A Representational Model of Grid Cells Based on Matrix Lie Algebras
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The grid cells in the mammalian medial entorhinal cortex exhibit striking hexagon firing patterns when the agent navigates in the open field. It is hypothesized that the grid cells are involved in path integral so that the agent is aware of its self-position by accumulating its self-motion. Assuming the grid cells form a vector representation of self-position, we elucidate a minimally simple recurrent model for path integral, which models the change of the vector representation given the self-motion, and we discern two matrix Lie algebras and their Lie groups that are naturally coupled together. This enables us to connect the path integral model to the dimension reduction model for place cells via group representation theory of harmonic analysis. By reconstructing the kernel functions for place cells, our model learns hexagon grid patterns that characterize the grid cells. The learned model is capable of near perfect path integral, and it is also capable of error correction.
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