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Grid cells in the entorhinal cortex of the mammalian brain exhibit strikinghexagon firing patterns in their response maps as the animal (e.g., a rat)navigates in a 2D open environment. The responses of the population of gridcells collectively form a vector in a high-dimensional neural activity space,and this vector represents the self-position of the agent in the 2D physicalspace. As the agent moves, the vector is transformed by a recurrent neuralnetwork that takes the velocity of the agent as input. In this paper, wepropose a simple and general conformal normalization of the input velocity forthe recurrent neural network, so that the local displacement of the positionvector in the high-dimensional neural space is proportional to the localdisplacement of the agent in the 2D physical space, regardless of the directionof the input velocity. Our numerical experiments on the minimally simple linearand non-linear recurrent networks show that conformal normalization leads tothe emergence of the hexagon grid patterns. Furthermore, we derive a newtheoretical understanding that connects conformal normalization to theemergence of hexagon grid patterns in navigation tasks.

Conformal Normalization in Recurrent Neural Network of Grid Cells