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 integration 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 grid cells' path integration based on two coupled matrix Lie algebras that underlie two coupled rotation systems that mirror the agent's self-motion : (1) When the agent moves along a certain direction, the vectoris rotated by a generator matrix. (2) When the agent changes direction, the generator matrix is rotated by another generator matrix. Our experiments show that our model learns hexagonal grid response patterns that resemble the firing patterns observed from the grid cells in the  brain. Furthermore, the learned model is capable of near exact path integration , and it is also capable of error correction. Our model is novel and simple, with explicit geometric and algebraic structures .
The purpose of this paper is to understand how the grid cells may perform path integration calculations. We study a general representational model in which the self-position is represented by a vector formed by the activities of a population of grid cells, and the self-motion is represented by the change of this vector which is transformed by a general recurrent network for path integration. We identify an isotropic condition so that the local change of this vector is a conformal embedding of the self-motion. We then study a minimally simple prototype model where the local change is a linear transformation of the vector. This linear model gives rise to explicit algebraic structure in terms of matrix Lie algebra and matrix Lie group, as well as explicit geometric structure where the self-motion is represented by the rotation of the vector. We connect the isotropic condition under the linear model to the hexagon grid patterns of the response maps of grid cells. Our numerical experiments demonstrate that our model learns hexagon grid patterns which share various observed properties of the grid cells in the rodent brain. Furthermore, the learned model is capable of near exact path integration .
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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 integration
purpose of this paper is to understand how the grid cells may perform path integration calculations. We study a general representational model in which the self-position is represented by a vector formed by the activities of a population of grid cells, and the self-motion is represented by the change of this vector which is transformed by a general recurrent network for path integration. We identify an isotropic condition
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agent is aware of its self-position by accumulating its
local change of this vector is a conformal embedding of the
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Assuming the grid cells form a vector representation of self-position, we elucidate a minimally simple recurrent model for grid cells' path integration based on two coupled matrix Lie algebras that underlie two coupled rotation systems that mirror the agent's
We then study a minimally simple prototype model where the local change is a linear transformation of the vector. This linear model gives rise to explicit algebraic structure in terms of matrix Lie algebra and matrix Lie group, as well as explicit geometric structure where the
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: (1) When the agent moves along a certain direction, the vectoris rotated by a generator matrix. (2) When the agent changes direction, the generator matrix is rotated by another generator matrix. Our experiments show
is represented by the rotation of the vector. We connect the isotropic condition under the linear model to the hexagon grid patterns of the response maps of grid cells. Our numerical experiments demonstrate
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hexagonal grid response patterns that resemble the firing patterns observed from
hexagon grid patterns which share various observed properties of
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, and it is also capable of error correction. Our model is novel and simple, with explicit geometric and algebraic structures