The brain of a mammal has to store vast amounts of information. The ability of animals to navigate through their environment, for example, depends on a map of the space around them being encoded in the electrical activity of a finite number of neurons. In 2014 the Nobel Prize in Physiology or Medicine was awarded to neuroscientists who had provided insights into this process. Two of the winners had shown that, in experiments on rats, the neurons in a specific region of the brain ‘fired’ whenever the rat was at any one of a number of points in space. When these points were plotted in two dimensions, they made a grid of interlocking hexagons, thereby providing the rat with a map of its environment.

However, many animals, such as bats and monkeys, navigate in three dimensions rather than two, and it is not clear whether these same hexagonal patterns are also used to represent three-dimensional space. Mathis et al. have now used mathematical analysis to search for the most efficient way for the brain to represent a three-dimensional region of space. This work suggests that the neurons need to fire at points that roughly correspond to the positions that individual oranges take up when they are stacked as tight as possible in a pile. Physicists call this arrangement a face-centered cubic lattice.

At least one group of experimental neuroscientists is currently making measurements on the firing of neurons in freely flying bats, so it should soon be possible to compare the predictions of Mathis et al. with data from experiments.

In mammals, the neural representation of space rests on at least two classes of neurons. ‘Place cells’ discharge when an animal is near one particular location in its environment (O'Keefe and Dostrovsky, 1971). ‘Grid cells’ are active at multiple locations that span an imaginary hexagonal lattice covering the environment (Hafting et al., 2005) and have been found in rats, mice, crawling bats, and human beings (Hafting et al., 2005; Fyhn et al., 2008; Yartsev et al., 2011; Jacobs et al., 2013). These cells are believed to build a metric for space.

In these experiments, locomotion occurs on a horizontal plane. Theoretical and numerical studies suggest that the hexagonal lattice structure is best suited for representing such a two-dimensional (2D) space (Guanella and Verschure, 2007; Mathis, 2012; Wei et al., 2013). In general, however, animals move in three dimensions (3D); this is particularly true for birds, tree dwellers, and fish. Their neuronal representation of 3D space may consist of a mosaic of lower-dimensional patches (Jeffery et al., 2013), as evidenced by recordings from climbing rats (Hayman et al., 2011). Place cells in flying bats, on the other hand, represent 3D space in a uniform and nearly isotropic manner (Yartsev and Ulanovsky, 2013).

As mammalian grid cells might represent space differently in 3D than in 2D, we study grid-cell representations in arbitrarily high-dimensional spaces and measure the accuracy of such representations in a population of neurons with periodic tuning curves. We measure the accuracy by the Fisher information (FI). Even though the firing fields between cells overlap, so as to ensure uniform coverage of space, we show how resolving the population's FI can be mapped onto the problem of packing non-overlapping spheres, which also plays an important role in other coding problems and cryptography (Shannon, 1948; Conway and Sloane, 1992; Gray and Neuhoff, 1998). The optimal lattices are thus the ones with the highest packing ratio—the densest lattices represent space most accurately. This remarkably simple and straightforward answer implies that hexagonal lattices are optimal for representing 2D space. In 3D, our theory makes the experimentally testable prediction that grid cells will have firing fields positioned on a face-centered-cubic lattice or its equally dense non-lattice variant—a hexagonal close packing structure.

Unimodal tuning curves with a single preferred stimulus, which are characteristic for place cells or orientation-selective neurons in visual cortex, have been extensively studied (Paradiso, 1988; Seung and Sompolinsky, 1993; Pouget et al., 1999; Zhang and Sejnowski, 1999; Bethge et al., 2002; Eurich and Wilke, 2000; Brown and Bäcker, 2006). This is also true for multimodal tuning curves that are periodic along orthogonal stimulus axes and generate repeating hypercubic (or hyper-rectangular) activation patterns (Montemurro and Panzeri, 2006; Fiete et al., 2008; Mathis et al., 2012). Our results extend these studies by taking more general stimulus symmetries into account and lead us to hypothesize that optimal lattices not only underlie the neural representation of physical space, but will also be found in the representation of other high-dimensional sensory or cognitive spaces.

