Gong Pulin, Steel Harrison, Robinson Peter, Qi Yang
School of Physics, University of Sydney, NSW 2006, Australia.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Oct;88(4):042821. doi: 10.1103/PhysRevE.88.042821. Epub 2013 Oct 30.
Formation of localized propagating patterns is a fascinating self-organizing phenomenon that happens in a wide range of spatially extended, excitable systems in which individual elements have resting, activated, and refractory states. Here we study a type of stochastic three-state excitable network model that has been recently developed; this model is able to generate a rich range of pattern dynamics, including localized wandering patterns and localized propagating patterns with crescent shapes and long-range propagation. The collective dynamics of these localized patterns have anomalous subdiffusive dynamics before symmetry breaking and anomalous superdiffusive dynamics after that, showing long-range spatiotemporal coherence in the system. In this study, the stability of the localized wandering patterns is analyzed by treating an individual localized pattern as a subpopulation to develop its average response function. This stability analysis indicates that when the average refractory period is greater than a certain value, there are too many elements in the refractory state after being activated to allow the subpopulation to support a self-sustained pattern; this is consistent with symmetry breaking identified by using an order parameter. Furthermore, in a broad parameter space, the simple network model is able to generate a range of interactions between different localized propagating patterns including repulsive collisions and partial and full annihilations, and interactions between localized propagating patterns and the refractory wake behind others; in this study, these interaction dynamics are systematically quantified based on their relative propagation directions and the resultant angles between them before and after their collisions. These results suggest that the model potentially provides a modeling framework to understand the formation of localized propagating patterns in a broad class of systems with excitable properties.
局部传播模式的形成是一种引人入胜的自组织现象，它发生在广泛的空间扩展可激发系统中，其中单个元素具有静止、激活和不应期状态。在这里，我们研究一种最近开发的随机三态可激发网络模型；该模型能够产生丰富多样的模式动力学，包括局部徘徊模式、具有新月形状的局部传播模式和长程传播模式。这些局部模式的集体动力学在对称性破缺之前具有反常亚扩散动力学，之后具有反常超扩散动力学，表明系统中存在长程时空相干性。在本研究中，通过将单个局部模式视为一个亚群体来开发其平均响应函数，分析了局部徘徊模式的稳定性。这种稳定性分析表明，当平均不应期大于某个值时，激活后处于不应期的元素过多，使得亚群体无法支持自持模式；这与使用序参量确定的对称性破缺是一致的。此外，在广泛的参数空间中，该简单网络模型能够产生不同局部传播模式之间的一系列相互作用，包括排斥碰撞、部分和完全湮灭，以及局部传播模式与其他模式后面的不应期尾迹之间的相互作用；在本研究中，基于它们的相对传播方向以及碰撞前后它们之间的夹角，系统地量化了这些相互作用动力学。这些结果表明，该模型可能为理解具有可激发特性的广泛系统中局部传播模式的形成提供一个建模框架。
