Generates structured synthetic sequences with NCAs for a training phase before natural-language pretraining. Studies downstream language modelling and reasoning, including how cellular dynamics and transferred attention layers affect the benefit.
@misc{web-2603-10055,
author = {Dan Lee and Seungwook Han and Akarsh Kumar and Pulkit Agrawal},
title = {{Training Language Models via Neural Cellular Automata}},
year = {2026},
howpublished = {arXiv (preprint)},
url = {https://arxiv.org/abs/2603.10055},
eprint = {2603.10055},
archivePrefix = {arXiv}
}
Ehsan Pajouheshgar; Yitao Xu; Ali Abbasi; Alexander Mordvintsev; Wenzel Jakob; Sabine Süsstrunk
Pairs an NCA operating on a coarse grid with a lightweight implicit decoder that converts local cell states and coordinates into appearance. Enables arbitrary-resolution rendering while retaining cellular growth, regeneration and dynamics across grid and mesh domains.
@inproceedings{web-2506-22899,
author = {Ehsan Pajouheshgar and Yitao Xu and Ali Abbasi and Alexander Mordvintsev and Wenzel Jakob and Sabine Süsstrunk},
title = {{Neural Cellular Automata: From Cells to Pixels}},
year = {2026},
booktitle = {SIGGRAPH 2026},
url = {https://arxiv.org/abs/2506.22899},
eprint = {2506.22899},
archivePrefix = {arXiv}
}
Learns a shared local cellular rule as a time-stepping surrogate for partial differential equations. Tests long-horizon prediction on heat, advection, Burgers, Allen–Cahn and Fisher–KPP dynamics against other learned solvers.
Analyses the features represented in hidden NCA channels and transfers a teacher’s internal states to a student during early training. Evaluates few-shot classification and scale generalisation on MNIST, separating feature extraction from collective label agreement.
Uses image-guided NCA updates to repair imperfect masks produced by other segmentation models. Learns to reconnect fragmented structures and remove loose regions, improving topological consistency for retinal vessels and myocardium.
@inproceedings{web-rnca2026,
author = {Malte Silbernagel and Albert Alonso and Jens Petersen and Bulat Ibragimov and Marleen de Bruijne and Madeleine K. Wyburd},
title = {{RNCA: Self-Repairing Segmentation Masks}},
year = {2026},
booktitle = {Medical Imaging with Deep Learning (MIDL 2026), PMLR 315},
url = {https://proceedings.mlr.press/v315/silbernagel26a.html}
}
Introduces adaptive neural cellular automata with input-dependent perceptive fields based on deformable convolutions. Evaluates iterative spatial reasoning on Sudoku, image-based Sudoku and shortest-path maze tasks using compact models with inspectable communication patterns.
Reviews neural cellular automaton architectures, training and applications using a common modular notation. Introduces NCAtorch as an open-source reference implementation for reproducible experiments.
@article{web-2604-24990,
author = {Martin Spitznagel and Janis Keuper},
title = {{A New Kind of Network? Review and Reference Implementation of Neural Cellular Automata}},
year = {2026},
journal = {Transactions on Machine Learning Research (TMLR)},
url = {https://arxiv.org/abs/2604.24990},
eprint = {2604.24990},
archivePrefix = {arXiv}
}
Nick Lemke; Niklas Ihm; John Kalkhof; Mirko Konstantin; Henry J. Krumb; Daniel M. Lang; Ehsan Pajouheshagar; Ario Sadafi; Arjan Kuijper; Karim Lekadir; Marco Lorenzi; Carsten Marr; Julia A. Schnabel; Anirban Mukhopadhyay
Surveys NCA architectural variations and practical applications, with particular attention to medical image analysis. Discusses efficiency, robustness, segmentation, registration, depth estimation and opportunities for further research.
@misc{web-2609-24595,
author = {Nick Lemke and Niklas Ihm and John Kalkhof and Mirko Konstantin and Henry J. Krumb and Daniel M. Lang and Ehsan Pajouheshagar and Ario Sadafi and Arjan Kuijper and Karim Lekadir and Marco Lorenzi and Carsten Marr and Julia A. Schnabel and Anirban Mukhopadhyay},
title = {{Applications of Neural Cellular Automata: State of the Art, Challenges and Opportunities}},
year = {2026},
howpublished = {arXiv (preprint)},
url = {https://arxiv.org/abs/2609.24595},
eprint = {2609.24595},
archivePrefix = {arXiv}
}
Introduces domain-specific adaptation layers for an NCA backbone used in continual hippocampus segmentation. Freezes the shared backbone after initial training and learns compact additions for new domains to limit forgetting and retraining costs.
