Emergent Behaviour Across Disciplines

Narrative

The 2020s have seen cellular automata transformed from theoretical curiosities into differentiable, learnable systems. Mordvintsev et al.'s Growing Neural Cellular Automata (Distill, 2020) established the paradigm of training local cell-update rules to produce global morphogenetic targets, generating self-regenerating patterns whose robustness to perturbation mirrors biological tissue. Subsequent NeurIPS 2022 work introduced attention mechanisms into the NCA architecture, and a 2025 arXiv paper applies NCAs to the ARC-AGI benchmark to expose hard limits on decentralised computation relative to large language models. Bert Wang-Chak Chan's Lenia (arXiv, 2018) independently expanded the CA frontier by introducing continuous space, time and state, cataloguing over 400 self-organising lifeform species and demonstrating formal equivalence with generalised reaction-diffusion systems. The Flow-Lenia extension (arXiv, 2022 and 2025) adds mass conservation and localised update-rule parameters, enabling multi-species co-existence and open-ended evolutionary dynamics.

On the reaction-diffusion front, the Gray-Scott model remains the canonical demonstration of pattern self-organisation. Recent arXiv work restores thermodynamic consistency to the classical model, revealing how energy constraints alter pattern regimes and connecting the model to physically grounded biology. A 2024 review extends Turing instability theory to networked and higher-order topologies, showing pattern formation across biological, economic and linguistic systems and arguing for unified mathematical treatment. The 2023 MDPI paper on Gray-Scott spatiotemporal dynamics documents empirically the full map of spot, stripe, mitosis and soliton regimes that emerge from variation in the feed-kill parameter pair.

In the physics of emergence, a 2021 Nature Communications paper engineered self-organised criticality into E. coli gene networks, directly linking SOC theory to living cells. A 2022 arXiv review by Romanczuk synthesises evidence that animal collectives (fish schools, bird flocks) operate near pseudo-critical points to optimise collective computation, while a 2025 arXiv paper demonstrates SOC and sharp phase transitions in diverse model ecosystems using directed-percolation universality classes. The strong-vs-weak emergence debate is addressed rigorously in a July 2025 arXiv survey and in Carlos Gershenson's March 2025 npj Complexity article, which maps self-organisation across a dozen disciplines including social science, ecology and AI.

The complexity-economics strand is anchored by W. Brian Arthur's 2021 Nature Reviews Physics article and the SFI Press 2020 proceedings volume, which together frame markets as non-equilibrium emergent systems driven by heterogeneous agents with bounded rationality. A 2025 Journal of Economic Interaction and Coordination review surveys ABM advances for complex multi-sectoral economic modelling, while an October 2025 SFI report documents efforts to encode agents with LLM priors to produce policy-grade ABMs. Critical scrutiny comes from a 2025 arXiv review of generative social simulations that finds validation largely absent and LLM opacity problematic for isolating emergent causal mechanisms, reflecting a live tension between the descriptive power of complexity metaphor and the rigour required for genuine predictive gain.