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 variant 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.

翻译：神经细胞自动机（NCA）模型展现出了通过局部协调形成复杂全局模式与行为的卓越能力。然而，在原始NCA实现中，细胞无法自主调整朝向，而需由模型设计者外部定向。最新各向同性变体（各向同性生长神经细胞自动机）通过消除对感知邻域空间状态梯度的依赖，使模型具备朝向无关性——细胞不再能辨别上下左右方向。本研究采用不同方法重新审视NCA：通过细胞可根据可调内部状态“转向”的机制，赋予每个细胞自主定向能力。由此产生的可操控NCA包含嵌入同一模式内具有不同朝向的细胞。我们观察到，各向同性NCA对朝向不敏感，而可操控NCA具有手性特征：呈现预定义的左右对称性。因此，我们证明可通过以下两种方式，以比各向同性变体更简洁的方式进行可操控NCA训练：（1）仅使用两种种子打破对称性，或（2）引入旋转不变训练目标并依赖细胞异步更新以打破系统的上下对称性。