Subcritical escape waves in fish

Research summary

One paper, summarised for reference. Subcritical escape waves in schooling fish — Winnie Poel, Bryan C. Daniels, Matthew M. G. Sosna, Colin R. Twomey, Simon P. Leblanc, Iain D. Couzin & Pawel Romanczuk, Science Advances (2022) 8, eabm6385. doi:10.1126/sciadv.abm6385.

The finding

Startle cascades (escape waves) in schools of juvenile golden shiners are subcritical — not maximally responsive to cues — and the school’s distance to criticality shrinks as perceived risk rises. The authors argue that being subcritical, and modulating the distance to criticality by context, is how the group manages the trade-off between sensitivity and robustness: criticality has a cost (amplifying noise into false alarms), usually ignored when only its benefits (maximal sensitivity) are counted.

System and model

Data (schools of $N=40$) come in two conditions — baseline and alarmed (an alarm substance raises perceived risk). Alarmed schools show larger cascades and higher density (lower nearest-neighbour distance, NND); crucially, the larger cascades cannot be explained by individual responsiveness alone — the group structure (density) is essential. A SIR-type behavioural-contagion model runs on visual interaction networks (links from a logistic regression on distance and visual angle); rescaling the positions predicts cascades at densities not observed, revealing a critical density (local→global cascade transition) at NND ≈ 0.3–0.4 body lengths. Criticality is located two ways that agree: a collective sensitivity $c = c_2 - c_1$ (responsiveness to two vs. one initial startlers — the finite-size susceptibility analog, peaking at the transition) and an analytic branching ratio $b = \tfrac{\tau_{\text{act}}}{\bar\theta}\sum_i w_{ij}/K_i$, with $b=1$ the critical line separating subcritical ($b<1$) from supercritical ($b>1$).

The result and the trade-off

Both observed conditions are subcritical, with the alarmed condition closer to the transition; their sensitivity could be raised 3.4–5.9× by becoming critical, but they don’t — and the fishes’ physical bodies cap density, so a school cannot reach maximal sensitivity by crowding alone. Distance to criticality is controlled by density and individual responsiveness (threshold). A hypothetical predator-detection model then weighs two error types — false positives (startle with no predator: energy/time cost) and false negatives (miss a real predator: injury/death) — into a relative payoff that depends on the environment’s relative noise cost:

• safe / noisy environments → optimum at low density, away from criticality (false positives dominate; near-criticality is detrimental);
• high-predation / low-noise → near (or past) criticality is best (false negatives dominate);
• intermediate → near-criticality maximises payoff.

Density also trades personal information (visual access, falls with density via occlusion) against social information (rises with density) — so the individual optimum is genuinely context-dependent.

The message

A real animal collective does not sit at the critical point; it tunes its distance to criticality (via density, with risk) to match the environment’s sensitivity/robustness demands. The paper is a pointed call to weigh the costs of criticality (amplified noise) alongside the benefits, to take an individual-based, context-dependent view, and to remember that the group-level optimum (criticality) need not be the individual’s (it cites Klamser & Romanczuk on exactly this).

Why it’s collected here

This is the empirical anchor for the “near, not at, criticality — and tune the distance” stance that runs through the Romanczuk & Daniels review and the Meijers result, and it bears directly on our own working hypothesis. Its subcriticality + cost-of-criticality framing is a ready-made lens for the possibility that our swarm has no accessible critical point (or sits deliberately below one): a collective that aggregates but doesn’t propagate cascades may be avoiding false alarms, not failing to be critical. Concretely reusable: the $b=1$ branching-ratio locator and the $c_2-c_1$ collective sensitivity are clean criticality measures; density/NND is the control parameter (our arena/spacing sweeps); and the false-positive/false-negative payoff is a genuinely different way to interpret performance than the raw forage score — one that could explain a subcritical optimum.

References

Poel et al. (2022), Sci. Adv. 8, eabm6385 · Sosna et al. (2019), PNAS 116, 20556 · Rosenthal et al. (2015), PNAS 112, 4690 · Daniels, Krakauer & Flack (2017), Nat. Commun. 8, 14301 · Romanczuk & Daniels (2022), Order, Disorder and Criticality · Klamser & Romanczuk (2021), PLoS Comput. Biol. 17, e1008832 · Dodds & Watts (2004), Phys. Rev. Lett. 92, 218701 · Muñoz (2018), Rev. Mod. Phys. 90, 031001.