Model

Population coding model for space

We consider the D-dimensional space ${\mathrm{ℝ}}^D$ in which spatial location is denoted by coordinates $x=\left(x_1,\dots ,x_D\right)\in {\mathrm{ℝ}}^D$. The animal's position in this space is encoded by N neurons. The dependence of the mean firing rate of each neuron i on x is called the neuron's tuning curve and will be denoted by Ω_i(x). To account for the trial-to-trial variability in neuronal firing, spikes are generated stochastically according to a probability $P_i\left(k_i|\tau {\text{Ω}}_i\left(x\right)\right)$ for neuron i to fire k_i spikes within a fixed time window τ. While two neurons can have correlated tuning curves Ω_i(x), we assume that the trial-to-trial variability of any two neurons is independent of each other. Thus, the conditional probability of the N statistically independent neurons to fire (k₁,…,k_N) spikes at position x summarizes the encoding model:

(1) $P\left(\left(k_1,\dots ,k_N\right)|x\right)={\displaystyle \prod }_{i=1}^N{P}_i\left(k_i|\tau {\text{Ω}}_i\left(x\right)\right).$

Decoding relies on inverting this conditional probability by asking: given a spike count vector K = (k₁,…,k_N), where is the animal? Such a position estimate will be written as $\widehat{x}\left(K\right)$. How precisely the decoding can be done is assessed by calculating the average mean square error of the decoder. The average distance between the real position of the animal x and the estimate $\widehat{x}\left(K\right)$ is

(2) $\mathit{\epsilon }\left(\widehat{x}|x\right)={\mathbb{E}}_{P\left(K|x\right)}\left(\parallel x-\widehat{x}\left(K\right)\parallel \right),$

given the population coding model $P\left(K|x\right)$. This error is called the resolution (Seung and Sompolinsky, 1993; Lehmann, 1998), whereby the term $\parallel .\parallel$ denotes Euclidean distance, $\parallel x\parallel =\sqrt{{\displaystyle \sum }_{\alpha }{x}_{\alpha }^2}$. More generally, the covariance matrix $\mathbf{\sum }\left(\widehat{x}|x\right)$ with coefficients $\mathbf{\sum }{\left(\widehat{x}|x\right)}_{\alpha ,\beta }={\mathbb{E}}_{P\left(K|x\right)}\left(\left(x_{\alpha }-\widehat{x_{\alpha }}\left(K\right)\right)\cdot \left(x_{\beta }-\widehat{x_{\beta }}\left(K\right)\right)\right)$ for spatial dimensions $\alpha ,\beta \in \left\{1,\dots ,D\right\}$, measures the covariance of the different error components, so that the sum of the diagonal elements of ∑ is just the resolution $\mathit{\epsilon }\left(\widehat{x}|x\right)$. In principle, the resolution depends on both the specific decoder and the population coding model. However, for unbiased estimators, that is, estimators that on average decode the location x as this location ${\mathbb{E}}_{P\left(K|x\right)}\left(\widehat{x}\left(K\right)\right)=x$, the FI provides an analytical measure to assess the highest possible resolution of any such decoder (Lehmann, 1998).

Resolution and Fisher Information

Given a response of K = (k₁,…,k_N) spikes across the population, we ask how accurately an ideal observer can decode the stimulus x. The FI measures how well one can discriminate nearby stimuli and depends on how P(x, K) changes with x. The greater the FI, the higher the resolution, and the lower the error $\mathit{\epsilon }\left(\widehat{x}|x\right)$, as these two quantities are inversely related. More precisely, the inverse of the FI matrix J(x),

(3) $J_{\alpha \beta }\left(x\right)={{\displaystyle \int }}^{\text{}}\left(\frac{\partial \text{\hspace{0.17em}ln\hspace{0.17em}}P\left(K,x\right)}{\partial x_{\alpha }}\right)\left(\frac{\partial \text{\hspace{0.17em}ln\hspace{0.17em}}P\left(K,x\right)}{\partial x_{\beta }}\right)P\left(K,x\right)\text{\hspace{0.17em}d}K,$

bounds the covariance matrix $\mathbf{\sum }\left(\widehat{x}|x\right)$ of the estimated coordinates x = (x₁,…,x_D)

