Growing Neural Cellular Automata

Contents Learning to Grow What persists, exists Learning to regenerate Rotating the perceptive field

This article is part of the Differentiable Self-organizing Systems Thread, an experimental format collecting invited short articles delving into differentiable self-organizing systems, interspersed with critical commentary from several experts in adjacent fields. Differentiable Self-organizing Systems Thread Self-classifying MNIST Digits

Most multicellular organisms begin their life as a single egg cell - a single cell whose progeny reliably self-assemble into highly complex anatomies with many organs and tissues in precisely the same arrangement each time. The ability to build their own bodies is probably the most fundamental skill every living creature possesses. Morphogenesis (the process of an organism’s shape development) is one of the most striking examples of a phenomenon called self-organisation. Cells, the tiny building blocks of bodies, communicate with their neighbors to decide the shape of organs and body plans, where to grow each organ, how to interconnect them, and when to eventually stop. Understanding the interplay of the emergence of complex outcomes from simple rules and homeostatic Self-regulatory feedback loops trying maintain the body in a stable state or preserve its correct overall morphology under external perturbations feedback loops is an active area of research . What is clear is that evolution has learned to exploit the laws of physics and computation to implement the highly robust morphogenetic software that runs on genome-encoded cellular hardware.

This process is extremely robust to perturbations. Even when the organism is fully developed, some species still have the capability to repair damage - a process known as regeneration. Some creatures, such as salamanders, can fully regenerate vital organs, limbs, eyes, or even parts of the brain! Morphogenesis is a surprisingly adaptive process. Sometimes even a very atypical development process can result in a viable organism - for example, when an early mammalian embryo is cut in two, each half will form a complete individual - monozygotic twins!

The biggest puzzle in this field is the question of how the cell collective knows what to build and when to stop. The sciences of genomics and stem cell biology are only part of the puzzle, as they explain the distribution of specific components in each cell, and the establishment of different types of cells. While we know of many genes that are required for the process of regeneration, we still do not know the algorithm that is sufficient for cells to know how to build or remodel complex organs to a very specific anatomical end-goal. Thus, one major lynch-pin of future work in biomedicine is the discovery of the process by which large-scale anatomy is specified within cell collectives, and how we can rewrite this information to have rational control of growth and form. It is also becoming clear that the software of life possesses numerous modules or subroutines, such as “build an eye here”, which can be activated with simple signal triggers. Discovery of such subroutines and a mapping out of the developmental logic is a new field at the intersection of developmental biology and computer science. An important next step is to try to formulate computational models of this process, both to enrich the conceptual toolkit of biologists and to help translate the discoveries of biology into better robotics and computational technology.

Imagine if we could design systems of the same plasticity and robustness as biological life: structures and machines that could grow and repair themselves. Such technology would transform the current efforts in regenerative medicine, where scientists and clinicians seek to discover the inputs or stimuli that could cause cells in the body to build structures on demand as needed. To help crack the puzzle of the morphogenetic code, and also exploit the insights of biology to create self-repairing systems in real life, we try to replicate some of the desired properties in an in silico experiment.

Model Those in engineering disciplines and researchers often use many kinds of simulations incorporating local interaction, including systems of partial derivative equation (PDEs), particle systems, and various kinds of Cellular Automata (CA).

We will focus on Cellular Automata models as a roadmap for the effort of identifying cell-level rules which give rise to complex, regenerative behavior of the collective. CAs typically consist of a grid of cells being iteratively updated, with the same set of rules being applied to each cell at every step. The new state of a cell depends only on the states of the few cells in its immediate neighborhood. Despite their apparent simplicity, CAs often demonstrate rich, interesting behaviours, and have a long history of being applied to modeling biological phenomena.

