Stability and Geometry of Attractors in Neural Cellular Automata

2026-04-14

AI Summary

This study investigates the types, stability, and perturbation responses of attractors learned by Neural Cellular Automata (NCA). Focusing on a deterministically updated growing gecko NCA as a case study, the work integrates dynamical systems theory to visualize attractor dynamics for the first time. Through systematic analysis—including Lyapunov spectrum computation, Fourier spectral analysis, long-term evolution simulations, and perturbation experiments—the research reveals that NCAs not only learn fixed-point attractors but also exhibit oscillatory, periodic, and quasiperiodic behaviors, which emerge early in training. Moreover, large perturbations can induce transitions to secondary patterns outside the original attractor basin. These findings offer novel insights and analytical tools for understanding the long-term dynamics and robustness of NCAs.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Evolutionary Learning
• Multiagent Systems: Agent-Based Simulation and Emergent Behavior

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Social networks and social learning
• Social Networks and Social Media: Applications and emergent phenomena

Abstract

Throughout the literature on Neural Cellular Automata (NCAs), it is often taken for granted that the systems learn attractors. This is shown through evolving the system for many timesteps and noting visual similarity to the goal state. There remain many questions after such an analysis. Namely, what kind of attractors do we have? Is their behavior ordered or chaotic? Can we estimate stability over very long time horizons? What really happens in the attractor when perturbations are applied? In this paper, we present a case study to help answer these questions, with methods drawn from the literature on dynamical systems theory. We use the growing gecko NCA of Mordvintsev et al. (2020) with deterministic cell updates as a case study. To the best of the authors' knowledge, we present the first visualizations of NCA attractor dynamics. We also analyze them using the Lyapunov and Fourier spectra, to reveal that the NCA displays oscillatory, periodic and quasi-periodic behavior, and that these behaviors arise early during training. This challenges the belief that NCAs learn fixed point attractors. Finally, we show that large perturbations to the attractor states can throw the NCAs into a secondary mode separate from the original attractor. We hope that this initial foray into NCA attractor dynamics expands the toolkit for NCA researchers to analyze the robustness and stability of their systems.

Problem

• Neural Cellular Automata
• attractors
• stability
• dynamical systems
• perturbations

Innovation

• Neural Cellular Automata
• attractor dynamics
• Lyapunov spectrum
• Fourier spectrum
• stability analysis