This paper proposes a representational model for grid cells. In this model, the 2D self-position of the agent is represented by a high-dimensional vector, and the 2D self-motion or displacement of the agent is represented by a matrix that transforms the vector. Each component of the vector is a unit or a cell. The model consists of the following three sub-models. (1) Vector-matrix multiplication. The movement from the current position to the next position is modeled by matrix-vector multiplication, i.e., the vector of the next position is obtained by multiplying the matrix of the motion to the vector of the current position. (2) Magnified local isometry. The angle between two nearby vectors equals the Euclidean distance between the two corresponding positions multiplied by a magnifying factor. (3) Global adjacency kernel. The inner product between two vectors measures the adjacency between the two corresponding positions, which is defined by a kernel function of the Euclidean distance between the two positions. Our representational model has explicit algebra and geometry. It can learn hexagon patterns of grid cells, and it is capable of error correction, path integral and path planning.
In representation theory in mathematics @cite_4 , the group elements are represented by matrices acting on the vectors. We learn a special system of representation for representing self-position and self-motion.
@cite_4
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Grid cells in the medial entorhinal cortex (MEC) are part of an environment-independent spatial coordinate system. To determine how information about location, direction, and distance is integrated in the grid-cell network, we recorded from each principal cell layer of MEC in rats that explored two-dimensional environments. Whereas layer II was predominated by grid cells, grid cells colocalized with head-direction cells and conjunctive grid × head-direction cells in the deeper layers. All cell types were modulated by running speed. The conjunction of positional, directional, and translational information in a single MEC cell type may enable grid coordinates to be updated during self-motion–based navigation.