Noise-Induced Collective Memory in Schooling Fish

Alyssa Chan; Eva Kanso

REVIEW 3 major objections 5 minor 51 references

Noise-Induced Collective Memory in Schooling Fish

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper argues that collective memory in the three-A fish-school model comes from noise near a transcritical bifurcation, not from two coexisting stable states.

desk verdict Claims noisy transcritical bifurcation for hysteresis in three-A fish model, but the Kramers fit has a sign error and the normal form is assumed, leaving a narrow subcritical pitchfork still viable. read the letter →

arxiv 2507.16102 v1 pith:N7MKS4VR submitted 2025-07-21 nlin.AO cond-mat.stat-mechphysics.bio-phphysics.flu-dyn

Alyssa Chan , Eva Kanso This is my paper · ORCID

classification nlin.AOcond-mat.stat-mechphysics.bio-phphysics.flu-dyn MSC 37N2560H1092D50

keywords collectivemotionfishschoolingmillinghysteresismemorytranscriticalbifurcationstochasticdifferentialequationsthree-Amodel

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Schooling fish in a popular agent-based model, the three-A model, appear to remember their past: slowly changing individual behavior from milling to schooling and back gives different patterns in each direction. The paper argues this hysteresis is not evidence that milling and schooling are two equally stable states. Instead, the two states exchange stability at a critical alignment-zone size, and finite noise lets the group linger in the state that has just lost stability, creating memory-like path dependence. The claim is supported by Monte Carlo simulations, a one-variable stochastic model for group polarization, and a Kramers escape-rate fit. If correct, it resolves a long-standing ambiguity about the origin of collective memory in a widely used model and cautions that observed bistability-like behavior can arise without structural bistability.

What carries the argument

The carrying object is the one-dimensional stochastic differential equation $dP = rP(1-P)\,dt + \sigma\,dW$ (Eq. 3), with effective potential $U = -r(P^2/2 - P^3/3)$: the normal form of a transcritical bifurcation perturbed by white noise. Here $P$ is group polarization, $r = z_l - z_l^*$ is the distance from the milling-to-schooling transition, and $\sigma$ is collective noise. The equation is posited as a phenomenological reduction, not derived from the agent-based update rule; at $r=0$ its drift vanishes, the effective potential is flat, and the dynamics are purely diffusive. It carries the argument by generating the observed finite-time bistability, the Kramers escape-rate scaling $\kappa = A\,r\,e^{-r/(3\sigma^2)}$, and the noise-dependent hysteresis, which lets the paper attribute the fish-model behavior to the bifurcation type of this SDE.

What would settle it

Measure the residence time in the milling state at alignment-zone sizes just above the transition, for several noise levels, and fit the escape rate to the Kramers form $\kappa = A\,r\,e^{-r/(3\sigma^2)}$. The paper's mechanism predicts the effective barrier $r/6$ vanishes as $z_l$ approaches $z_l^*$ from above, while structural bistability would keep a finite barrier at the transition; a nonzero extrapolated barrier would falsify the noisy-transcritical claim.

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Extended reading notes

Core claim

Rotationally milling and polarized schooling, measured by polarization $P$ and rotation $M$, appear to coexist in a narrow window of alignment-zone sizes around $z_l^* \approx 3.1$ (with attraction zone $z_a=10$ and noise $\sigma=0.01$): some simulations remain in milling while others switch to schooling. The paper's central claim is that this window is not structural bistability but the signature of a noisy transcritical bifurcation. It posits the phenomenological stochastic differential equation $dP = rP(1-P)\,dt + \sigma\,dW$, with $r = z_l - z_l^*$ and effective potential $U = -r(P^2/2 - P^3/3)$. In the deterministic limit, $P=0$ (milling) and $P=1$ (schooling) persist for all $r$ and exchange stability at $r=0$; with noise, the group can escape from the newly unstable milling state, with an escape rate matching Kramers' formula $\kappa = A\,r\,e^{-r/(3\sigma^2)}$. This reproduces the key observations: transient milling beyond the transition, a noise-dependent hysteresis loop that is strongest at intermediate noise and disappears at high noise, and collective memory without a double-well potential.

Load-bearing premise

The paper assumes, rather than derives, that group polarization obeys the simple stochastic equation whose deterministic part is the transcritical normal form; if the many-fish dynamics cannot be reduced to that single variable, the claim that the transition is a noisy transcritical bifurcation loses its foundation.

