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Noise-induced schooling of fish

Abstract

We report on the dynamics of collective alignment in groups of the cichlid fish Etroplus suratensis. Focusing on small- to intermediate-sized groups (10 N 100), we demonstrate that schooling (highly polarized and coherent motion) is noise induced, arising from the intrinsic stochasticity associated with finite numbers of interacting fish. The fewer the fish, the greater the (multiplicative) noise and therefore the greater the likelihood of alignment. Such rare empirical evidence tightly constrains the possible underlying interactions that govern fish alignment, suggesting that E. suratensis either spontaneously change their direction or copy the direction of another fish, without any local averaging (the otherwise canonical mechanism of collective alignment). Our study therefore highlights the importance of stochasticity in behavioural inference. Furthermore, rather than simply obscuring otherwise deterministic dynamics, noise can be fundamental to the characterization of emergent collective behaviours.

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Fig. 1: Capturing the stochastic dynamics of ordering in small- to medium-sized groups of fish.
Fig. 2: Steady-state statistics demonstrate high degree of schooling.
Fig. 3: Empirical fitting of the first and second jump moments reveals that schooling is noise induced.
Fig. 4: Pairwise versus ternary interactions: theory and stochastic simulations.

Data availability

All data necessary to reproduce Figs. 2–4 are available via https://doi.org/10.5281/zenodo.3596324.

Code availability

All codes, including readme files necessary to reproduce Figs. 2–4, are available via https://doi.org/10.5281/zenodo.3596324.

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Acknowledgements

V.G. thanks C. Jayaprakash for introducing him to the fascinating world of noise-induced phenomena. We acknowledge assistance from S. Chakraborty, E. M. Jos, A. Nabeel, A. Karichannavar and T. Goel. We thank Binoy V. V. for suggestions on schooling fish species native to India and their hatcheries. We also thank S. Ramaswamy for a critical reading of the manuscript and C. C. Ioannou for discussions. J.J. acknowledges support by the CSIR, India, through a research scholarship. R.G.M. acknowledges both the Simons Foundation (USA) and EMBL Australia for funding. M.D.R. acknowledges a DST India INSPIRE faculty award for funding. T.R. acknowledges the Royal Society (UK) for funding. V.G. acknowledges support from the DBT-IISc partnership programme, SERB (DST) and infrastructure support from DST-FIST.

Author information

Affiliations

Authors

Contributions

V.G. conceived of and oversaw the project. J.J., U.R.A.-K. and H.R. performed experiments. J.J. and R.G.M. analysed and interpreted the data. J.J. and M.D.R. performed simulations. T.R. and R.G.M. performed calculations. R.G.M. wrote the manuscript, with input from J.J., M.D.R. and V.G. J.J. and R.G.M. contributed equally to the manuscript.

Corresponding authors

Correspondence to Jitesh Jhawar or Richard G. Morris or Vishwesha Guttal.

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Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature Physics thanks Guy Theraulaz and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data

Extended Data Fig. 1 Auto-correlation of group polarisation.

Blue, yellow and green solid lines represent the data for N=15, 30, and 60, respectively. Two characteristic timescales are apparent; τ, which encapsulates the rate of initial decay of correlations to zero (solid black lines) and τenv, which is rate of decay of the envelope of quasi-periodic correlations (dotted grey lines).

Extended Data Fig. 2 Numerical check of the extracted SDE.

Milstein-method simulations of the SDE that was extracted from the data [Eq. (3) of the main manuscript]. The results are qualitatively in-line with experimental observations.

Extended Data Fig. 3 Second jump-moments: pairwise vs. ternary.

In agreement with theoretically derived expressions, the simulation-generated diagonal (non-zero) second jump-moments take a similar form for both pairwise and ternary models. Data points are generated by Gillespie simulation (using the stated parameter values), whilst both surfaces and the analytical expressions to which they correspond are taken from theory (Methods Sections II A & B).

Extended Data Fig. 4 Optimisation of higher-order copying interaction models.

Using a Genetic Algorithm in the context of repeated Gillespie simulations (Methods Section III C), we optimise a given model’s specific rates against the experimental data. The results – specifically, large values of r2 and negligible values of ri where i > 2 – imply that pairwise copying is the dominant mode of interaction and that higher order interactions are likely negligible.

Extended Data Fig. 5 Optimisation of higher-order Vicsek-like interaction models.

Using a Genetic Algorithm in the context of repeated Gillespie simulations for higher-order Vicsek-like interaction models (Methods section III D), we optimise a given model’s specific rates against the experimental data. The results confirm that direction-averaging (represented by ri where i > 2) is not represented by the data; this can be inferred from the values of the DKL corresponding to the optimized rates of interaction.

Supplementary information

Supplementary Information

Supplementary experimental details, boundary controls, discussion of one-dimensional toy models, configuration space mixing analysis and spatial schooling model with boundary.

Reporting Summary

Supplementary Video 1

A sample video of the experiments with schools of fish, E. suratensis, in groups of sizes 15, 30 and 60.

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Jhawar, J., Morris, R.G., Amith-Kumar, U.R. et al. Noise-induced schooling of fish. Nat. Phys. 16, 488–493 (2020). https://doi.org/10.1038/s41567-020-0787-y

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