Riecke Hermann, Roxin Alex, Madruga Santiago, Solla Sara A
Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois 60208, USA.
Chaos. 2007 Jun;17(2):026110. doi: 10.1063/1.2743611.
We study the dynamical states of a small-world network of recurrently coupled excitable neurons, through both numerical and analytical methods. The dynamics of this system depend mostly on both the number of long-range connections or "shortcuts", and the delay associated with neuronal interactions. We find that persistent activity emerges at low density of shortcuts, and that the system undergoes a transition to failure as their density reaches a critical value. The state of persistent activity below this transition consists of multiple stable periodic attractors, whose number increases at least as fast as the number of neurons in the network. At large shortcut density and for long enough delays the network dynamics exhibit exceedingly long chaotic transients, whose failure times follow a stretched exponential distribution. We show that this functional form arises for the ensemble-averaged activity if the failure time for each individual network realization is exponentially distributed.
我们通过数值和分析方法研究了由循环耦合的可兴奋神经元构成的小世界网络的动力学状态。该系统的动力学主要取决于远程连接或“捷径”的数量以及与神经元相互作用相关的延迟。我们发现，在捷径密度较低时会出现持续活动，并且当捷径密度达到临界值时，系统会经历向失效的转变。低于此转变的持续活动状态由多个稳定的周期性吸引子组成，其数量至少与网络中神经元的数量增长速度相同。在大捷径密度和足够长的延迟情况下，网络动力学表现出极其长的混沌瞬态，其失效时间遵循拉伸指数分布。我们表明，如果每个单独网络实现的失效时间呈指数分布，则对于系综平均活动会出现这种函数形式。
