On instructing cells to grow into differentiated bodies. This notebook has been resurrected from the trash bin years after I deleted it because of my great enjoyment of Mordvintsev et al. (2020).

Mordvintsev et al. (2020) is a fun paper. They improve upon boring old school cellular automata in several ways (not all of which are completely novel, but seem to be a novel combination.)

  1. Continuous states whose rules can be differentiably learned
  2. Use of Sobel filters for CA based on local gradients
  3. Framing the problem as “designing attractors of a dynamical systems”
  4. Clever use of noise in the training.

I am instinctively annoyed by the unfashionable loss function which is not any kind of optimal transport metric, but hey, it works so don’t listen to me about that. Logical extensions such as creating a model which can produce different patterns parametrically and interpolate between them also seem to leap out at me. I am also curious about an information bottleneck analysis which gives us restrictions on what patterns can be learned.

More general models of morphogenesis are out there, obviously. I will not even touch upon how this happens in real creatures as opposed to fake emoji monsters for now.

TBD: connection to bio computing, and the particular special case, models of pattern formation.


Hussein, Salome, Jerzy Maselko, and James T. Pantaleone. 2016. Growing a Chemical Garden at the Air–Fluid Interface.” Langmuir 32 (3): 706–11.
Kücken, Michael, Baruch Rinkevich, Lee Shaish, and Andreas Deutsch. 2011. Nutritional resources as positional information for morphogenesis in the stony coral Stylophora pistillata.” Journal of Theoretical Biology 275 (1): 70–77.
Kuffner, James J., and Steven M. LaValle. 2009. “Space-Filling Trees.”
Lee, Dongkyu, Soomi Shin, and Sungsoo Park. 2007. Computational Morphogenesis Based Structural Design by Using Material Topology Optimization.” Mechanics Based Design of Structures and Machines 35 (1): 39–58.
Mordvintsev, Alexander, Ettore Randazzo, Eyvind Niklasson, and Michael Levin. 2020. Growing Neural Cellular Automata.” Distill 5 (2): e23.
Pearce, Peter. 1980. Structure in nature is a strategy for design. Cambridge: MIT Press.
Thompson, D’Arcy Wentworth, and John Tyler. Bonner. 1997. On growth and form.
Turing, Alan Mathison. 1952. The Chemical Basis of Morphogenesis.” Philosophical Transactions of the Royal Society B: Biological Sciences 237 (641): 37–72.

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