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Grid cells in the entorhinal cortex of mammalian brains exhibit strikinghexagon grid firing patterns in their response maps as the animal (e.g., a rat)navigates in a 2D open environment. In this paper, we study the emergence ofthe hexagon grid patterns of grid cells based on a general recurrent neuralnetwork (RNN) model that captures the navigation process. The responses of gridcells collectively form a high dimensional vector, representing the 2Dself-position of the agent. As the agent moves, the vector is transformed by anRNN that takes the velocity of the agent as input. We propose a simple yetgeneral conformal normalization of the input velocity of the RNN, so that thelocal displacement of the position vector in the high-dimensional neural spaceis proportional to the local displacement of the agent in the 2D physicalspace, regardless of the direction of the input velocity. We apply thismechanism to both a linear RNN and nonlinear RNNs. Theoretically, we provide anunderstanding that explains the connection between conformal normalization andthe emergence of hexagon grid patterns. Empirically, we conduct extensiveexperiments to verify that conformal normalization is crucial for the emergenceof hexagon grid patterns, across various types of RNNs. The learned patternsshare similar profiles to biological grid cells, and the topological propertiesof the patterns also align with our theoretical understanding.

Emergence of Grid-like Representations by Training Recurrent Networks with Conformal Normalization