  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.