(4) $\mathbf{\sum }\left(\widehat{x}|x\right)\ge \mathit{J}{\left(x\right)}^{-1}.$

The resolution of any unbiased estimator of the encoded stimulus can achieve cannot be greater than J(x)⁻¹. This is known as the Cramér-Rao bound (Lehmann, 1998). Based on this bound, we will consider the FI as a measure for the resolution of the population code. In particular, we are interested in isotropic and homogeneous representations of space. These two conditions assure that the population has the same resolution at any location and along any spatial axis. Isotropy does not entail that the (global) spatial tuning of an individual neuron, Ω_i(x), has to be radially symmetric, but merely that the errors are (locally) distributed according to a radially symmetric distribution. For instance, the tuning curve of a grid cell with hexagonal tuning is not radially symmetric around the center of a field (it has three axes), but the posterior is radially symmetric around any given location for a module of such grid cells. Homogeneity requires that the FI J(x) be asymptotically independent of x (as the number of neurons N becomes large); spatial isotropy implies that all diagonal entries in the FI matrix J(x) are equal.

Periodic tuning curves

Grid cells have periodic tuning curves—they are active at multiple locations, called firing fields, and these firing fields are hexagonally arranged in the environment (Hafting et al., 2005). Their periodic structure is given by a hexagonal lattice. The periodic structure of the tuning curve Ω_i(x) reflects its symmetries, that is, the set of vectors that map the tuning curve onto itself. Since we want to understand how the periodic structure affects the resolution of the population code, we generalize the notion of a grid cell to allow different periodic structures other than just hexagonal. Mathematically, the symmetries of a periodic structure can be described by a lattice $\mathcal{L}$, which is constructed as follows: take a set of independent vectors (v_α)_1≤α≤D in D-dimensional space ${\mathrm{ℝ}}^D$, and consider all possible combinations of these vectors and their integer multiples—each such vector combination points to a node of the lattice, such that the union of these represents the lattice itself. For instance, the square lattice (Figure 1A, bottom) is given by basis vectors v₁ = (1, 0) and v₂ = (0, 1). Mathematically, the lattice $\mathcal{L}\subset {\mathrm{ℝ}}^D$ is

(5) $\begin{array}{lll}\mathcal{L}={\displaystyle \sum }_{\alpha =1}^D{k}_{\alpha }{v}_{\alpha }& \text{for}& k_{\alpha }\in \mathrm{\mathbb{Z}},v_{\alpha }\in {\mathrm{ℝ}}^D\end{array},$

for which (v_α)_1≤α≤D is a basis of ${\mathrm{ℝ}}^D$. We will not consider degenerate lattices. In this work, we follow the nomenclature from Conway and Sloane (1992). Applied fields might differ slightly in their terminology, especially regarding naming conventions for packings, which are generalizations of lattices (Whittaker, 1981; Nelson, 2002). We will address these generalizations of lattices below.

Figure 1

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Based on such a lattice $\mathcal{L}$, we construct periodic tuning curves as illustrated in Figure 1A. We start with a lattice $\mathcal{L}$ and a tuning shape $\text{Ω}:{\mathrm{ℝ}}^{+}\to [0,1]$ that decays from unity to zero; Ω(r) describes the firing rate of the periodified tuning curve at distance r from any lattice point and should be at least twice continuously differentiable. Each lattice point $p\in \mathcal{L}$ has a domain $V_p\subset {\mathrm{ℝ}}^D$ called the Voronoi region, which is defined as

(6) $V_p=\left\{x\in {\mathrm{ℝ}}^D|\parallel x-p\parallel <\parallel x-q\parallel \forall q\in \mathcal{L}\wedge p\ne q\right\},$

that contains all points x that are closer to p than to any other lattice point q. Note that V_p ∩ V_q = ϕ if p ≠ q and that for all $p,q\in \mathcal{L}$ there exists a unique vector $v\in \mathcal{L}$ with V_p = V_q + v.

