  In this paper, we first propose a new concept of almost periodic time scales,
a new definition of almost automorphic functions on almost periodic time
scales, and study some their basic properties. Then we prove a result ensuring
the existence of an almost automorphic solution for both the linear
nonhomogeneous dynamic equation on time scales and its associated homogeneous
equation, assuming that the associated homogeneous equation admits an
exponential dichotomy. Finally, as an application of our results, we establish
the existence and global exponential stability of almost automorphic solutions
to a class of shunting inhibitory cellular neural networks with time-varying
delays on time scales. Our results about the shunting inhibitory cellular
neural network with time-varying delays on time scales are new even for the
both cases of differential equations(the time scale $\mathbb{T}=\mathbb{R})$
and difference equations(the time scale $\mathbb{T}=\mathbb{Z})$.
