This paper investigates the conformal isometry hypothesis as a potential explanation for the emergence of hexagonal periodic patterns in the response maps of grid cells. The hypothesis posits that the activities of the population of grid cells form a high-dimensional vector in the neural space, representing the agent's self-position in 2D physical space. As the agent moves in the 2D physical space, the vector rotates in a 2D manifold in the neural space, driven by a recurrent neural network. The conformal isometry hypothesis proposes that this 2D manifold in the neural space is a conformally isometric embedding of the 2D physical space, in the sense that local displacements of the vector in neural space are proportional to local displacements of the agent in the physical space. Thus the 2D manifold forms an internal map of the 2D physical space, equipped with an internal metric. In this paper, we conduct numerical experiments to show that this hypothesis underlies the hexagon periodic patterns of grid cells. We also conduct theoretical analysis to further support this hypothesis. In addition, we propose a conformal modulation of the input velocity of the agent so that the recurrent neural network of grid cells satisfies the conformal isometry hypothesis automatically. To summarize, our work provides numerical and theoretical evidences for the conformal isometry hypothesis for grid cells and may serve as a foundation for further development of normative models of grid cells and beyond.

翻译：本文探究共形等距假说作为网格细胞响应图谱中六边形周期模式涌现的一种潜在解释。该假说认为，网格细胞群体的活动在神经空间中形成一个高维向量，表征智能体在二维物理空间中的自身位置。当智能体在二维物理空间中移动时，该向量在循环神经网络的驱动下于神经空间的二维流形中旋转。共形等距假说提出，神经空间中的该二维流形是二维物理空间的共形等距嵌入，其意义在于神经空间中向量的局部位移与智能体在物理空间中的局部位移成比例。因此，该二维流形构成了二维物理空间的内在图谱，并配备了一种内在度量。本文通过数值实验表明该假说是网格细胞六边形周期模式的基础，并进行了理论分析以进一步支持该假说。此外，我们提出对智能体输入速度进行共形调制，使得网格细胞的循环神经网络能自动满足共形等距假说。总而言之，我们的工作为网格细胞的共形等距假说提供了数值与理论证据，并可能为网格细胞及其他神经元的规范模型进一步发展奠定基础。