GoKawiilClick or tap the image to erase the part of the pattern and see it regenerate. Double clicking places a new seed cell on the grid.
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.
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