all segments )
experimental data simulated random walkers
...upper volatility bound σ0 fixed, μu and σω vary between RW segments
(13 / 25)

Results ...segment simulation
Example simulation (2x real-time)
(14 / 25)

Results ...calibrating individual fish
Trajectory comparison (individual fish) insufficient swimming data for F4 & F8
Average segment parameters used for calibration of (8) individuals
(15 / 25)

Modelling a shoal ... multi-agent models of collective behaviour
(16 / 25)
Pairs of zebrafish swimming in circular shallow tank
(45 cm radius, 10cm water depth)
Experiments:
• 18 observations of unique zebrafish pairs
• 20 min observations (30Hz sample freq.)
• automated tracking + manual repair
• samples proximal (< 2 BL) to walls omitted
pair
What kind of behaviour (rules) can we infer from observations of fish swimming together?
Attraction? Repulsion? Alignment?
Analysis of trajectory data can reveal
pairwise interactions in terms of
‘social forces’ (accelerations)
Jun – Aug 2014
Hold up! …start with two fish
(6 hrs)

Inferring interaction behaviour ... social forces
(17 / 25)
Katz et al. (2011) PNAS 108(46)Golden shiners (14 x 56 min @ 30Hz)
Zebrafish (2 x 20 min @ 30Hz)
5 cm
(juvenile)
3 cm
(adult)
Zienkiewicz et al. (incomplete data)

Inferring interaction behaviour ... social forces
(18 / 25)
Golden shiners (14 x 56 min @ 30Hz)
Zebrafish (2 x 20 min @ 30Hz)
5 cm
(juvenile)
3 cm
(adult)
Golden shiners
Zebrafish
Alignment :
an emergent phenomena?
Data suggests turning is dependent
only on relative position
….not orientation
 no explicit alignment ‘rule’

Interaction model ...toy model of speed / turning forces
(19 / 25)
Interpolate forces in primary response direction: Model potential:
Angular forceTangential (speed) force

Interaction model (2) ...conclusions
(20 / 25)
Speed:
Turning speed:
Modified SDEs:
Acceleration (force) due to pair-wise interactions:
(Similarly for angular acc. )
Interaction network adjacency matrix:
Voronoi neighbourhood
(Voronoi neighbourhood, radial proximity
networks, estimated visual networks)

Multi-agent simulations ...2-fish example
(21 / 25)

Global observables ...2-fish example
(22 / 25)
Polarisation (P)
Milling (M)
Cohesion (P)
Mean nearest-
neighbour
distance (MNND)
Live zebrafish Simulation
2 fish, 20 mins @ 30fps 2 agents, 20 mins @ 30 Hz
A set of measures / order parameters which describe the global, or macro-scale dynamics
Relative alignment
Rotation around a
common centre of mass
Rotation around a
common centre of mass

Multi-agent simulations ... 1000 fish example !!
(23 / 25)

Leadership and collective decision making
(24 / 25)
How can collective dynamics
be modulated by the
presence of a subset of
Informed individuals?
Emergent leadership in the
absence of explicit signals....
• zero turning (translation)
• constant turning (circular)
• ‘blind’ agent
• agent(s) with preferential
heading direction
Practical examples:
foraging, migration,
danger awareness
...artificial control ?

Conclusions
• direct calibration from experimental data, inc. boundary avoidance
- produce simulated trajectories with comparable curvature
- capture ‘passive’ wall following behaviour with ϑω /σω dependence
• describe zebrafish locomotion with an extended PTW model
- characterised by autoregressive, stochastic processes for
both speed and turning speed
- stochastic speed process more suitable for burst-and-coast
swimming mode of small, schooling fish
• model framework allows explicit inclusion of both speed and
turning speed modulation as responses to dynamic environment
- equilibrium bias of both speed and turning speed can be evolved
to simulate linear accelerations and torques (independently)
- infer interaction ‘forces’ to model group behaviour
(25 / 25)

Acknowledgements
My supervisors:
Mario di Bernardo
David Barton
Maurizio Porfiri
My sponsors (U.K.):
Dynamical Systems Laboratory:
(New York University Polytechnic School of Engineering)
Sachit Butail
Fabrizio Ladu
...thank you for listening

BCCS_seminar_Dec2014

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