We consider a scale-free network of inhibitory Hindmarsh–Rose (HR) bursting neurons, and make a computational study on coupling-induced cluster burst synchronization by varying the average coupling strength \(J_0\). For sufficiently small \(J_0\), non-cluster desynchronized states exist. However, when passing a critical point \(J^*_c~(\simeq 0.16)\), the whole population is segregated into 3 clusters via a constructive role of synaptic inhibition to stimulate dynamical clustering between individual burstings, and thus 3-cluster desynchronized states appear. As \(J_0\) is further increased and passes a lower threshold \(J^*_l~(\simeq 0.78)\), a transition to 3-cluster burst synchronization occurs due to another constructive role of synaptic inhibition to favor population synchronization. In this case, HR neurons in each cluster make burstings every 3rd cycle of the instantaneous burst rate \(R_w(t)\) of the whole population, and exhibit burst synchronization. However, as \(J_0\) passes an intermediate threshold \(J^*_m~(\simeq 5.2)\), HR neurons fire burstings intermittently at a 4th cycle of \(R_w(t)\) via burst skipping rather than at its 3rd cycle, and hence they begin to make intermittent hoppings between the 3 clusters. Due to such intermittent intercluster hoppings via burst skippings, the 3 clusters become broken up (i.e., the 3 clusters are integrated into a single one). However, in spite of such break-up (i.e., disappearance) of the 3-cluster states, (non-cluster) burst synchronization persists in the whole population, which is well visualized in the raster plot of burst onset times where bursting stripes (composed of burst onset times and indicating burst synchronization) appear successively. With further increase in \(J_0\), intercluster hoppings are intensified, and bursting stripes also become dispersed more and more due to a destructive role of synaptic inhibition to spoil the burst synchronization. Eventually, when passing a higher threshold \(J^*_h~(\simeq 17.8)\) a transition to desynchronization occurs via complete overlap between the bursting stripes. Finally, we also investigate the effects of stochastic noise on both 3-cluster burst synchronization and intercluster hoppings.

中文翻译：

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抑制性爆发神经元的无标度网络中的簇爆发同步

我们考虑了抑制性Hindmarsh-Rose（HR）爆发神经元的无标度网络，并通过改变平均耦合强度\（J_0 \）对耦合诱发的簇爆发同步进行了计算研究。对于足够小的\（J_0 \），存在非集群去同步状态。但是，当通过临界点\（J ^ * _ c〜（\ simeq 0.16）\）时，整个群体通过突触抑制的建设性作用被分为3个簇，以刺激各个爆发之间的动态簇，从而形成3个簇出现非同步状态。随着\（J_0 \）进一步增加并通过较低的阈值\（J ^ * _ 1〜（\ simeq 0.78）\），由于突触抑制的另一种建设性作用有利于群体同步，因此发生了向3群突发同步的过渡。在这种情况下，每个群集中的HR神经元每隔整个种群的瞬时猝发率\（R_w（t）\）的第三个周期就发生猝发，并表现出猝发同步。但是，当\（J_0 \）通过中间阈值\（J ^ * _ m〜（\ simeq 5.2）\）时，HR神经元会以\（R_w（t）\）的第四个周期间歇性地爆发通过突发跳过而不是在其第3个周期进行，因此它们开始在3个簇之间进行间歇性跳跃。由于通过突发跳跃的这种间歇性集群间跳跃，3个集群被分解（即，这3个集群被集成为一个集群）。然而，尽管3簇状态发生这种分裂（即消失），（非簇）猝发同步仍在整个种群中持续存在，这在猝发开始时间的栅格图中可以很好地看到，其中爆发条纹（由突发起始时间组成并表示突发同步的信号）相继出现。随着\（J_0 \）的进一步增加，簇间跳变加剧，并且由于突触抑制的破坏性作用破坏了爆发同步，爆发条纹也越来越分散。最终，当通过更高的阈值\（J ^ * _ h〜（\ simeq 17.8）\）时，通过突发条带之间的完全重叠会发生向不同步的过渡。最后，我们还研究了随机噪声对3集群突发同步和集群间跳变的影响。