The activity of the grid cell population in the medial entorhinal cortex (MEC) of the mammalian brain forms a vector representation of the self-position of the animal. Recurrent neural networks have been proposed to explain the properties of the grid cells by updating the neural activity vector based on the velocity input of the animal. In doing so, the grid cell system effectively performs path integration. In this paper, we investigate the algebraic, geometric, and topological properties of grid cells using recurrent network models. Algebraically, we study the Lie group and Lie algebra of the recurrent transformation as a representation of self-motion. Geometrically, we study the conformal isometry of the Lie group representation where the local displacement of the activity vector in the neural space is proportional to the local displacement of the agent in the 2D physical space. Topologically, the compact abelian Lie group representation automatically leads to the torus topology commonly assumed and observed in neuroscience. We then focus on a simple non-linear recurrent model that underlies the continuous attractor neural networks of grid cells. Our numerical experiments show that conformal isometry leads to hexagon periodic patterns in the grid cell responses and our model is capable of accurate path integration. Code is available at \url{https://github.com/DehongXu/grid-cell-rnn}.
翻译:哺乳动物大脑介质的内心皮层(MEC) 电网细胞群的活动构成动物自我定位的矢量表示。 提议通过根据动物的速度输入更新神经活动矢量来解释电网细胞的特性。 这样, 电网细胞系统可以有效地进行路径整合。 在本文中, 我们用经常性网络模型来调查电网细胞的代数、 几何和地形特性。 代数学上, 我们研究循环变异的利伊组和利代数, 以自我移动为代表。 从几何学上看, 我们研究利伊组和利代数, 以自我移动为代表。 我们研究利伊组代表的对应等同性等量度, 在神经空间2D物理空间中, 活动矢量性矢量矢量矢量矢量的矢量矢量矢量矢量矢量矢量矢量矢量矢量矢量矢量矢量矢量矢量。 我们的定数矩数路径实验显示, 在网格细胞的连续吸引性内心网状网络反应网状网状网状网状网状网状网状网状网状反应中, 我们的定的轨轨路图状阵列阵列阵列是可用的集体阵列阵列阵列阵列阵列阵列。