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Inferring the structure and dynamics of interactions in schooling fish
Open accessAbstract
Determining individual-level interactions that govern highly coordinated motion in animal groups or cellular aggregates has been a long-standing challenge, central to understanding the mechanisms and evolution of collective behavior. Numerous models have been proposed, many of which display realistic-looking dynamics, but nonetheless rely on untested assumptions about how individuals integrate information to guide movement. Here we infer behavioral rules directly from experimental data. We begin by analyzing trajectories of golden shiners (Notemigonus crysoleucas) swimming in two-fish and three-fish shoals to map the mean effective forces as a function of fish positions and velocities. Speeding and turning responses are dynamically modulated and clearly delineated. Speed regulation is a dominant component of how fish interact, and changes in speed are transmitted to those both behind and ahead. Alignment emerges from attraction and repulsion, and fish tend to copy directional changes made by those ahead. We find no evidence for explicit matching of body orientation. By comparing data from two-fish and three-fish shoals, we challenge the standard assumption, ubiquitous in physics-inspired models of collective behavior, that individual motion results from averaging responses to each neighbor considered separately; three-body interactions make a substantial contribution to fish dynamics. However, pairwise interactions qualitatively capture the correct spatial interaction structure in small groups, and this structure persists in larger groups of 10 and 30 fish. The interactions revealed here may help account for the rapid changes in speed and direction that enable real animal groups to stay cohesive and amplify important social information.
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“This knowledge can then be used to infer individual decision rules in collective motion (Katz, Tunstrom, Ioannou, Huepe, & Couzin, 2011;Nagy, Kos, Biro, & Vicsek, 2010) and to compute activity budgets for individuals without the need for continuous human observation or time-consuming video analysis.”
From Applications of machine learning in animal behaviour studies (2017) · cites this paper as Katz et al., 2011
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Mudaliar, R. K., & Schaerf, T. M. (2023) · PLoS ONE · doi:10.1371/journal.pone.0286810 · Cited by 7
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Genetic architecture of collective behaviors in zebrafishPreprint
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“It appears that the sometimes rapid changes in direction and mode result from propagation of local interactions (Katz et al., 2011).”
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Harris, S. J., & McDonald, N. R. (2023) · Acta Applicandae Mathematicae · doi:10.1007/s10440-023-00578-2 · Cited by 7
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Kleiman, M., Bodek, S., & Eilam, D. (2014) · Behavioural Processes · doi:10.1016/j.beproc.2014.10.017 · Cited by 11
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Variability and Heterogeneity in Natural Swarms: Experiments and Modeling
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Influential‘Selfish herders’ finish last in mobile animal groups
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“In interrelated calculations, we examined local group structure in the same relative coordinate system via graphs that illustrate the relative frequency that neighbours occupy given relative (x, y) coordinates (as in Lukeman et al. 2010; Katz et al. 2011; Schaerf et al. 2017), and statistics of the relative alignment…”
BackgroundResultData-driven discovery of stochastic dynamical equations of collective motion
Nabeel, A., Jadhav, V., Danny Raj, M., Sire, C., Théraulaz, G., Escobedo, R., Iyer, S. K., & Guttal, V. (2023) · Physical Biology · doi:10.1088/1478-3975/ace22d · Cited by 7
Investigating the Emergence of Collective States Within Rugby Sevens Gameplay
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Chen, Z., Zhang, H., Chen, X. X., Chen, D., & Zhou, T. (2015) · Europhysics Letters (EPL) · doi:10.1209/0295-5075/112/20008 · Cited by 7
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Studying feral horse behavior from the sky
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A fellow-following-principle based group model and its application to fish school analysis
Zhou, Z., Liu, J., Pan, J., Wang, J., & Yu, J. (2022) · Bioinspiration & Biomimetics · doi:10.1088/1748-3190/acab48 · Cited by 6
“Most studies have performed statistical analyses of small numbers of fish to obtain key information [14].”