The domain that contains the null (0) vector is called the fundamental domain and is denoted by L:= V₀. For each $x\in {\mathrm{ℝ}}^D$ there is a unique lattice point $p\in \mathcal{L}$ that maps x into the fundamental domain: $x-p\in L$. Let us call this mapping ${\pi }_{\mathcal{L}}$. With this notation one can periodify Ω onto $\mathcal{L}$ by defining a grid cell's tuning curve as ${\text{Ω}}^{\mathcal{L}}$:

(7) ${\text{Ω}}^{\mathcal{L}}\left(x\right):{\mathrm{ℝ}}^D\to {\mathrm{ℝ}}^{+},x\mapsto f_{max}\cdot \text{Ω}\left({\parallel {\pi }_{\mathcal{L}}\left(x\right)\parallel }^2\right),$

where f_max is the peak firing rate of the neuron. Note that throughout the paper we set f_max = τ = 1, for simplicity. As illustrated in Figure 1A, within the fundamental domain L, the tuning curve ${\text{Ω}}^{\mathcal{L}}$ defined above is radially symmetric. This pattern is repeated along the nodes of $\mathcal{L}$, akin to ceramic tiling.

A grid module is defined as an ensemble of M grid cells ${\text{Ω}}_i^{\mathcal{L}}$, $i\in \left\{1,\dots ,M\right\}$ with identical, but spatially shifted tuning curves, that is, ${\text{Ω}}_i^{\mathcal{L}}\left(x\right)={\text{Ω}}^{\mathcal{L}+c_i}\left(x\right)$ and spatial phases $c_i\in L$ (see Figure 1B). The various phases within a module can be summarized by their phase density $\rho \left(c\right)={{{\displaystyle \sum }}^{\text{}}}_{i=1}^M\delta \left(c-c_i\right)$. This definition is motivated by the observation of spatially shifted hexagonally tuned grid cells in the entorhinal cortex of rats (Hafting et al., 2005; Stensola et al., 2012).

Any grid module is uniquely characterized by its signature $\left(\text{Ω},\rho ,\mathcal{L}\right)$. To investigate the role of different periodic structures, we can fix the tuning shape Ω and density ρ and solely vary the lattice $\mathcal{L}$ to find the lattice that yields the highest FI.

To determine how the resolution of a grid module depends on the periodic structure $\mathcal{L}$, we compute the population FI J_ς(x) for a module of grid cells with signature $\varsigma =\left(\text{Ω},\rho ,\mathcal{L}\right)$, which describes the tuning shape, the density of firing fields, and the lattice. By fixing the tuning shape Ω and the number $\left|\rho \right|=M$ of spatial phases, we can compare the resolution for different periodic structures. (Table 1 contains a glossary of the variables.)

Packing ratio of lattices

The sphere packing problem is of general interest in mathematics (Conway and Sloane, 1992) and has wide-ranging applications from crystallography to information theory (Barlow, 1883; Shannon, 1948; Whittaker, 1981; Gray and Neuhoff, 1998; Gruber, 2004). When packing R-balls B_R in ${\mathrm{ℝ}}^D$ in a non-overlapping fashion, the density of the packing is defined as the fraction of the space covered by balls. For a lattice $\mathcal{L}$, it is given by

(15) $\frac{\text{vol}\left(B_R\left(0\right)\right)}{\text{det}\left(\mathcal{L}\right)},$

which is known as the packing ratio $\text{Δ}\left(\mathcal{L}\right)$ of the lattice. For a given lattice, this ratio is maximized by choosing the largest possible R, known as the packing radius, which is defined as the in-radius of a Voronoi region containing the origin (Conway and Sloane, 1992). Figure 2 depicts the disks with the largest in-radius for the hexagonal and the square lattice in blue and illustrates the packing ratio.

Figure 2

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FI and packing ratio

We now come to the main finding of this study: among grid modules with different lattices, the lattice with the highest packing ratio leads to the highest spatial resolution.

