We study the dynamics of excitable integrate-and-fire neurons in asmall-world network. At low densities $p$ of directed random connections, alocalized transient stimulus results in either self-sustained persistentactivity or in a brief transient followed by failure. Averages over thequenched ensemble reveal that the probability of failure changes from 0 to 1over a narrow range in $p$; this failure transition can be describedanalytically through an extension of an existing mean-field result. Exceedinglylong transients emerge at higher densities $p$; their activity patterns aredisordered, in contrast to the mostly periodic persistent patterns observed atlow $p$. The times at which such patterns die out are consistent with astretched-exponential distribution, which depends sensitively on thepropagation velocity of the excitation.
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机译：我们研究了一个小世界网络中可激发的集成激发神经元的动力学。在低密度的有向随机连接下，局部化的瞬态刺激会导致自我维持的持续性活动或短暂的瞬态继之以失败。淬火后的集合的平均值表明，失败概率在$ p $的狭窄范围内从0变为1；可以通过扩展现有均值场结果来分析描述这种失效过渡。在更高的密度$ p $处出现了非常长的瞬态现象；与在$ p $较低时观察到的大多数周期性持续模式相反，它们的活动模式是无序的。这种模式消失的时间与拉伸指数分布一致，该分布敏感地取决于激发的传播速度。
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