1. Computational and Systems Biology
  2. Neuroscience
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Probable nature of higher-dimensional symmetries underlying mammalian grid-cell activity patterns

  1. Alexander Mathis  Is a corresponding author
  2. Martin B Stemmler
  3. Andreas VM Herz
  1. Harvard University, United States
  2. Bernstein Center for Computational Neuroscience, Germany
  3. Ludwig-Maximilians-Universität München, Germany
Research Article
Cite this article as: eLife 2015;4:e05979 doi: 10.7554/eLife.05979
6 figures and 1 table

Figures

Grid cells and modules.

(A) Construction of a grid cell: Given a tuning shape Ω and a lattice L, here a square lattice generated by v1 and v2 with φ = π/2, one periodifies Ω with respect to L. One defines the value of ΩL(x) in the fundamental domain L as the value of Ω(r) applied to the distance from zero and then repeats this map over 2 like L tiles the space. This construction can be used for lattices L of arbitrary dimensions (Equation 7). (B) Grid module: The firing rates of three grid cells (orange, green, and blue) are indicated by color intensity. The cells' tuning is identical (Ω and L are the same), yet they differ in their spatial phases ci. Together, such identically tuned cells with different spatial phases define a grid module.

https://doi.org/10.7554/eLife.05979.003
Periodified grid-cell tuning curve L for two planar lattices, (A) the hexagonal (equilateral triangle) lattice H and (B) the square lattice Q, together with the basis vectors v1 and v2.

These are π/3 apart for the hexagonal lattice and π/2 for the square lattice. The fundamental domain, that is, the Voronoi cell around 0, is shown in gray. A few other domains that have been generated according to the lattice symmetries are marked by dashed lines. The blue disk shows the disk with maximal radius R that can be inscribed in the two fundamental domains. For equal and unitary node-to-node distances, that is, |v1|=|v2|=1, the maximal radius equals 1/2 for both lattices. The packing ratio Δ is Δ(H)=π/12 for the hexagonal and Δ(Q)=π/4 for the square lattice; the hexagonal lattice is approximately 15.5% denser than the square lattice.

https://doi.org/10.7554/eLife.05979.005
Figure 3 with 1 supplement
Fisher information for modules of two-dimensional grid cells.

(A) Top: Periodified bump-function Ω and square lattice L, for various parameter combinations θ1 and θ2. Here, θ1 modulates the decay and θ2 the support. Middle: Average trace trJL of the Fisher information (FI) for uniformly distributed grid cells ΩL. Hexagonal (H) and square (Q) lattices are considered for different θ1 and θ2 values. The FI of the hexagonal grid cells outperforms the quadratic grid when support is fully within the fundamental domain (θ2 < 0.5, see main text). Bottom: Ratio trJH/trJQ as a function of the tuning parameter θ2. For θ2 < 0.5, the hexagonal population offers 3/2 times the resolution of the square population, as predicted by the respective packing ratios. (B) Average trJL for grid cells distributed uniformly in lattices generated by basis vectors separated by an angle φ (basis depicted above graph). trJL behaves like 1/sin(φ) and has its maximum at π/3. (C) Distribution of 5000 realizations of trJLM/M at 0 for a population of M = 200 randomly distributed neurons. For both the hexagonal and square lattice, parameters are θ1 = 1/4 and θ2 = 0.4. The means closely match the average values in (A). However, due to the finite neuron number the FI varies strongly for different realizations, and in about 20% of the cases a square lattice module outperforms a hexagonal lattice.

https://doi.org/10.7554/eLife.05979.006
Figure 3—figure supplement 1
The firing rate and Fisher information of the bump tuning shape.

Upper left panel: Tuning shape Ω(r) with parameters θ2 = 0.5 and varying θ1. Lower left panel: Corresponding Fisher information (FI) integrand (r). Upper right panel: Tuning shape Ω(r) with parameters θ1 = 0.25 and varying θ2. Lower right panel: Corresponding FI integrand (r).

https://doi.org/10.7554/eLife.05979.007
Fisher information for modules of 3D grid cells.