To derive this result, let us fix a tuning shape Ω with supp(Ω) = [0, R], lattices ${\mathcal{L}}_j$ such that B_R(0) ⊂ L_j for 1 ≤ j ≤ K, and uniform densities ρ for each fundamental domain of equal cardinality M. Any linear order on the packing ratios,

(16) $\text{Δ}\left({\mathcal{L}}_1\right)\le \dots \le \text{Δ}\left({\mathcal{L}}_j\right)\le \dots \le \text{Δ}\left({\mathcal{L}}_K\right),$

is translated by Equation 14 into the same order for the traces of the FI

(17) $\text{tr}{\mathit{J}}_{{\text{Ω}}^{{\mathcal{L}}_1}}\le \dots \le \text{tr}{\mathit{J}}_{{\text{Ω}}^{{\mathcal{L}}_j}}\le \dots \le \text{tr}{\mathit{J}}_{{\text{Ω}}^{{\mathcal{L}}_K}},$

and thus the resolution of these modules: the higher the packing ratio, the higher the FI of a grid module.

The condition supp(Ω) = [0, R] with B_R(0) ⊂ L, although restrictive, is consistent with experimental observations that grid cells tend to stop firing between grid fields and that the typical ratio between field radius and spatial period is well below 1/2 (Hafting et al., 2005; Brun et al., 2008; Giocomo et al., 2011). Generally, the tuning width that maximizes the FI does not necessarily satisfy this condition; see Figures 3, 4, in which the optimal support radius of the tuning curve θ₂ is greater than the in-radius R = 1/2 of L. The same observation will hold in higher dimensions (D > 2), consistent with the finding that the optimal tuning width for Gaussian tuning curves increases with the number of spatial dimensions, whether space is infinite (Zhang and Sejnowski, 1999) or finite (Brown and Bäcker, 2006). When the radius R of the support of the tuning curve exceeds the in-radius, the optimal lattice can be different from the densest one as we will show numerically for specific tuning curves and Poisson noise. However, with well separated fields, like those observed experimentally, the densest lattice provides the highest resolution for any tuning shape Ω, as we just demonstrated.

Figure 3 with 1 supplement see all

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Figure 4

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The optimal packing ratio of lattices for low-dimensional space is well known. Having established our main result, we can now draw on a rich body of literature, in particular Conway and Sloane (1992), to discuss the expected firing-field structure of grid cells in 2D and 3D environments.

Equally optimal non-lattice solutions for grid-cell tuning

Fruit is often arranged in an $\mathcal{F}\mathcal{C}\mathcal{C}$ formation (Figure 5A). One arrives at this lattice by starting from a layer of hexagonally placed spheres. This requires two basis vectors to be specified and is the densest packing in 2D. To maximize the packing ratio in 3D, the next layer of hexagonally arranged spheres has to be stacked as tightly as possible. There are two choices for the third and final basis vector achieve this packing, denoted as γ₁ and γ₂ in Figure 5B (modulo hexagonal symmetry). If one chooses γ₁, then two layers below there is no sphere with its center at location γ₁, but instead there is one at γ₂ (and vice versa). This stacking of layers is shown in Figure 5C and generates the $\mathcal{F}\mathcal{C}\mathcal{C}$ lattice.

Figure 5

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One could achieve the same density by choosing γ₁ for both the top layer and the layer below the basis layer. Yet as this arrangement, called hexagonal close packing ($\mathcal{HCP}$), cannot be described by three vectors, it does not define a lattice (see Figure 5D), even though it is as tightly packed as the $\mathcal{F}\mathcal{C}\mathcal{C}$. Such packings, defined as an arrangement of equal non-overlapping balls (Conway and Sloane, 1992; Hales, 2012), generalize lattices.

