Turning avalanches in schooling fish

Research summary

One paper, summarised for reference. Signatures of criticality in turning avalanches of schooling fish — Andreu Puy, Elisabet Gimeno, David March-Pons, M. Carmen Miguel & Romualdo Pastor-Satorras, Phys. Rev. Research (2024) 6, 033270. doi:10.1103/PhysRevResearch.6.033270.

The study

A large-statistics empirical analysis of spontaneous turning avalanches — large heading changes propagating across a group — in freely swimming schools (black neon tetra, $N=8,16,32,50$; $>10^4$ avalanche events). It applies the full condensed-matter/seismology avalanche toolkit and finds scale-free behaviour — power laws, an exponent scaling relation, and data collapses — which it is careful to call a necessary (not sufficient) condition for criticality.

Definition and observables

An individual’s turning rate $\omega = |\mathbf v\times\mathbf a|/v^2$ (angular-velocity magnitude); above a threshold $\omega_{\text{th}}$ a fish is “active.” An avalanche is a run of consecutive frames with $\ge1$ active fish, characterised by duration $T$, size $S=\sum_t n_t$, and interevent time $t_i$; the activity rate $r$ is the fraction of frames inside an avalanche.

Distributions, scaling relations, collapses

At $\omega_{\text{th}}=0.1$: $P(T)\sim T^{-\alpha}$ ($\alpha\approx2.4$), $P(S)\sim S^{-\tau}$ ($\tau\approx1.97$), average size $\langle S\rangle_T\sim T^{m}$ with $m\approx1.41$ — obeying the crackling-noise scaling relation

$$m = \frac{\alpha-1}{\tau-1},$$

and interevent times $P(t_i)\sim t_i^{-\gamma}$ ($\gamma\approx1.62$). Distributions for different $N$ collapse (bar the finite-size tails). At fixed activity rate, duration and interevent-time distributions collapse across $N$; the interevent times obey the SOC form $P(t_i)=\tfrac1{\langle t_i\rangle}\Psi(t_i/\langle t_i\rangle)$; and the avalanche shape obeys $\langle n_t\rangle = T^{m-1}\,\mathcal F(t/T)$ — a universal temporal profile.

The key methodological point

The authors are explicit that power laws alone don’t demonstrate criticality (other mechanisms make them); the tighter evidence is the data collapses and the relation between exponents ($m=(\alpha-1)/(\tau-1)$, the shape collapse), which indicate quantitatively universal avalanche dynamics across scales. With no externally tuned parameter, a near-critical school would be an instance of self-organized criticality.

Boundaries, function, correlations

• Dragon kings. The distribution tails carry over-represented extreme events generated by a different mechanism — tank-wall interactions (large avalanches cluster at corners). Restricting to the tank centre removes them (statistical test: $p<10^{-15}$ overall vs. $p=0.1$ centred) and extends the clean power law to ~2 decades.
• Function. Large avalanches occur at the speed minima of the burst-and-coast cycle, temporarily reduce polarization and then recover it — i.e. they are collective decision-making about a new heading (U-turns are a special case). Boundary individuals are the frequent initiators (higher social influence / more risk-exposed).
• Aftershocks. Using seismology’s Gutenberg–Richter ($P(m)\sim e^{-bm}$, $m=\ln S$, $b=\tau-1$) and an Omori law $P(t)=K/(t+c)^p$, correlated aftershocks cluster below a half-turn timescale (~250 frames), with $p\approx2.2$ — a faster decay than earthquakes, and no collective memory beyond that timescale.

Why it’s collected here

This is, almost exactly, the empirical protocol for the reorientation-cascade experiment on our own next-set list — turning avalanches, defined by a heading-change threshold, analysed with $P(S)\sim S^{-\tau}$, $P(T)\sim T^{-\alpha}$, the $m=(\alpha-1)/(\tau-1)$ scaling relation, data collapse and avalanche-shape collapse. Three things to carry straight over: the exponents to compare against ($\tau\approx1.5$–2, $\alpha\approx2$); the insistence that collapses + exponent relations, not a bare power law, are the real test (the Muñoz/Watkins caution, operationalised); and the dragon-king / boundary lens — our torus has no walls, but the source is a boundary-like attractor, so the same “are the extreme events a different mechanism?” question applies. It also makes turning = collective decision-making, feeding the collective-decision-task idea in our plan.

References

Puy et al. (2024), Phys. Rev. Research 6, 033270 · Múgica et al. (2022), Sci. Rep. 12, 10783 · Poel et al. (2022), Sci. Adv. 8, eabm6385 · Gómez-Nava et al. (2023), Nat. Phys. 19, 663 · Sethna, Dahmen & Myers (2001), Nature 410, 242 · Friedman et al. (2012), Phys. Rev. Lett. 108, 208102 · Muñoz (2018), Rev. Mod. Phys. 90, 031001 · Baiesi & Paczuski (2004), Phys. Rev. E 69, 066106 · Sornette & Ouillon (2012), Eur. Phys. J. Spec. Top. 205, 1.