Reparameterizing Conway's Game of Life as a scalar reaction-diffusion equation Consider the set of functions $\mathbb Z^2\to\{0,1\}$, the state space of Conway's Game of Life. One common way to define the update rule is to use the following 3x3 neighbor-counting convolution kernel:
$$
k = \left[ \begin{array}{} 1 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 1 \end{array} \right]
$$
If $u_t$ is the state at time $t$, then $n_t := k*u_t$ denotes the convolution of $k$ with $u_t$, which yields the number of neighbors. If you want, it can be thought of as a map $n_t:\mathbb Z^2 \to \mathbb Z_{\geq 0}$.
Then we have a bivariate function $f:\{0,1\}\times\mathbb Z_{\geq 0}\to\{0,1\}$ as
$$
f(u,n) = \begin{cases}
1 & \text{if } (u,n) \in \{ (0,3), (1,2), (1,3) \} \\
0 & \text{else} 
\end{cases}
$$
This is a way of saying that dead cells with 3 live neighbors become alive, and alive cells with 2 or 3 neighbors stay alive, with all else becoming dead. The update rule can then be defined as $u_{t+1} := f\circ(u_t, k*u_t)$, with $f$ being applied point-wise.

So, here is the question (which I already have the answer to): can Conway's Game of Life be represented as a scalar reaction-diffusion system, without using a bivariate activation function?
 A: It can - basically, redefine the kernel $k$ as follows:
$$
k = \left[ \begin{array}{} 1 & 1 & 1 \\ 1 & 9 & 1 \\ 1 & 1 & 1 \end{array} \right]
$$
Then redefine $f$ as
$$
f(u) = \begin{cases}
1 & \text{if } u \in \{3, 11, 12\} \\
0 & \text{else} 
\end{cases}
$$
This corresponds to the previous parameterization via $(0,3)\to 3$, $(1,2)\to 11$, $(1,3)\to 12$. Now the system can be written simply as $u_{t+1} = f \circ (k * u_t)$.
This works because the value of 9 in the center cell outweighs the sum of its neighbors, so $k*u_t(x)\geq9$ guarantees $u_t(x)=1$, and likewise $k*u_t(x)<9$ guarantees $u_t(x)=0$, with $k*u_t(x)\bmod 9$ being the neighbor count.

Some comments: I began thinking about this after trying to implement Game of Life on this cool web app called neuralpatterns.io which only supports scalar reaction-diffusion systems. It suggests a family of similar systems with sharply mean-centric dispersion kernels and bimodal activation functions which display Life-like dynamics. Also, the scalar form is that of the somewhat obscure integro-difference equation.
