Effect of Inter-Modular Connection on Fast Sparse Synchronization in Clustered Small-World Neural Networks

We consider a clustered network with small-world sub-networks of inhibitory fast spiking interneurons, and investigate the effect of inter-modular connection on emergence of fast sparsely synchronized rhythms by varying both the inter-modular coupling strength $J_{inter}$ and the average number of inter-modular links per interneuron $M_{syn}^{(inter)}$. In contrast to the case of non-clustered networks, two kinds of sparsely synchronized states such as modular and global synchronization are found. For the case of modular sparse synchronization, the population behavior reveals the modular structure, because the intra-modular dynamics of sub-networks make some mismatching. On the other hand, in the case of global sparse synchronization, the population behavior is globally identical, independently of the cluster structure, because the intra-modular dynamics of sub-networks make perfect matching. We introduce a realistic cross-correlation modularity measure, representing the matching-degree between the instantaneous sub-population spike rates of the sub-networks, and examine whether the sparse synchronization is global or modular. Furthermore, we characterize the modular and global sparse synchronization by employing the realistic sub- and whole-population order parameters and statistical-mechanical measures. The roles of $J_{inter}$ and $M_{syn}^{(inter)}$ are thus found as follows. For large $J_{inter}$, due to strong inhibition it plays a destructive role to "spoil" the pacing between spikes, while for small $J_{inter}$ it plays a constructive role to "favor" the pacing between spikes. In contrast, $M_{syn}^{(inter)}$ seems to play a role just to favor the pacing between spikes. With increasing $M_{syn}^{(inter)}$, the pacing degree between spikes increases monotonically thanks to the increase in the degree of effectiveness of global communication between spikes.