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arxiv: 2605.14998 · v2 · pith:4P73ETCZnew · submitted 2026-05-14 · 💻 cs.AI · cs.SY· eess.SY· q-bio.QM

Learning Developmental Scaffoldings to Guide Self-Organisation

Milton L. Montero , Elias Najarro , Jakob Schauser , Sebastian Risi This is my paper

Pith reviewed 2026-05-19 16:42 UTC · model grok-4.3

classification 💻 cs.AI cs.SYeess.SYq-bio.QM
keywords self-organisationneural cellular automatapre-patternsdevelopmental scaffoldingpattern generationSIRENinformation theorysymmetry breaking
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The pith

Jointly learning self-organisation rules and pre-patterns improves robustness, encoding capacity, and symmetry breaking in pattern generation.

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

The paper examines how biological systems offload positional and symmetry-breaking information to initial conditions that then guide self-organising dynamics. It introduces a model pairing a Neural Cellular Automaton with a SIREN network that generates initial pre-patterns, training both components at the same time on target patterns. This joint setup permits direct measurement of how information is split between the starting pre-pattern and the subsequent local interactions. The results indicate gains in robustness to perturbations, greater overall encoding capacity, and more reliable symmetry breaking relative to systems that rely only on self-organisation. The analysis further shows that successful pre-patterns bias the developmental trajectory toward convergence rather than simply copying the final pattern.

Core claim

By jointly optimising a Neural Cellular Automaton for local self-organisation rules and a SIREN for generating coordinate-based pre-patterns, the combined system produces target patterns with higher robustness, larger encoding capacity, and improved symmetry breaking than purely self-organising baselines. Information-theoretic measures quantify the distribution of information between initial conditions and evolving dynamics, revealing that effective pre-patterns steer the self-organising process rather than merely approximating the desired outcome.

What carries the argument

A Neural Cellular Automaton (NCA) paired with a learned coordinate-based pattern generator (SIREN), trained simultaneously to generate a set of patterns.

Load-bearing premise

The NCA and SIREN architectures, when trained jointly on the pattern tasks, produce dynamics and information distributions that reflect broader principles of developmental scaffolding.

What would settle it

An experiment in which joint training yields no measurable gain in robustness or encoding capacity, or in which the learned pre-patterns function mainly by direct approximation of the target, would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.14998 by Elias Najarro, Jakob Schauser, Milton L. Montero, Sebastian Risi.

Figure 1
Figure 1. Figure 1: Model diagram and results examples. a) A pre-pattern function, defined using an implicit representation, transforms coordinates into initial values of the visible channels of the NCA. Development then unfolds using it as initial conditions towards the corresponding target. b) Optimal patterns for a one layer SIREN trained to directly predict the target showing that they are not capable of directly reconstr… view at source ↗
Figure 2
Figure 2. Figure 2: Information trade-off. We measure how information is balanced between the pre-pattern and the final reconstruction using three different measures. Left: We apply a pixel-wise linear probe and compute the R2 of the regression between the both pre and final patterns and the target. The plots show that at the start the pre-pattern is below chance, while at the end of development the patterns are very close to… view at source ↗
Figure 3
Figure 3. Figure 3: Robustness to noise. We compare the NCA with SIREN pre-patterns to a control model (GoalNCA, see main text), which can also generate multiple targets but uses self-organisation only, across multiple levels of reliability after training. a) The results for the NCA with pre-patterns on the left, on the right the control. As we decrease the reliability of the cells, the pre-patterned NCA suffers very little i… view at source ↗
Figure 4
Figure 4. Figure 4: Analysing model capacity. We compare re￾construction error when using pre-patterns generated with SIREN against GoalNCA — a purely self-organising control — as both are tasked with encoding an increasing number of patterns (1 to 16 in powers of two). At comparable total pa￾rameter budgets, the pre-patterned model degrades far more gracefully. This is the memory-compute trade-off in action: parameters that … view at source ↗
Figure 5
Figure 5. Figure 5: Capturing the structure of biological pre-patterns. a) A schematic of differential expression of homeobox genes during early development as a consequence of maternal-effect genes, based on [Schroeder et al.]. b) How a pre-pattern function (in this case SIREN) can generate the pre-patterns. c) The trained model showing 4 different examples where the pre-pattern exhibits the same kind of sinusoidal structure… view at source ↗
read the original abstract

From subcellular structures to entire organisms, many natural systems generate complex organisation through self-organisation: local interactions that collectively give rise to global structure without any blueprint of the outcome. Yet a significant portion of the information driving such processes is not produced by self-organisation itself, instead, it is often offloaded to initial conditions of the system. Biological development is a prime example, where maternal pre-patterns encode positional and symmetry-breaking information that scaffolds the self-organising process. From maternal morphogen gradients in early embryogenesis to tissue-level morphogenetic pre-patterns guiding organ formation, this transfer of information to initial conditions, analogous to a memory-compute trade-off in computational systems, is a fundamental part of developmental processes. In this work, we study this offloading phenomenon by introducing a model that jointly learns both the self-organisation rules and the pre-patterns, allowing their interplay to be varied and measured under controlled conditions: a Neural Cellular Automaton (NCA) paired with a learned coordinate-based pattern generator (SIREN), both trained simultaneously to generate a set of patterns. We provide information-theoretic analyses of how information is distributed between pre-patterns and the self-organising process, and show that jointly learning both components yields improvements in robustness, encoding capacity, and symmetry breaking over purely self-organising alternatives. Our analysis further suggests that effective pre-patterns do not simply approximate their targets; rather, they bias the developmental dynamics in ways that facilitate convergence, pointing to a non-trivial relationship between the structure of initial conditions and the dynamics of self-organisation.