BackgroundCollective response to local perturbations: how to evade threats without losing coherence
Loffredo, E., Venturelli, D., & Giardina, I. (2023) · Physical Biology · doi:10.1088/1478-3975/acc5cc · Cited by 6
Dynamic Motions of Topological Defects in Nematic Liquid Crystals under Spatial Confinement
Pawale, T., Swain, J., Hashemi, M. R., Tierra, G., & Li, X. (2023) · Advanced Materials Interfaces · doi:10.1002/admi.202300136 · Cited by 6
Schaerf, T. M., Wilson, A. D. M., Welch, M., & Ward, A. J. W. (2024) · Royal Society Open Science · doi:10.1098/rsos.231511 · Cited by 6
“…structure [25], comparison of local alignment and structure between observations and simulation models [26], social force matching [27], methods aimed at constructing data-driven models [28–32], social force mapping (a distinct technique to force matching) [31,33,34] and machine learning [35].”
BackgroundMethodSpatiotemporal organization of ant foraging from a complex systems perspective
Cristín, J., Fernández-López, P., Lloret‐Cabot, R., Genovart, M., Méndez, V., Bartumeus, F., & Campos, D. (2024) · Scientific Reports · doi:10.1038/s41598-024-63307-1 · Cited by 6
Experience-dependent modulation of collective behavior in larval zebrafish
Harpaz, R., Phillips-Batra, M., Goel, R., Bind, M.-A. C., Fishman, M., & Engert, F. (2026) · Nature Communications · doi:10.1038/s41467-026-74872-6
Promoting collective motion of self-propelled agents by discarding short-range interactions
Yang, H.-X., & Rong, Z. (2015) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2015.03.031 · Cited by 6
Determining collision potential as a measure of robustness in vehicular networks
Grunberg, T. W., & Gayme, D. F. (2017) · Proceedings of the ... American Control Conference/Proceedings of the American Control Conference · doi:10.23919/acc.2017.7963567 · Cited by 6
“Vehicular networks arise in a number of applications such as flocks of unmanned aerial vehicles, highway automation, and schools of fish [1]–[3].”
BackgroundBonnell, T. R., Henzi, S. P., & Barrett, L. (2018) · Journal of Human Evolution · doi:10.1016/j.jhevol.2018.10.011 · Cited by 6
Propagating wave based on transition of interaction within animal group
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A Kinetic Approach to Active Rods Dynamics in Confined Domains
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optiSwarm: Optical Swarm Robots Using Implicit Cooperation
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“The behaviors of birds [1], [2], [3], [4], [5], fish [6], [7], [8], [9], [10], bacteria [11], [12], [13], cells [14], [15], and even protein microtubules [16] are typical representatives.”
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“Tracking individual fish within a school is challenging, but that information is required to obtain information on the collective behavioral mechanisms inside fish schools [40].”
BackgroundAnalysis of multi-beam sonar echos of herring schools by means of simulation
Holmin, A. J. (2013) · NORA - Norwegian Open Research Archives · Cited by 6
Modeling collective behavior in groups of mice housed under semi-naturalistic conditionsPreprint
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“Additionally, semi-naturalistic environments, exemplified by fish in tanks ( Katz et al., 2011 ; Herbert-Read et al., 2011 ; Gautrais et al., 2012 ), marching locusts in arenas ( Buhl et al., 2006 ), flocks of sheep ( Ginelli et al., 2015 ), and hordes of rodents ( Kondrakiewicz et al., 2019 ; Pu ś…”
Individual and group behavioral responses to nutritional state and context in a social fish
Zeng, L.-Q., Ling, H., Fu, S., Pu, D.-Y., & Killen, S. S. (2024) · Behavioural Processes · doi:10.1016/j.beproc.2024.105059 · Cited by 5
Flow velocity and boundary effects on fish interaction
Mozzi, G., Comoglio, C., & Manes, C. (2025) · Scientific Reports · doi:10.1038/s41598-025-13332-5 · Cited by 5
“Nonetheless, correlation coefficients remain widely used due to their simplicity, interpretability, and utility in revealing broad patterns of animal interactions 13,14 .”
