Efficient and flexible representation of higher-dimensional cognitive variables with grid cells
All plots show the dynamic coding range of our model (see Methods section). (a) Exact coding range of the grid code for variables of dimension 3 to 6, assuming an overly conservative phase resolution of Δ = 0.2 to reduce time of computation. We show the geometric mean and standard deviation over 1000 different draws of the projection matrices A for each pair M, N. The entries of the matrices are sampled independently from a standard normal distribution. To compute the expected value of the benchmark in Eq 3, we also run this simulation with N = 1 (S1 Fig), solid line. The capacity grows exponentially with the number of modules; the benchmark provides an estimate of the expected capacity. (b) We use the benchmark to show the coding range for more realistic values of phase resolution (Δ = 0.2, …, 0.025). We chose the benchmark rather than measuring the exact range for practical reasons (the run-time scales with the volume of the coding range not its side-length). Results shown for M/N ≥ 1.