Editorial extensions

If this is right

The milling-to-schooling transition in the three-A model is controlled by the alignment-zone radius $z_l$, with the attraction-zone radius having little influence once cohesion is maintained; the abrupt switch occurs near $z_l^* \approx 3.1$ at $z_a=10$.
Hysteresis and collective memory in this model are noise-dependent: weak noise gives memory mainly on the backward sweep, intermediate noise gives a strong hysteresis loop, and strong noise destroys milling and the memory effect.
Observed bistability near the transition is a finite-time, noise-driven phenomenon; in infinite time the system would follow the newly stable schooling state for $r>0$, so bistability is degenerate and localized at the bifurcation point.
The escape rate from milling to schooling obeys Kramers' formula $\kappa = A\,r\,e^{-r/(3\sigma^2)}$, with a barrier that shrinks to zero at the bifurcation, quantitatively supporting the transcritical normal form.
Collective path dependence alone does not diagnose structural bistability: hysteresis can reflect noise near a stability exchange, so observed bistability-like behavior in animal groups must be interpreted through the underlying mechanism.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural next step is to construct the group-level effective potential directly from the agent-based trajectories by coarse-graining; the transcritical claim predicts a single-well potential whose minimum flattens and shifts near $z_l^*$, whereas structural bistability would show a double well. The paper gestures at this comparison but does not carry it out.
This suggests an experimental handle: if confined fish schools show intermittent milling-schooling transitions, adding or reducing sensory noise should sharpen or broaden the transition in a specific way under the transcritical mechanism, separating it from the double-well mechanism in real animals.
The same normal-form reasoning could be applied to self-propelled particle models with rotational inertia or boundary confinement; their hysteresis should scale differently with noise if their transitions are truly pitchfork-like, making noise dependence a fingerprint of the bifurcation type.
If Eq. (3) is taken literally, it predicts a specific residence-time distribution for the milling state past the transition; fitting that full distribution, rather than only the average escape rate, would test whether the assumed one-variable additive-noise reduction is the right one.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

What you should know: this paper argues that the well-known hysteresis in the three-A model is not structural bistability but a noise-induced effect near a transcritical bifurcation. That is a worthwhile and genuinely different way of reading the old Couzin et al. results. But the supporting evidence is not yet good enough to close the question.

The paper does several things well. The sweep over noise levels (Fig. 5) is clearly presented, and the observation that hysteresis is strongest at intermediate noise and disappears at high noise is a real, nontrivial result that deserves attention. The authors are also explicit that Eq. 3 is postulated, not derived, and they frame it as a phenomenological model. The conceptual point that hysteresis alone does not imply structural bistability is correct and worth making in this literature.

The soft spots are concentrated in the identification of the bifurcation type. First, the Kramers calculation in Eq. 4 has a sign problem: for r>0 the effective potential U=-r(P^2/2-P^3/3) has U(1)-U(0) = -r/6, not +r/6. So the 'barrier' they put into the escape rate is negative, meaning there is no activated escape from P=0 to P=1; the flow is already down hill. Fitting A*r*exp(-r/3σ^2) to the measured escape rate is therefore not a standard Kramers fit and cannot be taken as evidence of noise-activated crossing. Second, the normal form in Eq. 3 is assumed, and the comparison in Fig. 4D is a two-parameter fit of this same SDE to the fish data, so it cannot independently validate the mechanism. Third, the data in Fig. 4C show a narrow parameter window around z_l* where both states are visited; that is exactly what a subcritical pitchfork with a small bistable region would produce over finite simulation times. To distinguish the two, one would need very long runs to see whether the system eventually converges to a single globally stable state (transcritical) or keeps switching between two true wells (subcritical pitchfork). The paper does not provide that test.

So the central claim remains open. But it is an important question, and the paper is honest enough to warrant engagement. I would send it to peer review with a request for the authors to fix the sign error and add a discriminating simulation or a proper reduction from the microscopic rules. As it stands, the mechanistic attribution outruns the evidence.

Referee Report

3 major / 5 minor

Summary. This paper revisits the three-A model of fish schooling by Couzin et al. and examines the transition from milling to schooling as the alignment zone z_l is varied. Through Monte Carlo simulations, the authors find an abrupt transition near z_l* ≈ 3.1, with apparent bistability in a narrow window, and noise-dependent hysteresis. They propose that these features are signatures of a noisy transcritical bifurcation rather than structural bistability, and introduce a phenomenological stochastic differential equation for the polarization P, dP = rP(1-P)dt + σdW, with r = z_l - z_l*. They fit a Kramers escape-rate formula to the residence-time data and argue that the hysteresis behavior of the fish model matches that of the SDE.

