Department of Mathematics, University of Nice Sophia-Antipolis, JAD Laboratory and CNRS, Nice, France.
PLoS Comput Biol. 2009 Dec;5(12):e1000625. doi: 10.1371/journal.pcbi.1000625. Epub 2009 Dec 24.
We propose to use bifurcation theory and pattern formation as theoretical probes for various hypotheses about the neural organization of the brain. This allows us to make predictions about the kinds of patterns that should be observed in the activity of real brains through, e.g., optical imaging, and opens the door to the design of experiments to test these hypotheses. We study the specific problem of visual edges and textures perception and suggest that these features may be represented at the population level in the visual cortex as a specific second-order tensor, the structure tensor, perhaps within a hypercolumn. We then extend the classical ring model to this case and show that its natural framework is the non-Euclidean hyperbolic geometry. This brings in the beautiful structure of its group of isometries and certain of its subgroups which have a direct interpretation in terms of the organization of the neural populations that are assumed to encode the structure tensor. By studying the bifurcations of the solutions of the structure tensor equations, the analog of the classical Wilson and Cowan equations, under the assumption of invariance with respect to the action of these subgroups, we predict the appearance of characteristic patterns. These patterns can be described by what we call hyperbolic or H-planforms that are reminiscent of Euclidean planar waves and of the planforms that were used in previous work to account for some visual hallucinations. If these patterns could be observed through brain imaging techniques they would reveal the built-in or acquired invariance of the neural organization to the action of the corresponding subgroups.
我们建议使用分岔理论和模式形成作为理论探针，来检验关于大脑神经组织的各种假设。这使我们能够根据真实大脑活动中的观察模式来进行预测，例如通过光学成像，并为设计实验来检验这些假设打开了大门。我们研究了视觉边缘和纹理感知的具体问题，并提出这些特征可能在视觉皮层中以特定的二阶张量（结构张量）的形式代表，可能在超柱内。然后，我们将经典的环模型扩展到这种情况，并表明其自然框架是非欧几里得双曲几何。这带来了其等距群的美丽结构，以及某些子群的直接解释，这些子群与假设编码结构张量的神经群体的组织有关。通过研究在这些子群作用不变的假设下，结构张量方程（类比于经典的威尔逊和考恩方程）的解的分岔，我们预测了特征模式的出现。这些模式可以通过我们所谓的双曲或 H-平面图来描述，它们让人联想到欧几里得平面波，以及之前用于解释某些视觉幻觉的平面图。如果这些模式可以通过脑成像技术观察到，它们将揭示神经组织对相应子群作用的内在或获得的不变性。
