  We consider an inhomogeneous small-world network (SWN) composed of inhibitory
short-range (SR) and long-range (LR) interneurons. By varying the fraction of
LR interneurons $p_{long}$, we investigate the effect of network architecture
on emergence of sparsely synchronized rhythms, and make comparison with that in
the Watts-Strogatz SWN. Although SR and LR interneurons have the same average
in- and out-degrees, their betweenness centralities (characterizing the
potentiality in controlling communication between other interneurons) are
distinctly different. Hence, in view of the betweenness, SWNs we consider are
inhomogeneous, unlike the "canonical" Watts-Strogatz SWN with nearly same
betweenness centralities. As $p_{long}$ is increased, the average path length
becomes shorter, and the load of communication traffic is less concentrated on
LR interneurons, which leads to better efficiency of global communication
between interneurons. Eventually, when passing a critical value
$p_{long}^{(c)}$ $(\simeq 0.16)$, sparsely synchronized rhythms are found to
emerge. We also consider two cases of external time-periodic stimuli applied to
sub-groups of LR and SR interneurons, respectively. Dynamical responses (such
as synchronization suppression and enhancement) to these two cases of stimuli
are studied and discussed in relation to the betweenness centralities of
stimulated interneurons, representing the effectiveness for transfer of
stimulation effect in the whole network.
