The modeling of neural fields in the visual cortex involves geometrical structures which describe in mathematical formalism the functional architecture of this cortical area. The case of contour detection and orientation tuning has been extensively studied and has become a paradigm for the mathematical analysis of image processing by the brain. Ten years ago an attempt was made to extend these models by replacing orientation (an angle) with a second-order tensor built from the gradient of the image intensity, and it was named the structure tensor. This assumption does not follow from biological observations (experimental evidence is still lacking) but from the idea that the effectiveness of texture processing with the structure tensor in computer vision may well be exploited by the brain itself. The drawback is that in this case the geometry is not Euclidean but hyperbolic instead, which complicates the analysis substantially. The purpose of this review is to present the methodology that was developed in a series of papers to investigate this quite unusual problem, specifically from the point of view of tuning and pattern formation. These methods, which rely on bifurcation theory with symmetry in the hyperbolic context, might be of interest for the modeling of other features such as color vision or other brain functions.

中文翻译：

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用于主要视觉皮层中边缘和纹理检测的双曲模型。

视觉皮层中的神经场建模涉及几何结构，这些结构以数学形式主义描述了该皮质区域的功能结构。轮廓检测和方向调整的情况已被广泛研究，并已成为大脑对图像处理进行数学分析的范例。十年前，人们尝试通过将方向（角度）替换为根据图像强度梯度建立的二阶张量来扩展这些模型，并将其称为结构张量。这种假设并非来自生物学观察（仍然缺乏实验证据），而是基于这样的想法，即大脑本身很可能会利用计算机视觉中具有结构张量的纹理处理的有效性。缺点是在这种情况下，几何形状不是欧几里得而是双曲线，这使分析变得非常复杂。这篇综述的目的是介绍在一系列论文中开发的方法，以研究这个非常不寻常的问题，特别是从调整和模式形成的角度。这些方法依赖于双曲上下文中具有对称性的分叉理论，可能对于建模其他功能（例如色觉或其他脑功能）感兴趣。