论网格细胞的共形等距性：学习保持距离的位置嵌入
On Conformal Isometry of Grid Cells: Learning Distance-Preserving Position Embedding
AI总结：
本文提出并验证网格细胞六边形模式的共形等距假设，通过学习最大程度保持距离的位置嵌入，并理论证明六边形平环面的最优性。
摘要：
本文研究了共形等距假设，作为网格细胞响应图中六边形周期模式的一种潜在解释。我们假设网格细胞活动在神经空间中形成一个高维向量，编码智能体在二维物理空间中的位置。当智能体移动时，该向量在神经空间中的一个二维流形内旋转，由递归神经网络驱动。共形假设提出，该神经流形是二维物理空间的共形等距嵌入，其中局部物理距离通过嵌入得以保留，直至一个缩放因子（或度量单位）。这种保持距离的位置嵌入对于导航中的路径规划至关重要，尤其是规划局部直线路径段。我们进行了数值实验，表明通过学习最大程度保持距离的位置嵌入，该假设会导致六边形网格放电模式，且与递归神经网络的选择无关。此外，我们通过证明六边形平环面是最大程度保持距离的，提供了为何最小化我们的损失函数会出现六边形周期模式的理论解释。
英文摘要：
This paper investigates the conformal isometry hypothesis as a potential explanation for the hexagonal periodic patterns in grid cell response maps. We posit that grid cell activities form a high-dimensional vector in neural space, encoding the agent's position in 2D physical space. As the agent moves, this vector rotates within a 2D manifold in the neural space, driven by a recurrent neural network. The conformal hypothesis proposes that this neural manifold is a conformal isometric embedding of 2D physical space, where local physical distance is preserved by the embedding up to a scaling factor (or unit of metric). Such distance-preserving position embedding is indispensable for path planning in navigation, especially planning local straight path segments. We conduct numerical experiments to show that this hypothesis leads to the hexagonal grid firing patterns by learning maximally distance-preserving position embedding, agnostic to the choice of the recurrent neural network. Furthermore, we present a theoretical explanation of why hexagon periodic patterns emerge by minimizing our loss function by showing that hexagon flat torus is maximally distance preserving.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
