Title: Grid cell modeling with mapping representation of self-motion for path integration. Abstract: The representation of grid cells in the medial entorhinal cortex (MEC) region is crucial for path integration. In this paper, we proposed a grid cell modeling mechanism by mapping the agent’s self-motion in Euclidean space to the neuronal activity of grid cells. Our representational model can achieve hexagonal patterns of grid cells from recurrent neural network (RNN) and enables multi-scale path integration for 1D, 2D and 3D spaces. Different from the existing works which need to learn weights of RNN to get the vector representation of grid cells, our method can obtain weights by direct matrix operations. Moreover, compared with the classical models based on continuous attractor network (CAN), our model avoids the connection matrix’s symmetry limitation and spatial representation redundancy problems. In this paper, we also discuss the connection pattern between grid cells and place cells to demonstrate grid cells’  functioning as a metric for coding space. 