鸟类视网膜多超均匀斑图的最小几何力学模型
A minimal statistical-mechanical model for multihyperuniform patterns in avian retina
摘要：
鸟类以其极其敏锐的视觉而闻名。视网膜中五种不同类型的锥细胞的非常特殊的结构分布构成了这种精致的光采样能力的基础。最近发现，每个锥人口以及他们的总人口显示出无序的模式，其中长波长的密度波动消失。这种被称为超均匀性的特性也存在于完美的晶体中。在像鸟类视网膜这样的情况下，全局结构和每个组件的结构都显示出超均匀性，系统被称为多超均匀系统。在这项工作中，我们的目标是设计一个最小的几何力学模型，可以再现鸟类视网膜中的光感受器的空间分布的主要特征，即存在的障碍，多超均匀性和局部异质协调。最后一个特征是避免相同类型的光感受器的局部聚类的关键，这是光的有效采样的不期望的特征。为此，我们制定了一个简单的模型，明确表现出所需的结构特性，即等摩尔的三组分混合物（一个组件的样品每个原色，红色，绿色和蓝色）的非添加剂硬盘，其中一个长距离的对数排斥之间添加类似的颗粒。我们的理想化的光感受器系统的Voronoi分析表明，充满空间的Voronoi多边形有趣地显示出一个相当均匀的区域分布，对称地围绕着一个规则的晶格，结构特性也发现在人类视网膜。无序多超均匀性提供了一种替代方案，以产生具有最小长范围浓度和密度波动的感光体图案。这是克服缺乏晶体有序性的有效视觉系统的困难的关键。
英文摘要：
Birds are known for their extremely acute sense of vision. The very peculiar structural distribution of five different types of cones in the retina underlies this exquisite ability to sample light. It was recently found that each cone population as well as their total population display a disordered pattern in which long wave-length density fluctuations vanish. This property, known as hyperuniformity is also present in perfect crystals. In situations like the avian retina in which both the global structure and that of each component display hyperuniformity, the system is said to be multi-hyperuniform. In this work, we aim at devising a minimal statistical-mechanical model that can reproduce the main features of the spatial distribution of photoreceptors in avian retina, namely the presence of disorder, multi-hyperuniformity and local hetero-coordination. This last feature is key to avoid local clustering of the same type of photoreceptors, an undesirable feature for the efficient sampling of light. For this purpose we formulate a simple model that definitively exhibits the required structural properties, namely an equimolar three-component mixture (one component to sample each primary color, red, green, and blue) of non-additive hard disks to which a long-range logarithmic repulsion is added between like particles. A Voronoi analysis of our idealized system of photoreceptors shows that the space-filling Voronoi polygons interestingly display a rather uniform area distribution, symmetrically centered around that of a regular lattice, a structural property also found in human retina. Disordered multi-hyperuniformity offers an alternative to generate photoreceptor patterns with minimal long-range concentration and density fluctuations. This is the key to overcome the difficulties in devising an efficient visual system in which crystal-like order is absent.
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