Is the following restriction of cellular automata / tile assembly / CRN a known model? Consider the following model: we work in the $d$-dimensional grid $\mathbb{N}^d$, and we have an alphabet $\Sigma$.
The initial cell $(0,\ldots,0)\in \mathbb{N}^d$ is marked with some letter $\Sigma$, and from that we mark each cell with a letter from $\Sigma$ according to its immediate predecessors. That is, we have a function $f:\Sigma^d\to \Sigma$, and for each cell $(a_1,\ldots,a_d)$, we mark it with
$f(\sigma_1,\ldots,\sigma_d)$, where $\sigma_i$ is the labeling of the neighbor $(a_1,\ldots,a_i-1,\ldots, a_d)$.
This means that the marking is completely determined by $f$ and the initial symbol.
Note that at the edges, where there are fewer then $d$ neighbors, we ignore the missing values (technically, we can assume that anything outside $\mathbb{N}^d$ has some constant fixed marking).
I want to study certain properties of this model (specifically, I want to show that it is semilinear, in a sense). But I couldn't find any resources on it.
Specifically, I would like to know if this model is Presburger-definable.
However, it is a very natural restriction of several models:

Cellular automata: it can be thought of a single-state cellular automaton, that can only change the value of a cell once (i.e., a cellular transducer).
Tile Self-assembly: where the glue relations allow only a deterministic assembly, but the model is not two-dimensional.
Multidimensional subshifts: it's not really a shift, but it does have the same flavour of finite rules.
Chemical-reaction networks: I'm actually not sure if this is a restriction, but it seems somewhat related.

I'd appreciate any leads on this.
If you partition $\mathbb{Z}^d$ into the $(d-1)$-dimensional slices $S(n) := \{ \vec v | \sum_{k=1}^d \vec v_k = n \}$, then your model is essentially a spacetime diagram of the $(d-1)$-dimensional cellular automaton that maps $x|_{S(n)}$ to $x|_{S(n+1)}$. The initial configuration $x|_{S(0)}$ has a special quiescent state everywhere except at the origin. If the CA is linear over a field of characteristic $p$, then these diagrams are $p$-automatic; if not, they can be quite complex, since you don't restrict $f$ in any way. They are not necessarily semilinear in either case. For example, it's pretty easy to embed a computation history of an arbitrary Turing machine in the configuration.