While one can define a grid module for any lattice, as we showed above, one cannot define a grid module in a meaningful way for an arbitrary packing, due to the lack of symmetry. But for any given packing $\mathcal{P}$ of ${\mathrm{ℝ}}^D$ by balls $B_1$ of radius 1, one can define a ‘grid cell’ by generalizing the definition given for lattices (Equation 7). To this end, consider the Voronoi partition of ${\mathrm{ℝ}}^D$ by $\mathcal{P}$. For each location $x\in {\mathrm{ℝ}}^D$ there is a unique Voronoi cell V_p with node $p\in \mathcal{P}$. One defines the grid cell's tuning curve ${\text{Ω}}^{\mathcal{P}}\left(x\right)$ by assigning the firing rate according to $\text{Ω}\left({\parallel p-x\parallel }^2\right)$ for tuning shape Ω and distance $\parallel p-x\parallel$. Depending on the specific packing, this tuning curve ${\text{Ω}}^{\mathcal{P}}$ may or may not be periodic. Because a packing $\mathcal{P}$ often has fewer symmetries than a lattice $\mathcal{L}$, the ‘grid cells’ in an arbitrary $\mathcal{P}$ cannot generally be used to define a ‘grid module’. To explain why, consider an arbitrary packing and the unique Voronoi cell V₀ that contains the point 0. Choose M uniformly distributed phases c₁,…,c_M within V₀. Locations within V₀ will then be uniformly covered by shifted tuning curves ${\text{Ω}}_i\left(x\right):={\text{Ω}}^{\mathcal{P}}\left(x-c_i\right)$. However, typically the different Voronoi cells will neither be congruent, nor have similar volumes. Thus, the Ω_i will typically not cover each Voronoi cell with the same density and will therefore fail to define a proper grid module. This problem does not exist for lattices. Here, the equivalence classes $c_i+\mathcal{L}$ cover each cell with the same density.

Highly symmetric packings, on the other hand, do permit the definition of grid modules. For example, the hexagonal close packing $\mathcal{HCP}$ can be used to define a grid cell ${\text{Ω}}^{\mathcal{HCP}}\left(x\right)$. Using the same symmetry argument from Equations 9–11, implies for the FI:

(18) ${\mathit{J}}_{\left(\text{Ω},\rho ,\mathcal{H}\mathcal{C}\mathcal{P}\right)}\left(x\right)\approx {\mathit{J}}_{\left(\text{Ω},\rho ,\mathcal{H}\mathcal{C}\mathcal{P}\right)}\left(0\right)\approx \frac{M}{\text{vol}\left(V_0\right)}{\displaystyle {\int }_{V_0}{\mathit{J}}_{{\text{Ω}}^{\mathcal{H}\mathcal{C}\mathcal{P}}}\left(c\right)}\text{d}c.$

The maximal in-radius R for the $\mathcal{HCP}$ with grid size λ = 1 is equal to 1/2. Like for lattices, we assume that supp(Ω) = [0, R] and B_R(0) ⊂ V₀. Then the integrand vanishes for distances larger than 1/2 from 0. Hence, we obtain:

(19) ${\mathit{J}}_{\left(\text{Ω},\rho ,\mathcal{H}\mathcal{C}\mathcal{P}\right)}\left(0\right)\approx \frac{M}{\text{vol}\left(V_0\right)}{\displaystyle {\int }_{B_{1/2}\left(0\right)}{\mathit{J}}_{{\text{Ω}}^{\mathcal{H}\mathcal{C}\mathcal{P}}}\left(c\right)}\text{d}c.$

Considering the same tuning shape Ω and number of phases M for an $\mathcal{F}\mathcal{C}\mathcal{C}$ lattice, which also has maximal in-radius 1/2, Equation 13 gives us the following expression for the $\mathcal{F}\mathcal{C}\mathcal{C}$ lattice:

(20) ${\mathit{J}}_{\left(\text{Ω},\rho ,\mathcal{F}\mathcal{C}\mathcal{C}\right)}\left(0\right)\approx \frac{M}{\text{det}\left(\mathcal{F}\mathcal{C}\mathcal{C}\right)}{\displaystyle {\int }_{B_{1/2}\left(0\right)}{\mathit{J}}_{{\text{Ω}}^{\mathcal{F}\mathcal{C}\mathcal{C}}}\left(c\right)}\text{d}c.$

