Other types of uniformity for circuits (incl. by small modifications)
When you want to define (fully) uniform versions of circuit classes, log-space or poly-time uniformity is only sensible for classes of circuits whose power exceeds log-space or poly-time, respectively. This means that poly-time uniformity is next to useless, and log-space uniformity is good for classes like P, NC, $\mathrm{NC}^{k+1}$, $\mathrm{AC}^k$ for $k\ge1$, but not, say, for $\mathrm{NC}^1$.

The standard notion of uniformity for small complexity classes is DLOGTIME-uniformity, where furthermore one has to be careful what exact kind of description of the circuit is asked to be computable in DLOGTIME: in cases where it matters (e.g., $\mathrm{NC}^1$), one needs to use the extended connection language (cf. Ruzzo) of the circuit. This is known as $U_E$-uniformity.