Activation fronts in active systems

Research summary

One paper, summarised for reference. A generic coupling between internal states and activity leads to activation fronts and criticality in active systems — Hadrien-Matthieu Gascuel, Parisa Rahmani, Richard Bon & Fernando Peruani, arXiv:2311.06208 (2024). arXiv:2311.06208.

The question

How does the onset of collective motion — a static, polarized group set in motion by a single initiator (a startled individual, a predator reaction) — propagate through a group, and when does it become critical? Most collective-motion work models constantly moving units; this paper studies agents that switch self-propulsion on and off, with the internal state coupled to motion, and shows criticality emerges generically from that coupling.

The model

Each agent has a position $x_i$ and a behavioural state $q_i\in\{I \text{ (inactive)}, A \text{ (active)}\}$, with the coupling $\dot x_i = v$ if active, $0$ if inactive. Starting from a polarized static group (all $I$ except one initiator in $A$), the only transition is a mimetic activation,

$$I + A \xrightarrow{\;\gamma(|\Delta x|)\;} 2A, \qquad \gamma(|\Delta x|)=\alpha\,K(|\Delta x|/d),$$

so a newly active agent adopts the active neighbour’s velocity — inheriting the initiator’s chosen $v$. The sign and size of $v$ controls the physics.

Fronts and regimes

A hydrodynamic (Kolmogorov-forward) treatment gives, for $v=0$, an FKPP-like activation front; for $v>0$ the leading-edge speed is linear but non-trivial, $c(v)\approx c_0 + m\,v$ with $m\neq1$ ($m=4/3$ for exponential $K$). Two regimes follow: a slow regime ($0<v<c_0$) where the whole group activates and moves keeping its structure, and a high-speed / “selfish” regime ($v\ge c_0$) where the initiator outruns its neighbours, penetrates the group, and recruits agents behind it — Hamilton’s selfish-herd scenario, achieved simply by choosing $v>c_0$.

From two states to an effective three — and criticality

For $v<0$ the fluctuating dynamics maps onto the SIR / forest-fire model ($S+I\to2I$, $I\to R$): an inactive agent may be “passed” before it activates, ending a cascade — so the 2-state active system behaves as an effective 3-state (excitable) one. In 1D this gives a sharp transition in the order parameter $\phi=\langle s\rangle/N$ (fraction ever activated) at $v_c=0$ — exponentially distributed cascades below, full activation above. In 2D, because the 3-state SIR model is itself critical, the active system shows genuine critical behaviour at a critical speed $v_c<0$: power-law activity avalanches $P(s)\propto s^{-\beta}$ with $\beta\approx1$ (small to system-spanning), a susceptibility $\chi(\phi)=N(\langle\phi^2\rangle-\langle\phi\rangle^2)$ that grows and narrows with system size, and no characteristic correlation length. (The 2D SIR mapping is approximate — moving active agents can reactivate inactive “islands” that SIR would leave protected.)

Why it’s collected here

This is close to a template for a next-step experiment on our own swarm. It is exactly a reorientation / activation cascade through a group, with an internal-state→motion coupling (the reservoir-controls-body idea in miniature), analysed as an excitable / SIR avalanche process with a power-law cascade-size test for criticality — the same machinery as the naive-agent cascade-scaling experiment we have planned (power law with a finite-size cutoff, susceptibility growing with $N$). It also gives a clean, mechanistic account of behavioural contagion (complementing the Romanczuk & Daniels review) and shows how a single control — the initiator’s speed — selects between “whole group moves together” and “selfish” restructuring.

References

Gascuel et al. (2024), arXiv:2311.06208 · Ginelli, Peruani et al. (2015), PNAS 112, 12729 · Gómez-Nava, Bon & Peruani (2022), Nat. Phys. 18, 1494 · Poel et al. (2022), Sci. Adv. 8, eabm6385 · Muñoz (2018), Rev. Mod. Phys. 90, 031001 · Klamser & Romanczuk (2021), PLoS Comput. Biol. 17, e1008832 · Grassberger (1983), Math. Biosci. 63, 157 · Hamilton (1971), J. Theor. Biol. 31, 295.