Combines neural cellular automata with diffusion models for skin lesion segmentation. Multilevel refinement and spatial and channel attention aim to reduce model size while retaining segmentation quality.
Reviews neural cellular automata for biological and bioengineering applications. Discusses learned local interactions as models of development, regeneration and adaptive behaviour across biological scales.
Separates visible cell states from private memory channels to model decentralised memory transfer. Interacting automata encode, propagate and grow different morphologies from shared internal information.
@misc{AUAEA9N8,
author = {Etienne Guichard and Felix Reimers and Mia Kvalsund and Mikkel Lepperød and Stefano Nichele},
title = {{EngramNCA: a Neural Cellular Automaton Model of Memory Transfer}},
year = {2025},
howpublished = {arXiv (preprint)},
url = {https://arxiv.org/abs/2504.11855},
eprint = {2504.11855},
archivePrefix = {arXiv}
}
John Kalkhof; Niklas Ihm; Tim Köhler; Bjarne Gregori; Anirban Mukhopadhyay
Extends the Med-NCA and M3D-NCA line of work to a broader evaluation across eight anatomies and multiple imaging modalities. Studies compact segmentation models, robustness, quality control and tools for visualising cellular inference.
@article{web-10-1016-j-media-2025-103601,
author = {John Kalkhof and Niklas Ihm and Tim Köhler and Bjarne Gregori and Anirban Mukhopadhyay},
title = {{MED-NCA: Bio-inspired medical image segmentation}},
year = {2025},
journal = {Medical Image Analysis},
url = {https://doi.org/10.1016/j.media.2025.103601},
doi = {10.1016/j.media.2025.103601}
}
Applies gradient-trained cellular update rules to abstract grid transformations from ARC-AGI. Evaluates few-shot generalisation and examines how architectural and training choices affect precise transformations of input grids.
Presents a modular JAX library for hardware-accelerated cellular automaton research. Supports discrete, continuous and neural automata, with examples covering growth, classification and other emergent behaviours.
Uses internally encoded genomic signals to generate several textures with a single compact NCA. Studies interpolation, coexistence and editing of learned textures while preserving regeneration after damage.
@article{web-10-1038-s41598-025-23997-7,
author = {Mirela-Magdalena Catrina and Ioana Cristina Plajer and Alexandra Băicoianu},
title = {{Multi-texture synthesis through signal responsive neural cellular automata}},
year = {2025},
journal = {Scientific Reports},
url = {https://doi.org/10.1038/s41598-025-23997-7},
doi = {10.1038/s41598-025-23997-7}
}
Alex D. Richardson; Tibor Antal; Richard A. Blythe; Linus J. Schumacher
Learns local update rules from image time series and partial differential equation trajectories. Extends NCA modelling to transient and stable spatial patterns, including Turing pattern dynamics.
@misc{YE2PXS46,
author = {Alex D. Richardson and Tibor Antal and Richard A. Blythe and Linus J. Schumacher},
title = {{Learning spatio-temporal patterns with Neural Cellular Automata}},
year = {2024},
howpublished = {arXiv (preprint)},
url = {https://arxiv.org/abs/2310.14809},
eprint = {2310.14809},
archivePrefix = {arXiv}
}
Moves neural cellular automaton updates into the latent space of a pretrained autoencoder. Applies this approach to image restoration to lower computational requirements while preserving reconstruction quality.
Generalises cellular updates to vertices of triangle meshes to synthesise dynamic surface textures without UV maps. Supports image, text and motion supervision, generalisation to unseen meshes and interactive texture grafting.
Investigates whether texture NCAs learn continuous dynamics or overfit their training discretisation. Uses noisy initial states to improve consistency across spatial resolutions and time steps, enabling continuous controls over pattern scale and growth speed.
Uses neural cellular automata as compact diffusion-model backbones. Adds Fourier-based diffusion to support communication over larger spatial distances, with image generation, inpainting and super-resolution examples.
Introduces generative cellular automata inspired by growth from a single cell. Evaluates cellular image synthesis as data augmentation for retinal disease classification in fundus and OCT images.
@inproceedings{WPCQB3ZE,
author = {Marawan Elbatel and Konstantinos Kamnitsas and Xiaomeng Li},
title = {{An Organism Starts with a Single Pix-Cell: A Neural Cellular Diffusion for High-Resolution Image Synthesis}},
year = {2024},
booktitle = {MICCAI 2024},
url = {https://arxiv.org/abs/2407.03018},
eprint = {2407.03018},
archivePrefix = {arXiv}
}
Paul Seibert; Alexander Raßloff; Yichi Zhang; Karl Kalina; Paul Reck; Daniel Peterseim; Markus Kästner
Trains an NCA to reconstruct material microstructures from statistical descriptors, which can be obtained from a single reference. Separates training cost from output size so that large or repeated reconstructions can be generated efficiently.
