Let $p_t(w)$ and $s_t(w)$ denote the prefix and suffix of length $t$ of the word $w$, respectively. If $|w| < t$, then $p_t(w) = s_t(w) = w$. Furthermore, let $i_t(w)$ be the set of infixes of length $t$ of $w$. (If $|w| < t$, then $i_t(w)$ is empty.)
For $k \in \mathbb{N}_0$, a language $L \subseteq \Sigma^\ast$ is said to be strictly $k$-testable if there is are sets $P, S \subseteq \Sigma^{\le k}$ and $I \subseteq \Sigma^k$ such that, for every word $w$, $w \in L$ if and only if $p_k(w) \in P$, $s_k(w) \in S$, and $I_k(w) \subseteq I$.
A language is strictly locally testable if it is strictly $k$-testable for some $k$. A language is locally testable if it can be expressed by the union, intersection, or complement of finitely many strictly locally testable languages. It is known that the locally testable languages are a strict subset of the regular languages.
(The above definitions are slightly adapted from here and here.)
Are there any papers which analyse the case in which $k$ is no longer a constant? That is, when $k$ is allowed to scale with the length $|w|$ (but we still have $k \in o(|w|)$; e.g., $k = \log |w|$)? Clearly, the resulting languages will not necessarily be still regular, but they seem to be strongly related to parallel computing models such as cellular automata.
(My Google Scholar searches have been fruitless so far...)