Figure 4. Fisher information for modules of 3D grid cells.

(A) The three lattices considered: face-centered cubic ($\mathcal{F}\mathcal{C}\mathcal{C}$), body-centered cubic ($\mathcal{B}\mathcal{C}\mathcal{C}$), and cubic ($\mathcal{C}$). (B) $\text{tr}{\mathit{J}}_{\mathcal{L}}$ for the periodified bump-function Ω for the three lattices and various parameter combinations θ₁ and θ₂. The Fisher information (FI) of the $\mathcal{F}\mathcal{C}\mathcal{C}$ grid cells outperforms the other lattices when the support is fully within the fundamental domain (θ₂ < 0.5, see main text). For larger θ₂ the best lattice depends on the relation between the Voronoi cell's boundary and the tuning curve. (C) Ratio $\text{tr}{\mathit{J}}_{\mathcal{L}}/\text{tr}{\mathit{J}}_{\mathcal{C}}$ as a function of θ₂ for $\mathcal{L}\in \left\{\mathcal{F}\mathcal{C}\mathcal{C},\mathcal{B}\mathcal{C}\mathcal{C},\mathcal{C}\right\}$. For θ₂ < 0.5, the hexagonal population has 3/2 times the resolution of the square population, as predicted by the packing ratios. (D) Average $\text{tr}{\mathit{J}}_{{\mathcal{L}}_{\phi ,\psi }}$ for uniformly distributed grid cells within a lattice ${\mathcal{L}}_{\phi ,\psi }$ generated by basis vectors separated by angles φ and ψ (as shown above; θ₁ = θ₂ = 1/4). $\text{tr}{\mathit{J}}_{{\mathcal{L}}_{\phi ,\psi }}$ behaves like 1/(sinφ⋅sinψ) and has its maximum for the lattice with the smallest volume. (E) Distribution of 5000 realizations of $\text{tr}{\mathit{J}}_{\mathcal{L}}^M/M$ at 0 for a population of M = 200 randomly distributed neurons. Parameters: θ₁ = 1/4, θ₂ = 0.4. The means closely match the averages in (B). Due to the finite neuron number, the FI varies strongly for different realizations.

DOI: http://dx.doi.org/10.7554/eLife.05979.008