**What kind of definitions of t-structures
on stable model categories have been investigated in the literature?**

Of course, one can always define a t-structure on a stable model category as a t-structure on its homotopy category; this is analogous to how Lurie defines t-structures on stable quasicategories (Definition 1.2.1.4 in Higher Algebra).

However, in the setting of model categories one normally wants a more strict presentation that could be exploited to perform computations more easily.

In particular, one can conceive of various strictifications of k-connective objects C_{≥k}, k-coconnective objects C_{≤k}, and their truncating functors τ_{≥k} and τ_{≤k}.

For example, in the case of symmetric simplicial spectra one can say that
a spectrum X is *strictly connective* if for each n the nth spectral level X_n
is a simplicial set with exactly one k-simplex for all k

Using coskeletal simplicial sets one can also conceive of a similar picture for coconnective spectra and coconnective truncations.

Is there anything like this in the literature? Of course, I'm not just interested in symmetric simplicial sets, but also (say) in motivic symmetric spectra and other stable model categories.