@article{web-10-1007-s40192-023-00335-1,
author = {Paul Seibert and Alexander Raßloff and Yichi Zhang and Karl Kalina and Paul Reck and Daniel Peterseim and Markus Kästner},
title = {{Reconstructing Microstructures From Statistical Descriptors Using Neural Cellular Automata}},
year = {2024},
journal = {Integrating Materials and Manufacturing Innovation},
url = {https://doi.org/10.1007/s40192-023-00335-1},
doi = {10.1007/s40192-023-00335-1}
}
Studies how the number of state channels and hidden neurons in the update network relate to emergent motion. Proposes an architectural design principle for generating dynamic rather than stationary NCA patterns.
Introduces DyNCA for real-time synthesis of dynamic textures with controllable motion. Learns appearance and motion from separate targets and supports arbitrary video duration and frame size after training.
Allows each cell to adjust its own orientation through an internal state. Studies how this steerable perception supports pattern growth, rotational symmetry breaking and chirality.
Uses neural cellular automata to predict how metal microstructures develop during solidification. Learned local updates provide a data-driven alternative to conventional cellular automaton simulations.
@article{4XBBPM8D,
author = {Jian Tang and Siddhant Kumar and Laura De Lorenzis and Ehsan Hosseini},
title = {{Neural cellular automata for solidification microstructure modelling}},
year = {2023},
journal = {Computer Methods in Applied Mechanics and Engineering},
url = {https://doi.org/10.1016/j.cma.2023.116197},
doi = {10.1016/j.cma.2023.116197}
}
Uses multilevel patchification to apply compact NCAs to three-dimensional medical image segmentation. Derives a quality metric from prediction variability to identify segmentation errors without relying solely on a final mask.
Introduces a two-stage NCA pipeline that communicates global context at low resolution before segmenting high-resolution patches. Evaluates compact models for hippocampus and prostate segmentation, including robustness and deployment on limited hardware.
Explores hierarchical neural cellular automata as models of organisation across multiple scales. Investigates how locally interacting units can form higher-level, goal-directed structures.
Introduces isotropic neural cellular automata whose update rules do not depend on a globally shared orientation. Uses structured seeds or rotation- and reflection-invariant training objectives to learn asymmetric patterns.
Uses an NCA as a hypernetwork that grows the weights of a neural-network controller. Evaluates the resulting developmental networks on reinforcement-learning tasks and explores transformations between related tasks.
Uses attention to parameterise interactions between neighbouring cells. Explores attention-based local update rules as an alternative to convolution-based neural cellular automata for pattern generation.
Combines a variational latent representation with a cellular growth process to generate diverse images. Studies reconstruction, probabilistic generation and recovery of learned stable patterns after damage.
Introduces goal encodings that steer cellular behaviour throughout growth rather than fixing a single target in advance. Investigates changing goals and maintaining task performance when only some cells receive the control signal.
@inproceedings{web-2205-06806,
author = {Shyam Sudhakaran and Elias Najarro and Sebastian Risi},
title = {{Goal-Guided Neural Cellular Automata: Learning to Control Self-Organising Systems}},
year = {2022},
booktitle = {ICLR 2022 — From Cells to Societies workshop},
url = {https://arxiv.org/abs/2205.06806},
eprint = {2205.06806},
archivePrefix = {arXiv}
}
Sidney Pontes-Filho; Kathryn Walker; Elias Najarro; Stefano Nichele; Sebastian Risi
Presents a neural cellular robot substrate in which a single NCA develops a modular robot’s body and controls its activity. Uses evolutionary optimisation and quality-diversity search to explore robot morphologies across benchmark tasks.
Extends learned cellular automaton rules from regular grids to arbitrary graphs using graph neural networks. Evaluates rule learning on irregular tessellations, flocking behaviour and convergence to target states.
@inproceedings{TZVDTWZ7,
author = {Daniele Grattarola and Lorenzo Livi and Cesare Alippi},
title = {{Learning graph cellular automata}},
year = {2021},
booktitle = {Advances in Neural Information Processing Systems},
url = {https://proceedings.nips.cc/paper_files/paper/2021/hash/af87f7cdcda223c41c3f3ef05a3aaeea-Abstract.html}
}
Ettore Randazzo; Alexander Mordvintsev; Eyvind Niklasson; Michael Levin
Investigates how crafted perturbations can redirect the behaviour of a trained neural cellular automaton. Demonstrates reprogramming of self-organising dynamics without changing the learned update rule.
