Collaborative Research: Quantifying Human Retinotopic Maps by Conformal Geometry

视网膜映射是视觉输入从视网膜到大脑神经元的映射。视网膜定位图是在大脑中特定位置发生的神经元活动的特定发生的视觉。通过分析与刺激相关的功能性磁共振成像（fMRI）反应，生成人类视皮层的视网膜定位图。有人假设，人类视网膜的地图是共形映射，但迄今为止还没有理论模型已经开发出量化这些地图。本计画利用适形几何学与功能性磁振造影资料来研究视网膜定位图，试图建立大脑视觉皮质组织的模型。该项目将：（i）计算将确定和/或验证人类视网膜中的适形性的内在几何特征;（ii）对纹外视觉区中的视网膜定位图与初级视觉皮层中的视网膜定位图之间的关系进行建模;（iii）开发量化个体受试者的视网膜定位图的方法。在这个项目中应用的数学模型结合联合收割机工具，从拓扑，共形几何，复杂的分析，优化和准共形Teichmüller理论。据估计，视觉处理区域占据了灵长类动物新皮层总表面的一半以上。然而，由于其生物学的复杂性，对视觉皮层组织的理解仍然是难以捉摸的。因此，人类视觉皮层的视网膜定位图可以成为研究大脑回路的重要工具。这个项目将产生理论上合理和实际有效的方法来量化视网膜定位图。这些方法将导致视觉功能的非侵入性生物标志物，并可能导致视觉缺陷的治疗。在这个项目中开发的计算理论和算法将在其他领域中有应用，包括计算机视觉，计算机图形学，传感器网络和几何建模。该项目通过增加我们对人脑视觉皮质组织的了解以及开发用于从大脑成像数据中发现知识的实验室基础设施，为BRAIN计划做出了贡献。该项目将为研究生和本科生创造一个跨学科的环境，并将在数学和神经科学的接口开发跨学科课程。

Retinotopy is the mapping of visual inputs from the retina to neurons in the brain. A retinotopic map is a visual of a particular occurrence of neuron activity taking place on a specific location in the brain. By analyzing the stimulus-referred functional magnetic resonance imaging (fMRI) response, retinotopic maps of the human visual cortex are generated. It has been hypothesized that human retinotopic maps are conformal mappings, but to date no theoretical models have been developed to quantify these maps. This project uses conformal geometry and fMRI data to study retinotopic maps in an attempt to model visual cortical organizations in the brain. This project will: (i) compute the intrinsic geometrical features that will determine and/or validate the conformality in the human retinotopy; (ii) model the relationships between the retinotopic maps in extrastriate visual areas and those in the primary visual cortex; (iii) develop methods to quantify retinotopic maps of individual subjects. The mathematical models applied in this project combine tools from topology, conformal geometry, complex analysis, optimization, and quasiconformal Teichmüller theory. Visual processing areas have been estimated to occupy more than half of the total surface of the primate neocortex. However, an understanding of visual cortical organizations still remains elusive due to their biological complexity. Retinotopic mapping of the human visual cortex, therefore, can be an important tool for studying the brain's circuitry. This project will produce theoretically sound and practically efficient methods for quantifying retinotopic maps. These methods will lead to non-invasive biomarkers of visual functions and may lead to cures for visual deficits. The computational theories and algorithms developed in this project will have applications in other fields, including computer vision, computer graphics, sensor networks, and geometric modeling. This project contributes to the BRAIN Initiative by increasing our knowledge of visual cortical organizations in the human brain and by developing laboratory infrastructure for knowledge discovery from brain imaging data. This project will create an interdisciplinary environment for graduate and undergraduate students and will develop interdisciplinary courses at the interface of mathematics and neuroscience.