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.

翻译：从亚细胞结构到整个生物体，许多自然系统通过自组织生成复杂组织：即局部相互作用在没有结果蓝图的情况下集体产生全局结构。然而，驱动此类过程的大部分信息并非由自组织本身产生，而是常被转移到系统的初始条件中。生物发育是一个典型例子，其中母体预模式编码了位置和对称破缺信息，为自组织过程提供了支架。从早期胚胎发生中的母体形态素梯度到引导器官形成的组织水平形态发生预模式，这种将信息转移至初始条件的过程（类似于计算系统中的存储-计算权衡）是发育过程的基本组成部分。在本研究中，我们通过引入一个共同学习自组织规则和预模式的模型来研究这种转移现象，从而在受控条件下测量和调节两者间的相互作用：该模型由神经细胞自动机（NCA）与基于坐标的学习模式生成器（SIREN）配对组成，两者同时训练以生成一组模式。我们提供了信息论分析，阐明信息如何在预模式与自组织过程之间分布，并表明共同学习这两个组成部分在鲁棒性、编码能力和对称破缺方面优于纯自组织替代方案。我们的分析进一步表明，有效的预模式并非简单近似目标；相反，它们以促进收敛的方式偏置发育动力学，揭示了初始条件结构与自组织动态之间的非平凡关系。