When doing "derived commutative algebra" over a discrete commutative ring $R$ that *doesn't* contain $\mathbb{Q}$, it's fairly well known that you generally have two flavors of "totally commutative object," given by simplicial commutative $R$-algebras on the one hand, and $E_\infty$-algebras in connective complexes over $R$ (equivalently connective $E_\infty$-algebra spectra over $HR$) on the other. The former category maps to the latter, and there are various ways to describe its image, as detailed in the answers to the question "What is a simplicial commutative ring from the point of view of homotopy theory? "

I'm interested in the dual question involving cosimplicial commutative $R$-algebras and coconnective $E_\infty$-algebra spectra over $HR$. There seems again to be a natural functor given by co-Dold-Kan, but I have no feeling for the objects it picks out. Are there similar intrinsic characterizations of the image of this functor as in the connective case? I'd be happy to know anything about this, even/especially when $R$ is a finite field.