Effect of Network Architecture on Burst and Spike Synchronization in A Scale-Free Network of Bursting Neurons

We investigate the effect of network architecture on burst and spike synchronization in a directed scale-free network (SFN) of bursting neurons, evolved via two independent $\alpha-$ and $\beta-$processes. The $\alpha-$process corresponds to a directed version of the Barab\'{a}si-Albert SFN model with growth and preferential attachment, while for the $\beta-$process only preferential attachments between pre-existing nodes are made without addition of new nodes. We first consider the "pure" $\alpha-$process of symmetric preferential attachment (with the same in- and out-degrees), and study emergence of burst and spike synchronization by varying the coupling strength $J$ and the noise intensity $D$ for a fixed attachment degree. Characterizations of burst and spike synchronization are also made by employing realistic order parameters and statistical-mechanical measures. Next, we choose appropriate values of $J$ and $D$ where only the burst synchronization occurs, and investigate the effect of the scale-free connectivity on the burst synchronization by varying (1) the symmetric attachment degree and (2) the asymmetry parameter (representing deviation from the symmetric case) in the $\alpha-$process, and (3) the occurrence probability of the $\beta-$process. In all these three cases, changes in the type and the degree of population synchronization are studied in connection with the network topology such as the degree distribution, the average path length $L_p$, and the betweenness centralization $B_c$. It is thus found that not only $L_p$ and $B_c$ (affecting global communication between nodes) but also the in-degree distribution (affecting individual dynamics) are important network factors for effective population synchronization in SFNs.