Logical characterization of $NC^1$
Morioka considers uniform versions of his circuit classes:


  Throughout this paper we write $\mathbf{NC^1}$ to mean $\mathbf{Dlogtime}$-uniform $\mathbf{NC^1}$, which is equivalent to the class $\mathbf{Alogtime}$ of languages accepted by an alternating Turing machine in $O(\log n)$ time.


The paper you mention should imply the equivalence of these two definitions. The statement you quote, however, is just true by definition.