Empowered Neural Cellular Automata

TL;DR

This work extends empowerment, an information-theoretic intrinsic motivation, to neural cellular automata (NCA) and shows that including empowerment alongside morphogenesis improves the ability of NCAs to grow into target shapes. By treating each cell as an agent with a signaling-based sensorimotor loop, the authors maximize the mutual information $\,\mathfrak{E} = I(A_0^{N/2}, S_{N/2}^{N})$ between early actions and later sensor states within development, using multiobjective AFPO to balance loss and empowerment. The results demonstrate a synergistic relationship between morphogenesis and empowerment, with multiobjective treatments producing more accurate shapes and higher empowerment than single-objective baselines, and patterns suggesting broader applicability to development-like processes. These findings point to general, task-independent information-driven coordination strategies potentially relevant to biology and complex coordinated growth in distributed systems.

Abstract

Information-theoretic fitness functions are becoming increasingly popular to produce generally useful, task-independent behaviors. One such universal function, dubbed empowerment, measures the amount of control an agent exerts on its environment via its sensorimotor system. Specifically, empowerment attempts to maximize the mutual information between an agent's actions and its received sensor states at a later point in time. Traditionally, empowerment has been applied to a conventional sensorimotor apparatus, such as a robot. Here, we expand the approach to a distributed, multi-agent sensorimotor system embodied by a neural cellular automaton (NCA). We show that the addition of empowerment as a secondary objective in the evolution of NCA to perform the task of morphogenesis, growing and maintaining a pre-specified shape, results in higher fitness compared to evolving for morphogenesis alone. Results suggest there may be a synergistic relationship between morphogenesis and empowerment. That is, indirectly selecting for coordination between neighboring cells over the duration of development is beneficial to the developmental process itself. Such a finding may have applications in developmental biology by providing potential mechanisms of communication between cells during growth from a single cell to a multicellular, target morphology. Source code for the experiments in this paper can be found at: \url{https://github.com/caitlingrasso/empowered-nca}.

Figures (8)

• Figure 1: (a): Neural cellular automata (NCA) are evolved to grow from an initial seed into a desired target shape (green). The CA's signaling channel (greyscale) can be used to share information between cells but remains constant during simulation. (b) NCA are evolved for empowerment maximize mutual information between cells' past actions (cell's signal) and cells' future sensors (neighboring signals) resulting in a greater diversity of action, sensor pairs and coordination of the CA's signaling throughout space and time. (c) We here report that NCAs evolved for both shape matching and empowerment are better able to match the target shape than those evolved for shape matching alone.
• Figure 2: A single update of the NCA model. The NCA at an arbitrary time step $n$ (a) is comprised of two channels: a binary live/dead channel (b) and a signaling channel (c). At each time step, a neural network (d) is applied sequentially to each cell in the CA. For cell $i,j$, the network takes as input the neighboring cells' live/dead values and signals. The output of the network produces the updated the CA grid at time step $n+1$ (e). In computing empowerment, each cell is considered an agent. The sensor state of cell $i,j$ at time step $n$, $s_{ijn}$, is the average signal of the cell's neighbors (f). Cell $i,j$ produces an action at time step $n$, $a_{ijn}$, which updates its own signal value for neighboring cells to sense (g).
• Figure 3: (top): Loss of the lowest loss NCA in the population at each generation averaged over all 25 runs (95% C.I.) for each AFPO treatment. (bottom): Empowerment of the lowest loss NCA in the population at each generation averaged over all 25 runs (95% C.I.) for each AFPO variation. Empowerment is measured in bits.
• Figure 4: The most lowest loss NCAs resulting from each of the 25 evolutionary runs for each AFPO variation in the loss-empowerment space. Arrows corresponding to each treatment indicate the direction of selection pressure for that treatment. Shown in the top-left in red are the lowest loss NCAs from 25 different populations at generation zero. No evolution occurs on these individuals, thus, there is no corresponding arrow.
• Figure 5: Morphogenesis of the lowest loss NCA over all runs from each treatment starting from the initial seed at $n=0$ to the end of simulation, $n=50$ as it attempts to match the target shape (green). Local mutual information corresponding to each cell's contribution to overall empowerment is depicted with the blue heatmap. Darker blue indications higher mutual information.