Significance. If the mechanism were established, the paper would resolve a long-standing ambiguity about the origin of hysteresis in a widely used model of collective motion. The manuscript is also valuable for its systematic phase diagram and for characterizing how hysteresis depends on noise level. However, the central identification is not currently established: the SDE in Eq. (3) is posited rather than derived, the Kramers comparison in Eq. (4) has a sign-inconsistent barrier, and the presented observables do not discriminate between a transcritical bifurcation and a narrow subcritical pitchfork. The manuscript therefore requires additional analysis before its central claim is supportable.

major comments (3)

[Section 'Signature of Noisy Transcritical Bifurcation', Eq. (3)] The central mechanism is not derived from the agent-based update rule in Eq. (1). The text explicitly states that a formal proof would require deriving the equations for P and M and that, instead, the authors 'sought a phenomenological model' and 'postulate an effective potential.' Because Eq. (3) is assumed, the agreement of its predictions with fish simulations (bimodal histograms, noise-dependent hysteresis) is not a validation of the mechanism; the escape-rate comparison is a two-parameter fit (Eq. (4) and Fig. 4F), so it is not an independent prediction. Moreover, the bimodal histograms in Fig. 4C and the 0/50/100% schooling fractions in Fig. 4B are also what a subcritical pitchfork with a narrow bistable interval would produce in finite simulation time, especially with σ = 0.01. To establish the transcritical mechanism, the paper needs a test that discriminates between the two normal forms, such as measuring the deterministic drift from simulation data or demonstrating that the apparent bistability vanishes in the infinite-time limit (or persists as genuine two-well behavior).
[Section 'Residence Time and Escape Rate', Eq. (4)] The Kramers formula is applied with an incorrect potential barrier. For U = -r(P^2/2 - P^3/3), one has U(1) - U(0) = -r/6, not +r/6 as stated in the text. For r > 0, P = 1 is the global minimum and P = 0 is a local maximum; there is no potential barrier to escape from P = 0 to P = 1. The expression κ = A r e^{-r/(3σ^2)} is therefore not the standard Kramers escape rate from a metastable well, and the fit in Fig. 4F does not provide evidence supporting the transcritical normal form. This is a load-bearing error because the escape-rate comparison is one of the two quantitative links between the fish simulations and the SDE in Eq. (3).
[Eq. (3) and interpretation of P] The additive noise in Eq. (3) has no reflecting or absorbing boundaries, so P can leave the interval [0,1] where the polarization order parameter defined in Eq. (2) is confined. The Kramers escape-rate calculation in Eq. (4) likewise presumes well-defined local minima of a confining potential; with unbounded diffusion, P = 0 is a singular point rather than a metastable state. This reinforces that Eq. (3) is a coarse phenomenological 'cartoon' (the authors' own characterization is 'phenomenological model') and the paper should specify how the SDE is meant to be interpreted, e.g., with boundaries or as an approximation valid only in the interior of [0,1]. Without such specification, the quantitative agreement in Fig. 4F and the hysteresis loops in Fig. 5D-E are not well-defined predictions.

minor comments (5)

[Section heading, p. 7] The section heading 'T ransition from Milling to Schooling' contains a stray space before 'r'.
[Reference 51] Reference 51 (Kramers, Nature 1924) is the wrong citation for the escape-rate formula used in Eq. (4); the standard reference is Kramers, Physica 7, 284-304 (1940).
[Fig. 4D caption] The caption calls panel D a 'Transcritical bifurcation plot,' but the panel shows Monte Carlo trajectories of the SDE in Eq. (3) in finite time, not a deterministic bifurcation diagram; this labeling could mislead readers into thinking the panel is derived from the fish simulations.
[Fig. 2 caption] The caption states 'In all simulations, noise intensity σ = 0.01,' but the paper later reports results at σ = 0.05 and 0.10; the caption should say 'unless otherwise stated' or specify the default value.
[Reference 12] Reference 12 has a truncated DOI: 'https://doi.org/10.1073/pnas.0711' is incomplete.

Circularity Check

3 steps flagged · score 6.0 of 10

The transcritical mechanism is assumed via a posited normal-form SDE and then 'confirmed' by fitting its own Kramers formula to the escape-rate data; the fish data themselves do not discriminate between transcritical and narrow subcritical-pitchfork bistability.

fitted input called prediction [Residence Time and Escape Rate, Eq. (4), Fig. 4F]
"The proportionality constant A and collective noise intensity σ are obtained by fitting this formula to the numerical simulations in Fig. 4F. With this numerical fit (Fig. 4F, red line), Kramers' formula captures the increase in the transition rate from milling to schooling past the bifurcation point. These findings support our central argument that the bifurcation governing the transition from milling to schooling in Fig. 4 is that of a noisy transcritical bifurcation of the form proposed in (3)."

