In the primary visual cortex, the processing of information uses the distribution of orientations in the visual input: neurons react to some orientations in the stimulus more than to others. In many species, orientation preference is mapped in a remarkable way on the cortical surface, and this organization of the neural population seems to be important for visual processing. Now, existing models for the geometry and development of orientation preference maps in higher mammals make a crucial use of symmetry considerations. In this paper, we consider probabilistic models for V1 maps from the point of view of group theory; we focus on Gaussian random fields with symmetry properties and review the probabilistic arguments that allow one to estimate pinwheel densities and predict the observed value of π. Then, in order to test the relevance of general symmetry arguments and to introduce methods which could be of use in modeling curved regions, we reconsider this model in the light of group representation theory, the canonical mathematics of symmetry. We show that through the Plancherel decomposition of the space of complex-valued maps on the Euclidean plane, each infinite-dimensional irreducible unitary representation of the special Euclidean group yields a unique V1-like map, and we use representation theory as a symmetry-based toolbox to build orientation maps adapted to the most famous non-Euclidean geometries, viz. spherical and hyperbolic geometry. We find that most of the dominant traits of V1 maps are preserved in these; we also study the link between symmetry and the statistics of singularities in orientation maps, and show what the striking quantitative characteristics observed in animals become in our curved models.

中文翻译：

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V1和非欧几里得几何中的方向图。

在主要的视觉皮层中，信息的处理使用视觉输入中方向的分布：神经元对刺激中某些方向的反应要多于对其他方向的反应。在许多物种中，取向偏好以显着方式映射到皮层表面上，并且这种神经种群的组织对于视觉处理似乎很重要。现在，在高级哺乳动物中，用于定位偏好图的几何形状和发展的现有模型对对称性的考虑进行了关键性的利用。在本文中，我们从群体理论的角度考虑了V1映射的概率模型。我们将重点放在具有对称性的高斯随机场上，并回顾一下概率论，这些论点允许人们估计风车密度并预测π的观测值。然后，为了测试一般对称性参数的相关性并介绍可用于对弯曲区域进行建模的方法，我们根据对称性的经典数学表示法理论重新考虑了该模型。我们表明，通过欧氏平面上复值映射空间的Plancherel分解，特殊欧氏群的每个无穷维不可约unit表示都产生了一个唯一的类似于V1的映射，并且我们将表示理论用作基于对称的工具箱来构建适应最著名的非欧几里得几何形状的方向图，即。球面和双曲线几何。我们发现，V1映射的大多数主要特征都保留在其中；我们还研究了方向图中对称性与奇异性统计之间的联系，