Since both fundamental domains have the same volumes, that is, $\text{det}\left(\mathcal{F}\mathcal{C}\mathcal{C}\right)=\text{vol}\left(V_0\right)$, and the integrands restricted to these balls are identical, that is, $J_{{\text{Ω}}^{\mathcal{F}\mathcal{C}\mathcal{C}}}{|}_{B_{1/2}\left(0\right)}=J_{{\text{Ω}}^{\mathcal{H}\mathcal{C}\mathcal{P}}}{|}_{B_{1/2}\left(0\right)}$, we can conclude that grid modules comprising $\mathcal{F}\mathcal{C}\mathcal{C}$ or $\mathcal{HCP}$-like symmetries have the same FI. We also numerically calculate the trace of the average FI for a module of $\mathcal{HCP}$ grid cells and compare it to the $\mathcal{F}\mathcal{C}\mathcal{C}$ case. For bump-like tuning curves Ω, both FIs are identical (Figure 5E) as expected from the radial symmetry of Ω. As a consequence, grid cells defined by either $\mathcal{HCP}$ or $\mathcal{F}\mathcal{C}\mathcal{C}$ symmetries provide optimal resolution.

Figure 5D,E shows that the cyclic sequences (γ₀, γ₁) and (γ₁, γ₀, γ₂) lead to $\mathcal{HCP}$ and $\mathcal{F}\mathcal{C}\mathcal{C}$, respectively. The centers γ₀, γ₁, and γ₂ can also be used to make a final point on packings: there are infinitely many distinct packings with the same density $\pi /(3\sqrt{2})$. They can be constructed by inequivalent words, generated by finitewalks through the triangle with letters γ₀, γ₁, and γ₂ (Hales, 2012), with each letter representing one of three orientations for the layers. For instance, (γ₀, γ₁, γ₀, γ₂) describes another packing with the same density. All packings share one feature: around each sphere there are exactly 12 spheres, arranged in either $\mathcal{HCP}$ or $\mathcal{F}\mathcal{C}\mathcal{C}$ lattice fashion (Hales, 2012). These packings can also be used to define a grid module, because the density of phases will be uniform in all cells. Furthermore, as in the calculation of the FI for the $\mathcal{HCP}$ and $\mathcal{F}\mathcal{C}\mathcal{C}$ (Equation 18–20) only local integration was necessary, such mixed packings will have equally large, uniform FI as the pure $\mathcal{HCP}$ or $\mathcal{F}\mathcal{C}\mathcal{C}$ packings.

Only in recent years has it been proven that no other arrangement has a higher packing ratio than the $\mathcal{F}\mathcal{C}\mathcal{C}$, a problem known as Kepler's conjecture (Hales, 2005, 2012). Based on these results and our comparison of $\text{tr}{\mathit{J}}_{\mathcal{HCP}}$ and $\text{tr}{\mathit{J}}_{\mathcal{F}\mathcal{C}\mathcal{C}}$ (Figure 5E), we predict that 3D grid cells will correspond to one of these packings. While there are equally dense packings as the densest lattice in 3D, this is not the case in 2D. Thue proved that the hexagonal lattice is unique in being the densest amongst all planar packings (Thue, 1910); grid cells in 2D should possess a hexagonal lattice structure.

Grid cells are active when an animal is near one of any number of multiple locations that correspond to the vertices of a planar hexagonal lattice (Hafting et al., 2005). We generalize the notion of a grid cell to arbitrary dimensions, such that a grid cell's stochastic activity is modulated in a spatially periodic manner within ${\mathrm{ℝ}}^D$. The periodicity is captured by the symmetry group of the underlying lattice $\mathcal{L}$. A grid module consists of multiple cells with equal spatial period but different spatial phases. Using information theory, we then asked which lattice offers the highest spatial resolution.

We find that the resolution of a grid module is related to the packing ratio of $\mathcal{L}$—the lattice with highest packing ratio corresponds to the grid module with highest resolution. Well-known results from mathematics (Lagrange, 1773; Gauss, 1831; Conway and Sloane, 1992) then show that the hexagonal lattice is optimal for representing 2D, whereas the $\mathcal{F}\mathcal{C}\mathcal{C}$ lattice is optimal for 3D. In 3D, but not in 2D, there are also non-lattice packings with the same resolution as the densest lattice (Thue, 1910; Hales, 2012). A common feature of these highly symmetric optimal solutions in 3D is that each grid field is surrounded by 12 other grid fields, arranged in either $\mathcal{F}\mathcal{C}\mathcal{C}$ lattice or hexagonal close packing fashion. These solutions emerge from the set of all possible packings simply by maximizing the resolution, as we showed. However, resolution alone, as measured by the FI, does not distinguish between optimal packing solutions with different symmetries. Whether a realistic neuronal decoder, such as one based on population vector averages, favors one particular solution is an interesting open question.

