Afgoustidis Alexandre
Institut de Mathématiques de Jussieu-Paris Rive Gauche, Universite Paris 7 Denis Diderot, 75013, Paris, France,
J Math Neurosci. 2015 Dec;5(1):24. doi: 10.1186/s13408-015-0024-7. Epub 2015 Jun 17.
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.
在初级视觉皮层中，信息处理利用视觉输入中方向的分布：神经元对刺激中的某些方向反应比对其他方向更强烈。在许多物种中，方向偏好以一种显著的方式映射在皮层表面，并且这种神经群体的组织似乎对视觉处理很重要。现在，高等哺乳动物中方向偏好图的几何形状和发育的现有模型关键地利用了对称性考虑。在本文中，我们从群论的角度考虑V1图的概率模型；我们关注具有对称性质的高斯随机场，并回顾那些能让人估计风车密度并预测π观测值的概率论证。然后，为了检验一般对称性论证的相关性并引入可用于对弯曲区域建模的方法，我们根据群表示理论（对称性的规范数学）重新考虑这个模型。我们表明，通过欧几里得平面上复值映射空间的普兰切尔分解，特殊欧几里得群的每个无限维不可约酉表示都产生一个独特的类似V1的图，并且我们将表示理论用作基于对称性的工具箱来构建适用于最著名的非欧几里得几何（即球面几何和双曲几何）的方向图。我们发现这些图中保留了V1图的大多数主要特征；我们还研究了对称性与方向图中奇点统计之间的联系，并展示了在动物中观察到的显著定量特征在我们的弯曲模型中变成了什么。
