  The balance between excitation and inhibition is crucial for neuronal
computation. It is observed that the balanced state of neuronal networks exists
in many experiments, yet its underlying mechanism remains to be fully
clarified. Theoretical studies of the balanced state mainly focus on the
analysis of the homogeneous Erd$\ddot{\text{o}}$s-R\'enyi network. However,
neuronal networks have been found to be inhomogeneous in many cortical areas.
In particular, the connectivity of neuronal networks can be of the type of
scale-free, small-world, or even with specific motifs. In this work, we examine
the questions of whether the balanced state is universal with respect to
network topology and what characteristics the balanced state possesses in
inhomogeneous networks such as scale-free and small-world networks. We discover
that, for a sparsely but strongly connected inhomogeneous network, despite that
the whole network receives external inputs, there is a small active subnetwork
(active core) inherently embedded within it. The neurons in this active core
have relatively high firing rates while the neurons in the rest of the network
are quiescent. Surprisingly, the active core possesses a balanced state and
this state is independent of the model of single-neuron dynamics. The dynamics
of the active core can be well predicted using the Fokker-Planck equation with
the mean-field assumption. Our results suggest that, in the presence of
inhomogeneous network connectivity, the balanced state may be ubiquitous in the
brain, and the network connectivity in the active core is essentially close to
the Erd$\ddot{\text{o}}$s-R\'enyi structure. The existence of the small active
core embedded in a large network may provide a potential dynamical scenario
underlying sparse coding in neuronal networks.