(A) The three lattices considered: face-centered cubic (FCC), body-centered cubic (BCC), and cubic (C). (B) trJL for the periodified bump-function Ω for the three lattices and various parameter combinations θ1 and θ2. The Fisher information (FI) of the FCC 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 trJL/trJC as a function of θ2 for L{FCC,BCC,C}. 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 trJLφ,ψ for uniformly distributed grid cells within a lattice Lφ,ψ generated by basis vectors separated by angles φ and ψ (as shown above; θ1 = θ2 = 1/4). trJLφ,ψ behaves like 1/(sinφ⋅sinψ) and has its maximum for the lattice with the smallest volume. (E) Distribution of 5000 realizations of trJLM/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.

https://doi.org/10.7554/eLife.05979.008
Lattice and non-lattice solutions in 3D.

(A) Stacking of spheres as in an FCC lattice. In this densest lattice in 3D, each sphere touches 12 other spheres and there are four different planar hexagonal lattices through each node. (B) Over a layer of hexagonally arranged spheres centered at γ0 (in black) one can put another hexagonal layer by starting from one of six locations, two of which are highlighted, γ1 and γ2. (C) If one arranges the hexagonal layers according to the sequence (…,γ1, γ0, γ2,…) one obtains the FCC. Note that spheres in layer I are not aligned with those in layer III. (D) Arranging the hexagonal layers following the sequence (…,γ0, γ1, γ0,…) leads to the hexagonal close packing HCP. Again, each sphere touches 12 other spheres. However, there is only one plane through each node for which the arrangement of the centers of the spheres is a regular hexagonal lattice. This packing has the same packing ratio as the FCC, but is not a lattice. (E) trJL for bump-function Ω with L=FCC and HCP for various parameter combinations θ1 and θ2; θ1 modulates the decay and θ2 the support. The two packings have the same packing ratio and for this tuning curve also provide identical spatial resolution. FI: Fisher information.

https://doi.org/10.7554/eLife.05979.009
Author response image 1

Average trace trJL of FI for uniformly distributed grid cells ΩL for Poisson noise and bump tuning shape Ω. Left Panel: trJL for hexagonal (H in green) and square (Q in blue) lattices are shown for different θ2 values, with θ1 varying from 0.25 (lowest pair of lines), 0.1 (middle pair of lines) to 0.01 (top pair of lines). Thus, with decreasing θ1 the FI grows and reaches the maximum at the in-radius of both lattices θ2 = 0.5. Right panel: trJL for face-centered cubic (FCC in green), body-centered cubic (BCC in red) and cubic (C in blue) are shown for various θ2 and θ1 varying from 0.25 (lowest triple of lines), 0.1 (middle triple of lines) to 0.01 (top triple of lines). Again, the FI of the smallest θ1 is best and reaches its peak at θ2 = 0.5.

https://doi.org/10.7554/eLife.05979.012

Tables

Table 1

List of acronyms, variables, and terms

https://doi.org/10.7554/eLife.05979.004
DDimension of the stimulus space D
FIFisher information, usually denoted by J (Equation 3)
LNon-degenerate point lattice describing periodic structure (Equation 5)
LFundamental domain of L, which is the Voronoi cell containing 0 (Equation 6)
ΩTuning shape
supp(Ω)Support of Ω, that is, the subset where Ω does not vanish
ΩLPeriodified tuning curve on D, where L is a D-dimensional lattice and Ω a tuning curve. Simply referred to as a ‘grid cell’ (Equation 7)
ρPhase density of grid cells' phases ci within a module ρ(c)=i=1Mδ(cci)
MNumber of phases in grid module Lρ=M
ς=(Ω,ρ,L)Signature defining a grid module, which is an ensemble of grid cells differing in spatial phases ci, defined by ρ and tuning curves given by ΩL
det(L)Determinant of lattice L (equal to volume of L)
BR(0)Subset of D containing all points with distance less than R from 0
Δ(L)Packing ratio of a lattice, that is, the volume of the largest BR(0) that fits inside L divided by det(L) (Equation 15)
H, QHexagonal and square planar lattice of unit node-to-node distance (Figure 2)
FCC, BCC, CFace-centered, body-centered, and cubic lattice of unit node-to-node distance, respectively (Figure 4).
trJTrace of the FI, that is, the sum of diagonal elements
JςPopulation FI of grid module with signature ς
trJL, trJQ, trJHTrace of FI per neuron for lattice L (Q and H, respectively) with fixed bump-like Ω defined in Equation 26
trJLMTrace of FI for lattice L for M randomly distributed phases in L for the same bump function

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