Learning Grid Cells as Vector Representation of Self-Position Coupled with Matrix Representation of Self-Motion
This addresses a neuroscience problem for understanding spatial navigation in the brain, but it appears incremental as it builds on existing representational frameworks.
The paper tackles the problem of modeling grid cells by proposing a representational model where self-position is a vector and self-motion is a matrix, enabling explicit algebra and geometry. The result is that this model can learn hexagon patterns and perform tasks like error correction, path integral, and path planning, though no concrete numbers are provided.
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.