Equation (4) is derived from the same postulated SDE (3) whose transcritical form is the claim under test. The two free parameters A and σ are fit to the very escape-rate data that the formula is then said to 'capture,' so the agreement is statistically forced rather than an independent prediction. Moreover, for r>0 the potential U = -r(P^2/2 - P^3/3) has a maximum at P=0 and a minimum at P=1, so ΔU = U(1)-U(0) = -r/6, not +r/6; Kramers' formula is applied to escape from an unstable state, not from a metastable well. The fit therefore cannot discriminate a noisy transcritical bifurcation from a narrow structural bistability.
self definitional [Transition from Milling to Schooling, Fig. 4B-C]
"We thus considered the bifurcation point as the value z∗l = 3.1, for which nearly half of the 300 MC simulations converged to schooling while the other half to milling. To underscore that bistability is a phenomenon localized near the bifurcation point, we computed histograms of P and M based on the 300 Monte Carlo simulations. ... Near the transition, both milling and schooling co-exist, as evident from the bimodal histograms in Fig. 4C."

The 'bistability' at the transition is built into the definition of z_l*: choosing the value where half the simulations school and half mill guarantees a 50/50 split and hence bimodal histograms at that point. This observation is therefore not independent evidence for an exchange of stability. A subcritical pitchfork with a narrow bistable interval would produce the same 50/50 point and the same bimodal histograms, so this step does not distinguish the two mechanisms; the apparent 'coexistence at the bifurcation point' is a consequence of the definition, not a diagnostic of transcriticality.

1 more flagged steps

other [Signature of Noisy Transcritical Bifurcation, Eq. (3)]
"we sought a phenomenological model that describes the normal forms that P and M must obey to reproduce qualitatively the transition from milling to schooling reported in Fig. 4A,B. ... which we consider to follow the prototypical stochastic differential equation, dP = rP(1-P)dt + σdW P, (3) Here, we postulate an effective potential function of the form U = −r(P 2/ 2 − P 3/ 3)."

The transcritical normal form is put in by hand: Eq. (3) is posited, not derived from the agent-based update rule (1), and the paper explicitly states that a formal proof would require deriving the P and M equations. The subsequent 'predictions'—the flat potential at r=0, noise-induced bimodality, and hysteresis—are deductive properties of this assumed SDE. Using those properties to 'support' the conclusion that the fish transition is a noisy transcritical bifurcation closes the loop: the mechanism is assumed, and the model's behavior is then cited as corroboration of the assumption.

full rationale

The paper is transparent that Eq. (3) is a phenomenological postulate rather than a reduction of the agent-based model, and much of the fish-simulation phenomenology (abrupt transition, transient milling past z_l*, noise-dependent hysteresis) is real and independently measured. However, the central claim—that the milling-to-schooling transition is a noisy transcritical bifurcation rather than structural bistability—is not established by an independent derivation. The load-bearing chain is: define z_l* as the 50/50 schooling point (which guarantees the observed 'coexistence'), posit the transcritical SDE (3), derive Kramers' escape formula (4) from that same SDE, fit its two free parameters to the escape-rate data, and then treat the fit as support for the transcritical classification. This is a partial circularity: the predicted quantity is generated by the assumed model and fitted to the target data, so it cannot validate the model form. The hysteresis comparison with Eq. (3) is likewise a consistency check of an assumed normal form, not an independent test. No load-bearing self-citation chain was found: refs. [24] and [50] include the present authors, but they are cited for general remarks on numerical approximation and normal forms, not to force the transcritical conclusion. The honest finding is that the mechanism claim is plausible but underdetermined; the derivation chain reduces in part to the assumed normal form and its fitted escape rate, warranting a score of 6 rather than 0.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The central argument rests on a postulated normal form (Eq. 3) and a fitted Kramers escape formula, plus a stated assumption that the rotational order M mirrors P. No new physical entities are introduced; the effective potential U is a mathematical reduction, not a new agent or force. Two fit parameters (A, sigma) and the estimated bifurcation point z_l* are used to connect the reduced model to the fish simulations.

free parameters (3)