Let’s try to develop a cellular automata update rule that, starting from a single cell, will produce a predefined multicellular pattern on a 2D grid. This is our analogous toy model of organism development. To design the CA, we must specify the possible cell states, and their update function. Typical CA models represent cell states with a set of discrete values, although variants using vectors of continuous values exist. The use of continuous values has the virtue of allowing the update rule to be a differentiable function of the cell’s neighbourhood’s states. The rules that guide individual cell behavior based on the local environment are analogous to the low-level hardware specification encoded by the genome of an organism. Running our model for a set amount of steps from a starting configuration will reveal the patterning behavior that is enabled by such hardware.

So - what is so special about differentiable update rules? They will allow us to use the powerful language of loss functions to express our wishes, and the extensive existing machinery around gradient-based numerical optimization to fulfill them. The art of stacking together differentiable functions, and optimizing their parameters to perform various tasks has a long history. In recent years it has flourished under various names, such as (Deep) Neural Networks, Deep Learning or Differentiable Programming.

A single update step of the model.

Cell State We will represent each cell state as a vector of 16 real values (see the figure above). The first three channels represent the cell color visible to us (RGB).

The target pattern has color channel values in range [0.0,1.0][0.0, 1.0] and an α\alpha equal to 1.0 for foreground pixels, and 0.0 for background.

The alpha channel (α\alpha) has a special meaning: it demarcates living cells, those belonging to the pattern being grown. In particular, cells having α>0.1\alpha > 0.1 and their neighbors are considered “living”. Other cells are “dead” or empty and have their state vector values explicitly set to 0.0 at each time step. Thus cells with α>0.1\alpha > 0.1 can be thought of as “mature”, while their neighbors with α≤0.1\alpha \leq 0.1 are “growing”, and can become mature if their alpha passes the 0.1 threshold.

state⃗→0.00\vec{state} \rightarrow 0.00 when no neighbour with α>0.10\alpha > 0.10

Hidden channels don’t have a predefined meaning, and it’s up to the update rule to decide what to use them for. They can be interpreted as concentrations of some chemicals, electric potentials or some other signaling mechanism that are used by cells to orchestrate the growth. In terms of our biological analogy - all our cells share the same genome (update rule) and are only differentiated by the information encoded the chemical signalling they receive, emit, and store internally (their state vectors).

Cellular Automaton rule Now it’s time to define the update rule. Our CA runs on a regular 2D grid of 16-dimensional vectors, essentially a 3D array of shape [height, width, 16]. We want to apply the same operation to each cell, and the result of this operation can only depend on the small (3x3) neighborhood of the cell. This is heavily reminiscent of the convolution operation, one of the cornerstones of signal processing and differential programming.

Convolution is a linear operation, but it can be combined with other per-cell operations to produce a complex update rule, capable of learning the desired behaviour. Our cell update rule can be split into the following phases, applied in order:

Perception. This step defines what each cell perceives of the environment surrounding it. We implement this via a 3x3 convolution with a fixed kernel. One may argue that defining this kernel is superfluous - after all we could simply have the cell learn the requisite perception kernel coefficients. Our choice of fixed operations are motivated by the fact that real life cells often rely only on chemical gradients to guide the organism development. Thus, we are using classical Sobel filters to estimate the partial derivatives of cell state channels in the x⃗\vec{x} and y⃗\vec{y} directions, forming a 2D gradient vector in each direction, for each state channel. We concatenate those gradients with the cells own states, forming a 16∗2+16=4816*2+16=48 dimensional perception vector, or rather percepted vector, for each cell.

def perceive(state_grid): sobel_x = [[-1, 0, +1], [-2, 0, +2], [-1, 0, +1]] sobel_y = transpose(sobel_x) # Convolve sobel filters with states # in x, y and channel dimension. grad_x = conv2d(sobel_x, state_grid) grad_y = conv2d(sobel_y, state_grid) # Concatenate the cell’s state channels, # the gradients of channels in x and # the gradient of channels in y. perception_grid = concat( state_grid, grad_x, grad_y, axis=2) return perception_grid

Update rule. Each cell now applies a series of operations to the perception vector, consisting of typical differentiable programming building blocks, such as 1x1-convolutions and ReLU nonlinearities, which we call the cell’s “update rule”.