Editorial analysis

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Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces a joint optimization framework pairing a Neural Cellular Automaton (NCA) with a SIREN coordinate-based generator to simultaneously learn self-organization rules and initial pre-patterns for target pattern generation. It performs information-theoretic measurements of how information is partitioned between the learned initial conditions and the update dynamics, and reports that this joint approach improves robustness, encoding capacity, and symmetry breaking relative to purely self-organizing baselines.

Significance. If the empirical gains prove robust and the information-offloading interpretation holds under broader controls, the work supplies a controllable computational testbed for studying developmental scaffolding and memory-compute trade-offs in self-organizing systems. The explicit use of end-to-end differentiable models to vary and quantify the distribution of information between pre-patterns and dynamics is a constructive contribution to both artificial life and developmental biology modeling.

major comments (2)
  1. [Experimental results] Experimental results section: the central claim that joint learning produces improvements in robustness, encoding capacity, and symmetry breaking is supported only by the specific NCA+SIREN pair trained on the chosen pattern set; no architecture-swap ablations or out-of-distribution pattern tests are reported, leaving open the possibility that observed gains arise from the composite inductive bias rather than from the scaffolding principle itself.
  2. [Information-theoretic analysis] Information-theoretic analysis section: the reported measures of information distribution between pre-patterns and dynamics rest on post-training estimates whose sensitivity to the particular choice of entropy estimators or sampling procedure is not quantified, which is load-bearing for the claim that pre-patterns bias dynamics in a non-trivial, facilitative manner rather than simply approximating targets.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'memory-compute trade-off' is introduced without a brief parenthetical reference to the relevant computational literature, which would help readers outside the developmental-biology community.
  2. [Methods] Methods: the precise loss function combining pattern reconstruction and any regularization on the SIREN or NCA parameters is not stated explicitly; adding the equation would improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and insightful comments. We address each major comment below and outline revisions that will strengthen the manuscript while preserving its core contributions on developmental scaffolding via joint optimization of self-organization rules and initial conditions.

read point-by-point responses

  1. Referee: [Experimental results] Experimental results section: the central claim that joint learning produces improvements in robustness, encoding capacity, and symmetry breaking is supported only by the specific NCA+SIREN pair trained on the chosen pattern set; no architecture-swap ablations or out-of-distribution pattern tests are reported, leaving open the possibility that observed gains arise from the composite inductive bias rather than from the scaffolding principle itself.

    Authors: We appreciate this point. The primary experimental contrast is between the jointly trained NCA+SIREN model and purely self-organizing NCA baselines that receive no learned pre-patterns; this isolates the effect of offloading information to initial conditions. We nevertheless agree that additional controls would more convincingly attribute gains to the scaffolding principle rather than the specific inductive biases of the chosen pair. In the revised manuscript we will add architecture-swap ablations (e.g., substituting alternative coordinate-based generators for SIREN) and out-of-distribution pattern evaluations to address this concern. revision: yes

  2. Referee: [Information-theoretic analysis] Information-theoretic analysis section: the reported measures of information distribution between pre-patterns and dynamics rest on post-training estimates whose sensitivity to the particular choice of entropy estimators or sampling procedure is not quantified, which is load-bearing for the claim that pre-patterns bias dynamics in a non-trivial, facilitative manner rather than simply approximating targets.

    Authors: We agree that robustness of the information-theoretic claims requires explicit sensitivity checks. The current results rely on standard post-training entropy estimators applied to the learned distributions. In the revision we will add a dedicated sensitivity analysis that varies both the entropy estimator (including k-nearest-neighbor and histogram-based variants) and sampling procedures. This will confirm that the observed partitioning of information between pre-patterns and dynamics is stable and supports a non-trivial facilitative bias rather than mere target approximation. revision: yes

Circularity Check

0 steps flagged

No circularity: results from empirical joint training and post-hoc information measures

full rationale

The paper's core contribution consists of jointly optimizing an NCA update rule together with a SIREN initial-condition generator on pattern-generation tasks, followed by separate information-theoretic measurements of how information is partitioned between the learned pre-patterns and the dynamics. No equation or claim reduces a prediction to a fitted parameter by construction, nor does any load-bearing step invoke a self-citation chain, uniqueness theorem, or ansatz that is merely renamed. The reported gains in robustness, encoding capacity, and symmetry breaking are measured outcomes of the trained systems rather than tautological re-expressions of the training loss; the work therefore remains self-contained against external empirical benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The approach rests on standard domain assumptions about cellular automata and implicit neural representations; no new physical entities are postulated.

free parameters (1)
  • NCA and SIREN architecture sizes and training hyperparameters
    Chosen to enable joint optimization on the target pattern set; values are not reported in the abstract.
axioms (2)
  • domain assumption Local update rules in a grid can produce global self-organized patterns when iterated.
    Foundational premise of Neural Cellular Automata models invoked throughout the work.
  • domain assumption SIREN networks can represent smooth spatial pre-patterns suitable for initializing developmental dynamics.
    Relies on prior properties of sinusoidal representation networks for coordinate-based function approximation.

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Reference graph

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