Thanks for the reply! I'm familiar with (and am skeptical of) the basic information theoretic argument as to why genome size should constrain the complexity of whatever algorithm the brain is running, but my question here is more specific. What I'm not clear on is how those two numbers (20,000 genes and a few thousand neuron types) specifically relate to each other in your model of brain functioning. Is the idea that each neuron type roughly corresponds to the expression of one or two specific genes, and thus you'd expect <20,000 neuron types?
"For sure, the genome could build a billion different “cell types” by each cell having 30 different flags which are on and off at random in a collection of 100 billion neurons. But … why on earth would the genome do that?"
Interestingly, the genome does do this! Protocadherins in vertebrates and DSCAM1 are expressed in exactly this way, and it's thought to help neurons to distinguish themselves from other neurons, which is essential for neuronal self avoidance: https://en.wikipedia.org/wiki/Neuronal_self-avoidance#Molecular_basis_of_self-avoidance
Of course in an emulation you could probably just tell the neurons to not interact with themselves so this crazy system wouldn't be necessary, but it is a nice example of how biology does things you might a priori think would never happen.
"They found low-thousands of neuron types in the mouse, which makes sense on priors given that there are only like 20,000 genes encoding the whole brain design and everything in it, along with the rest of the body."
I'm a bit puzzled by this statement; how would the fact that there are ~20,000 genes in the mouse/human genome constrain the number of neuron types to the low thousands? From a naive combinatorics standpoint it seems like 20,000 genes is sufficiently large to place basically zero meaningful constraints on the number of potential cell types. E.g. if you assume that only 15,000 genes vary meaningfully between cell types, and that there are 3000 of those variable genes expressed per cell, chatgpt tells me that the number of potential combinations is too large for it to even estimate its order of magnitude. And that's with a simple, extremely unrealistic binary on/off model of gene expression.
Interesting...I think I vaguely understand what you're talking about, but I'm doubtful that these concepts really apply to biology. Especially since your example is about constraints on evolvability rather than functioning. In practice that is pretty much how everything tends to work, with absolutely wild amounts of pleiotropy and epistasis, but that's not a problem unless you want to evolve a new function. Which is probably why the strong strong evolutionary default is towards stasis, not change.
I guess my priors are pretty different because my background is in virology, where our expectation (after decades of painful lessons) is that the default is for proteins to be wildly multifunctional, with many many many "design degrees of freedom." Granted viruses are a bit of a special case, but I do think they can provide a helpful stress test/simpler model for information theoretic models of genome function.