Bhowmick, A., Mitra, S., & Mohanty, P. K. (2025) · Physical review. E · doi:10.1103/nw6q-868p · Cited by 5
“Some examples include flocks of birds [4], fish schools [5], actin filaments [6], and microtubules [7] in the context of the study of locomotion and colloidal systems [8–11].”
Networked collective dynamics in animal ecology and cell biology
Sun, Y., Wang, H., Liu, X., Wen, G., Lin, W., & Maini, P. K. (2026) · Physics of Life Reviews · doi:10.1016/j.plrev.2026.02.003 · Cited by 5
Inverse problem for multi-body interaction of nonlinear waves
Marruzzo, A., Tyagi, P., Antenucci, F., Pagnani, A., & Leuzzi, L. (2017) · Scientific Reports · doi:10.1038/s41598-017-03163-4 · Cited by 5
LaViers, A. (2019) · PLoS ONE · doi:10.1371/journal.pone.0215671 · Cited by 5
Unsupervised manifold learning of collective behavior
Titus, M., Hagstrom, G. I., & Watson, J. R. (2021) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1007811 · Cited by 5
“group polarization and rotation, Op and Or, and classifying behavior as in [24] we find that 353 group G1 contains 85.”
Influential
Metadata
- Authors
- Katz, Y., Tunstrøm, K., Ioannou, C. C., Huepe, C., & Couzin, I. D.
- Affiliations
- Princeton University (US)
- 614 North Paulina Street, Chicago, IL 60622
- Venue
- Proceedings of the National Academy of Sciences
- Publisher
- National Academy of Sciences
- Published
- 2011-07-27
- Volume
- 108
- Issue
- 46
- Pages
- 18720-18725
- ISSN
- 0027-8424 (ISSN-L)
- 1091-6490
- PMC
- PMC3219116
- OpenAlex
- W2002266200
- Cited by
- 927
- References
- 54
- Citations by year
- 2026: 50 · 2025: 69 · 2024: 76 · 2023: 80 · 2022: 72 · 2021: 73 · 2020: 76 · 2019: 58 · 2018: 69 · 2017: 57 · 2016: 53 · 2015: 54 · 2014: 46 · 2013: 46 · 2012: 40
- FWCI
- 58.66
- Type
- journal article
- Language
- en
- Open access
- green
- www.ncbi.nlm.nih.gov
- Locations
- Proceedings of the National Academy of Sciences (publishedVersion)
- Europe PMC (PubMed Central) (publishedVersion)
- PubMed Central (publishedVersion)
- PubMed (publishedVersion)
- Bristol Research (University of Bristol) (publishedVersion)
- KOPS (University of Konstanz) (submittedVersion)
- MPG.PuRe (Max Planck Society) (submittedVersion)
- Funders
- National Science Foundation (0848755, 0905970, N00014-09-1-1074)
- Princeton University
- Norges Forskningsråd
- Advanced Research Projects Agency
- Office of Naval Research (N00014-09-1-1074, N00014)
- Topics
- Physical Sciences › Engineering › Control and Systems Engineering › Distributed Control Multi-Agent Systems
- Social Sciences › Safety Research › Evacuation and Crowd Dynamics
- Life Sciences › Agricultural and Biological Sciences › Ecology, Evolution, Behavior and Systematics › Animal Behavior and Reproduction
- Keywords
- fish schooling, collective behavior, golden shiner, Notemigonus crysoleucas, fish shoals, effective force, speed regulation, alignment, attraction and repulsion, three-body interactions, pairwise interactions, trajectory analysis, social information, group cohesion
- MeSH terms
- Animals, Behavior, Animal, Cyprinidae, Fishes, Models, Biological, Movement, Social Behavior, Software, Swimming, Time Factors
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
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