@article{PQJYUSSF,
author = {Ettore Randazzo and Alexander Mordvintsev and Eyvind Niklasson and Michael Levin},
title = {{Adversarial Reprogramming of Neural Cellular Automata}},
year = {2021},
journal = {Distill},
url = {https://distill.pub/selforg/2021/adversarial/}
}
Eyvind Niklasson; Alexander Mordvintsev; Ettore Randazzo; Michael Levin
Trains neural cellular automata to synthesise textures through repeated local updates. Explores texture formation as a self-organising process that can run on grids of different sizes.
@article{IAFV8T4H,
author = {Eyvind Niklasson and Alexander Mordvintsev and Ettore Randazzo and Michael Levin},
title = {{Self-Organising Textures}},
year = {2021},
journal = {Distill},
url = {https://distill.pub/selforg/2021/textures/}
}
Models crystal surface growth with cellular automata incorporating Ehrlich–Schwoebel barriers. Explores the emergence of step bunches, nanowires and other surface structures.
@article{NC5R9TSU,
author = {Magdalena Załuska-Kotur and Hristina Popova and Vesselin Tonchev},
title = {{Step bunches, nanowires and other vicinal “creatures”—ehrlich–schwoebel effect by cellular automata}},
year = {2021},
journal = {Crystals},
url = {https://www.mdpi.com/2073-4352/11/9/1135}
}
Maximilian Otte; Quentin Delfosse; Johannes Czech; Kristian Kersting
Introduces GANCA, combining neural cellular automata with adversarial training. A single automaton generates different outputs from different initial conditions and is evaluated on inputs beyond its training distribution.
Extends neural cellular automata to three-dimensional growth using Minecraft as an experimental environment. Demonstrates construction of complex structures and regeneration of parts of functional moving machines.
@inproceedings{web-2103-08737,
author = {Shyam Sudhakaran and Djordje Grbic and Siyan Li and Adam Katona and Elias Najarro and Claire Glanois and Sebastian Risi},
title = {{Growing 3D Artefacts and Functional Machines with Neural Cellular Automata}},
year = {2021},
booktitle = {ALIFE 2021},
url = {https://arxiv.org/abs/2103.08737},
eprint = {2103.08737},
archivePrefix = {arXiv}
}
Alexander Mordvintsev; Ettore Randazzo; Eyvind Niklasson; Michael Levin
Trains a shared neural update rule to grow a target image from a single seed cell. Explores persistent patterns and regeneration after damage as models of morphogenesis.
@article{Y9LPUEMP,
author = {Alexander Mordvintsev and Ettore Randazzo and Eyvind Niklasson and Michael Levin},
title = {{Growing Neural Cellular Automata}},
year = {2020},
journal = {Distill},
url = {https://distill.pub/2020/growing-ca/}
}
Ettore Randazzo; Alexander Mordvintsev; Eyvind Niklasson; Michael Levin; Sam Greydanus
Trains cells to recognise handwritten digits using only local communication. Digit labels emerge across the grid as cells exchange information through repeated neural updates.
@article{6FXKQCRS,
author = {Ettore Randazzo and Alexander Mordvintsev and Eyvind Niklasson and Michael Levin and Sam Greydanus},
title = {{Self-classifying MNIST Digits}},
year = {2020},
journal = {Distill},
url = {https://distill.pub/2020/selforg/mnist/}
}
Mark Sandler; Andrey Zhmoginov; Liangcheng Luo; Alexander Mordvintsev; Ettore Randazzo; et al.
Formulates image segmentation as an iterative cellular process. Cells exchange local information and learn update rules that collectively produce pixel-level segmentation predictions.
@misc{CXNI728H,
author = {Mark Sandler and Andrey Zhmoginov and Liangcheng Luo and Alexander Mordvintsev and Ettore Randazzo and others},
title = {{Image segmentation via cellular automata}},
year = {2020},
howpublished = {arXiv preprint arXiv:2008.04965},
url = {https://arxiv.org/abs/2008.04965},
eprint = {2008.04965},
archivePrefix = {arXiv}
}
Connects cellular automaton rules with convolutional neural network representations. Demonstrates how convolutional models can reproduce and learn local cellular dynamics.
Introduces convolution and region-of-interest pooling with learned spatial offsets. These modules adapt their sampling locations to image geometry rather than using a fixed sampling grid.
