Neural Cellular Automata (NCA) models have shown remarkable capacity for pattern formation and complex global behaviors stemming from local coordination. However, in the original implementation of NCA, cells are incapable of adjusting their own orientation, and it is the responsibility of the model designer to orient them externally. A recent isotropic variant of NCA (Growing Isotropic Neural Cellular Automata) makes the model orientation-independent - cells can no longer tell up from down, nor left from right - by removing its dependency on perceiving the gradient of spatial states in its neighborhood. In this work, we revisit NCA with a different approach: we make each cell responsible for its own orientation by allowing it to “turn” as determined by an adjustable internal state. The resulting Steerable NCA contains cells of varying orientation embedded in the same pattern. We observe how, while Isotropic NCA are orientation-agnostic, Steerable NCA have chirality: they have a predetermined left-right symmetry. We therefore show that we can train Steerable NCA in similar but simpler ways than their Isotropic counterpart by (1) breaking symmetries using only two seeds, or (2) introducing a rotation-invariant training objective and relying on asynchronous cell updates to break the up-down symmetry of the system.

Modeling the ability of multicellular organisms to build and maintain their bodies through local interactions between individual cells (morphogenesis) is a long-standing challenge of developmental biology. Recently, the Neural Cellular Automata (NCA) model was proposed as a way to find local system rules that produce a desired global behaviour, such as growing and persisting a predefined target pattern, by repeatedly applying the same rule over a grid starting from a single cell. In this work, we argue that the original Growing NCA model has an important limitation: anisotropy of the learned update rule. This implies the presence of an external factor that orients the cells in a particular direction. In other words, “physical” rules of the underlying system are not invariant to rotation, thus prohibiting the existence of differently oriented instances of the target pattern on the same grid. We propose a modified Isotropic NCA (IsoNCA) model that does not have this limitation. We demonstrate that such cell systems can be trained to grow accurate asymmetrical patterns through either of two methods: ( 1 ) by breaking symmetries using structured seeds or ( 2 ) by introducing a rotation-reflection invariant training objective and relying on symmetry-breaking caused by asynchronous cell updates.

Cellular Automata have intrigued curious minds for the better part of the last century, with significant contributions to their field from the likes of Von Neumann et al. (1966), John Conway (Gardner (1970)), and Wolfram and Gad-el Hak (2003). They can simulate and model phenomena in biology, chemistry, and physics (Chopard and Droz (1998)). Recently, Neural Cellular Automata (NCA) have demonstrated a capacity to learn complex behaviour, including constructing a target morphology (Mordvintsev et al. (2020)), classifying the shape they occupy (Randazzo et al. (2020)), or segmentation of images (Sandler et al. (2020)). As a computational model, NCA have appealing properties. They are parallelisable, fault tolerant and partially robust to operating on manifolds other than those used during training. A strong parallel exists between training NCA and system identification of a partial differential equation (PDE) satisfying certain boundary value conditions. In the original work by Mordvintsev et al. (2020), asynchronicity in cell updates is justified by a desire to have purely local communication between cells. We demonstrate that asynchronicity is not just an ideological feature of the model and is in fact necessary to learn a well-behaved PDE and to allow the model to be used in arbitrary integrators.

Reaction-Diffusion (RD) systems provide a computational framework that governs many pattern formation processes in nature. Current RD system design practices boil down to trial-and-error parameter search. We propose a differentiable optimization method for learning the RD system parameters to perform example-based texture synthesis on a 2D plane. We do this by representing the RD system as a variant of Neural Cellular Automata and using task-specific differentiable loss functions. RD systems generated by our method exhibit robust, non-trivial “life-like” behavior.

Self-replication is a fundamental skill present in every living system. Successful living systems must be able to produce offspring that is both capable of performing a set of required tasks and of producing offspring with these same two properties, which we define as fertility. Moreover, species generally produce offspring as a fertile variation of its parents. Despite the widespread use of deep learning and neural networks in industry and academia over the last decades, self-replication with neural networks still remains largely unexplored. In this paper we train neural networks capable of encoding specific images and of producing fertile offspring with and without variation for arbitrary lengths of a genealogy. We accomplish stable self-replication by creating contractions in the parameter space of the self-replication function, and train replication with meaningful variation to give the agents a possibility of escaping these contractions in the search for other configurations that do not diverge to chaotic behaviours.