Collective noise intensity sigma = not reported
In Eq. (4), sigma is obtained by fitting Kramers' formula to the fish simulation escape rates (Fig. 4F); this is a collective-level parameter distinct from the fish-level noise in Table 1.
Kramers prefactor A = not reported
Fitted along with sigma to the escape-rate data; absorbs unknown time-scale and geometric factors in the reduced model.
Bifurcation point z_l* = about 3.1
Estimated as the alignment-zone radius at which roughly half of the 300 Monte Carlo runs converge to schooling (Fig. 4B); sets the origin of the reduced bifurcation parameter r.

assumptions (3)

ad hoc to paper Collective polarization P follows the transcritical normal form dP = rP(1-P)dt + sigma dW (Eq. 3).
Postulated without derivation from the microscopic update rule (Eq. 1); the central conclusion that the fish model undergoes a noisy transcritical bifurcation rests entirely on this assumed normal form.
domain assumption Rotational order M is a mirror of P, so a one-variable reduced description suffices.
The paper states that the behavior of M is nearly a mirror opposite of P, but does not quantify the quality of this reduction or justify ignoring additional degrees of freedom.
domain assumption Kramers escape-rate formula applies to the milling-to-schooling escape for r>0.
Kramers' formula describes barrier crossing from a metastable well; for r>0 in Eq. (3) the state P=0 is a maximum, not a well, so the formula's applicability is questionable. The barrier is also assigned the wrong sign: U(1)-U(0) = -r/6, not +r/6.

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Cite this review

Pith. "Pith review of Noise-Induced Collective Memory in Schooling Fish." pith.science (2026). https://pith.science/paper/N7MKS4VR

@misc{pith2026250716102,
  author       = {Pith},
  title        = {Pith review of: Noise-Induced Collective Memory in Schooling Fish},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N7MKS4VR}},
  note         = {Machine review of arXiv:2507.16102}
}

read the original abstract

Schooling fish often self-organize into a variety of collective patterns, from polarized schooling to rotational milling. Mathematical models support the emergence of these large-scale patterns from local decentralized interactions, in the absence of individual memory and group leadership. In a popular model where individual fish interact locally following rules of avoidance, alignment, and attraction, the group exhibits collective memory: changes in individual behavior lead to emergent patterns that depend on the group's past configurations. However, the mechanisms driving this collective memory remain obscure. Here, we combine numerical simulations with tools from bifurcation theory to uncover that the transition from milling to schooling in this model is driven by a noisy transcritical bifurcation where the two collective states intersect and exchange stability. We further show that key features of the group dynamics - the bifurcation character, transient milling, and collective memory - can be captured by a phenomenological model of the group polarization. Our findings demonstrate that collective memory arises from a noisy bifurcation rather than from structural bistability, thus resolving a long-standing ambiguity about its origins and contributing fundamental understanding to collective phase transitions in a prevalent model of fish schooling.

Figures

Figures reproduced from arXiv: 2507.16102 by the authors.

**Figure 1.** Two possible mechanisms for the transition between milling and schooling as represented by the dependence of group-level polarization P on individual-level parameter zl: A. Subcritical pitchfork bifurcation: P admits two equilibria, P = 1, which persists for all zl and P = 0, which exist only for zl below the bifurcation point z ∗ l . As a result, the system is bistable for zl < z∗ l and becomes monostable at zl > z… view at source ↗

**Figure 2.** Collective behavior of the fish group: A. Fragmentation at zl = 1.0 and za = 4.0, B. Swarming at zl = 1.0 and za = 10.0, C. Milling at zl = 2.0 and za = 10.0, D. Schooling at zl = 4.0 and za = 10.0. E. Phase diagram as a function of alignment zone zl and attraction zone za, evaluated at steady-state using the group polarization P and rotation M, identified with the two-dimensional color map to the right. Black marke… view at source ↗

**Figure 3.** A. Time evolution of P and M indicates stable milling at zl = 2, bistable behavior at the transition point z ∗ l = 3.1, where in one realization, the group remains in the milling state while in another realization, it transitions to schooling, and schooling at zl = 4 after a short transience in the milling state. B. Snapshots showing that as time evolves, the group transitions from milling to schooling. Parameter va… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: A. P and M as a function of zl: at each zl, P and M values at steady state are averaged over 300 Monte Carlo simulations, with initial conditions randomly sampled in a three-dimensional unit cube. Behavior is bistable near the transition z ∗ l = 3.1. B. Percentage of M…

**Figure 5.** Figure 5: Hysteresis plot at noise level A. σ = 0.01, B. σ = 0.05, C. σ = 0.10. Each value is the average of 20 Monte Carlo simulations, za = 10, ∆zl = 0.25. Each value of zl is run for 400 time units before it is increased (black line) or decreased (blue line). D. Hysteresis pl…

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