Heterogeneous criticality in a fish school

Research summary

One paper, summarised for reference. Information structure of heterogeneous criticality in a fish school — Takayuki Niizato, Kotaro Sakamoto, Yoh-ichi Mototake, Hisashi Murakami & Takenori Tomaru, Scientific Reports (2024) 14, 29758. doi:10.1038/s41598-024-79232-2.

The problem it targets

Applying theoretical criticality to real animal groups runs into two obstacles the authors call out directly: choosing the unit of analysis (what counts as the “system” — the whole group? which subgroup?) and handling nested criticality (small- and large-scale criticality coexisting). These stem from conflating internal fluctuation (heterogeneous, context-dependent, generated by the agents’ own interactions) with external fluctuation (uniform, random, system-independent — the “temperature” of a spin model). Standard order-parameter/avalanche methods bake in a fixed unit and a uniform external noise; this paper avoids both.

The method: Integrated Information (Φ)

Integrated Information Theory measures $\Phi$ — how much a system is more than the sum of its parts — via the Minimum Information Partition (MIP): the cut splitting the system into the two subsystems with the weakest information link ($\Phi=0$ means the parts are independent, i.e. external fluctuation to each other). Crucially, IIT does not pre-assume the unit or separate internal from external fluctuation — the MIP finds them. A main complex is a subgroup at a local maximum of $\Phi$ (an “information core”); the maximum main complex (MMC) is the strongest. Applied to 10 ayu (Plecoglossus altivelis) tracked in 2D, with $\Phi$ computed separately on turning rate (direction) and acceleration (speed).

The key result: criticality is heterogeneous, not homogeneous

$\Phi$ is maximised at the critical state — validated against the self-propelled-particle (SPP) model, where $\max\Phi_{\text{dir}}$ peaks near the critical noise. But the contrast with the model is the point: in the SPP model the MMC at criticality is the whole group (indecomposable — matching the classical “a critical system is inseparable”), whereas in the real fish the MMC is fragmented (sizes 2–10), with multiple critical subgroups coexisting, each with high $\Phi$. And the whole-group $\Phi$ is also critical. So global and local (subgroup) criticality coexist — empirical criticality is heterogeneous and nested, unlike the homogeneous criticality of the toy models (which a simple Boid model also fails to reproduce). The $\Phi$ time series is pink noise (Hurst $H\approx0.25$, scale-free, long autocorrelation), while the peak-value series is Brownian.

Core fish and the avalanche hypothesis

MMC membership is heterogeneous (a stretched-exponential lifespan, $\alpha\approx0.61$ — long-tailed, not memoryless): some fish are frequently “core,” most are not. The core fish have small variance in direction and speed — they are less affected by internal/external perturbation, and are not leaders (never at the group’s leading edge). The authors’ reading: most individuals sit in a supercritical (disordered) state, and the group needs a few stable, subcritical core individuals to knit the varied demands together — so global criticality is a mixture of roles. They conjecture that the interplay between supercritical (responsive) and subcritical (stable) agents is what drives sandpile-type information avalanches in collective motion, rather than tuning a single noise threshold (paralleling coexisting local critical states in gene-expression dynamics).

Why it’s collected here

This is the paper that names, and offers a tool for, the exact problem our harness ran into: what is the right unit/scale for an ensemble-criticality measure? IIT’s MIP finds the unit instead of assuming it, and it makes nested criticality (subgroup + whole-group, coexisting) a measurable object — a fresh angle on our two-scale (network/ensemble) question. Two more direct hooks: its internal-vs-external fluctuation distinction is precisely what our circular-shift null test tries to separate (agent coupling vs common external drive); and its stable “core” individuals are a data-driven version of the informed-minority / lookout idea, but defined by low responsiveness rather than leadership. $\Phi$ is also simply a different measure to have in the kit — one that, unlike a bare avalanche power law, is constructed to be irreducible-by-definition. (Note the heavy compute: exhaustive MIP limits it to small groups.)

References

Niizato et al. (2024), Sci. Rep. 14, 29758 · Tononi (2004), BMC Neurosci. 5, 42 · Oizumi et al. (2016), PLoS Comput. Biol. 12, e1004654 · Vicsek et al. (1995), Phys. Rev. Lett. 75, 1226 · Cavagna et al. (2010), PNAS 107, 11865 · Tsuchiya et al. (2015), PLoS ONE 10, e0128565 · Sánchez-Puig et al. (2022), Front. Complex Syst. 1, 22 · Múgica et al. (2022), Sci. Rep. 12, 10783 · Puy et al. (2024), Phys. Rev. Res. 6, 033270.