Conformal Normalization in Recurrent Neural Network of Grid Cells

Grid cells in the entorhinal cortex of the mammalian brain exhibit striking hexagon 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 grid cells collectively form a vector in a high-dimensional neural activity space, and this vector represents the self-position of the agent in the 2D physical space. As the agent moves, the vector is transformed by a recurrent neural network that takes the velocity of the agent as input. In this paper, we propose a simple and general conformal normalization of the input velocity for the recurrent neural network, so that the local displacement of the position vector in the high-dimensional neural space is proportional to the local displacement of the agent in the 2D physical space, regardless of the direction of the input velocity. Our numerical experiments on the minimally simple linear and non-linear recurrent networks show that conformal normalization leads to the emergence of the hexagon grid patterns. Furthermore, we derive a new theoretical understanding that connects conformal normalization to the emergence of hexagon grid patterns in navigation tasks.