@misc{54PCYE7M,
author = {Jifeng Dai and Haozhi Qi and Yuwen Xiong and Yi Li and Guodong Zhang and Han Hu and Yichen Wei},
title = {{Deformable Convolutional Networks}},
year = {2017},
howpublished = {arXiv (preprint)},
url = {https://arxiv.org/abs/1703.06211},
eprint = {1703.06211},
archivePrefix = {arXiv}
}
Haralampos Hatzikirou; David Basanta; Matthias Simon; Karl Schaller; Andreas Deutsch
Tests whether switching between migration and proliferation, together with oxygen shortage, can explain invasive tumour growth. Uses a lattice-gas cellular automaton to investigate the “go or grow” hypothesis.
@article{75KEELR5,
author = {Haralampos Hatzikirou and David Basanta and Matthias Simon and Karl Schaller and Andreas Deutsch},
title = {{‘Go or grow’: the key to the emergence of invasion in tumour progression?}},
year = {2012},
journal = {Mathematical medicine and biology: a journal of the IMA},
url = {https://doi.org/10.1093/imammb/dqq011},
doi = {10.1093/imammb/dqq011}
}
Provides an introduction to cellular automata, their dynamics and their use in modelling complex systems. Covers one- and two-dimensional rules, chaos and patterns arising from simple local interactions.
@book{6RVYVQUP,
author = {Joel L Schiff},
title = {{Cellular automata: a discrete view of the world}},
year = {2011},
publisher = {John Wiley \& Sons},
url = {https://doi.org/10.1002/9781118032381},
doi = {10.1002/9781118032381}
}
Examines mathematical descriptions of seashell geometry and pigmentation. Connects shell growth and visible patterns with simple generative and pattern-forming mechanisms.
@misc{ABYJYDT4,
author = {Stephen Coombes},
title = {{The geometry and pigmentation of seashells}},
year = {2009},
howpublished = {Nottingham: Department of Mathematical Sciences, University of Nottingham},
url = {https://www.maths.nottingham.ac.uk/plp/pmzsc/pdfs/Seashells09.pdf}
}
Proves that the elementary cellular automaton Rule 110 can perform universal computation. Establishes that a simple one-dimensional local rule can support arbitrarily complex computation.
A review of Wolfram’s proposal that simple computational rules can account for complex natural behaviour. The library citation refers to the 2003 Applied Mechanics Reviews entry; the summary source links to the underlying book.
@article{LVMT9RQE,
author = {Stephen Wolfram and M Gad-el-Hak},
title = {{A new kind of science}},
year = {2003},
journal = {Appl. Mech. Rev.},
url = {https://doi.org/10.1115/1.1553433},
doi = {10.1115/1.1553433}
}
Presents a cellular automaton for spatial pattern formation in chemical systems. Local excitation and recovery rules reproduce organised waves and related structures in reaction media.
@article{A39C8XKJ,
author = {Martin Gerhardt and Heike Schuster},
title = {{A cellular automaton describing the formation of spatially ordered structures in chemical systems}},
year = {1989},
journal = {Physica D: Nonlinear Phenomena},
url = {https://doi.org/10.1016/0167-2789(89)90081-X},
doi = {10.1016/0167-2789(89)90081-X}
}
Discusses fibroblasts and biological form through the perspective of cellular automata. Connects local cellular organisation with the modelling of morphogenesis.
Studies cellular automata as discrete models of statistical physics. Examines how local deterministic rules generate ordered, periodic and complex collective behaviour.
@article{NNR5TXP5,
author = {Stephen Wolfram},
title = {{Statistical mechanics of cellular automata}},
year = {1983},
journal = {Reviews of modern physics},
url = {https://doi.org/10.1103/RevModPhys.55.601},
doi = {10.1103/RevModPhys.55.601}
}
Introduces Conway’s Game of Life through its local birth and survival rules. Shows how simple interactions produce moving, oscillating and growing configurations.
@article{RZRACJHH,
author = {Martin Gardner},
title = {{MATHEMATICAL GAMES - The fantastic combinations of John Conway's new solitaire game "life"}},
year = {1970},
journal = {Scientific American},
url = {https://www.ibiblio.org/lifepatterns/october1970.html}
}
Develops a formal account of machines capable of constructing copies of themselves. Its cellular automaton construction connects self-reproduction, universal computation and descriptions of machines.
@book{UI29T3TY,
author = {John Von Neumann and Arthur Walter Burks and others},
title = {{Theory of self-reproducing automata}},
year = {1966},
publisher = {University of Illinois Press},
url = {https://cba.mit.edu/events/03.11.ASE/docs/VonNeumann.pdf}
}
Keuper-Labs