As we have demonstrated, using the FI makes finding the optimal $\mathcal{L}$ analytically tractable for all dimensions D and singles out densest lattices as optimal tuning shapes under assumptions that are restrictive, but are consistent with experimental measurements (Hafting et al., 2005; Brun et al., 2008; Giocomo et al., 2011). The assumption that the tuning curves must have finite support within the fundamental domain of the lattice corresponds to grid cells being silent outside of the firing field. Indeed, our numerical simulations also showed that for broader tuning curves, grid modules with quadratic lattices can provide more FI than the hexagonal lattice (Figure 3A, θ₂ ≈ 0.6 and θ₁ = 1/4) and that grid cells with a $\mathcal{C}$ or $\mathcal{B}\mathcal{C}\mathcal{C}$ lattice can provide more FI than the $\mathcal{F}\mathcal{C}\mathcal{C}$ (Figure 4B, θ₂ > 0.65 and θ₁ = 1/4). For the planar case, Guanella and Verschure (2007) show numerically that triangular tessellations yield lower reconstruction errors under maximum-likelihood decoding than equivalently scaled square grids. Complementing this numerical analysis, Wei et al. (2013) provide a mathematical argument that hexagonal grids are optimal. To do so, they define the spatial resolution of a single module representing 2D space as the ratio R = (λ/l)², where λ is the grid scale and l is the diameter of the circle in which one can determine the animal's location with certainty. For a fixed resolution R, the number of neurons required is N = d sin(φ) R in their analysis, where d is the number of tuning curves covering each point in space. As φ ∈ [π/3, π/2] for the lattice to have unitary length (Figure 3B), minimizing N for a fixed resolution R yields φ = π/3; thus, hexagonal lattices should be optimal. Furthermore, Wei et al. show that this result also holds when considering a Bayesian decoder (Wei et al., 2013). While Wei et al. minimize N for fixed l, we minimize l (in their notation). Like Wei et al., we assume that the tuning curve Ω is isotropic (notwithstanding the fact that the lattice has preferred directions); unlike these authors, we show that there are conditions under which the firing fields should be arranged in a square lattice, and not hexagonally.

Using the FI gives a theoretical bound for the local resolution of any unbiased estimator (Lehmann, 1998). In particular, this local resolution does not take into account the ambiguity introduced by the periodic nature of the lattice. Our analysis is restricted to resolving the animal's position within the fundamental domain. For large neuron numbers N and expected peak spike counts f_maxτ the resolution of asymptotically efficient decoders, like the maximum likelihood decoder, or the minimum mean square estimator, can indeed attain the resolution bound given by the FI (Seung and Sompolinsky, 1993; Bethge et al., 2002; Mathis et al., 2013). Thus, for these decoders and conditions the results hold. In contrast, for small neuron numbers and peak spike counts, the optimal codes could be different, just as it has been shown in the past that the optimal tuning width in these cases cannot be predicted by the FI (Bethge et al., 2002; Yaeli et al., 2010; Berens et al., 2011; Mathis et al., 2012).

Maximizing the resolution explains the observed hexagonal patterns of grid cells in 2D, and predicts an $\mathcal{F}\mathcal{C}\mathcal{C}$ lattice (or equivalent packing) for grid-cell tuning curves of mammals that can freely explore the 3D nature of their environment. Quantitatively, we demonstrated that these optimal populations provide 15.5% (2D) and about 41% (3D) more resolution than grid codes with quadratic or cubic grid cells for the same number of neurons. Although better, this might not seem substantial, at least not at the level of a single grid module. However, as medial entorhinal cortex harbors a nested grid code with at least 5 and potentially 10 or more modules (Stensola et al., 2012), this translates into a much larger gain of ${1.155}^{5\dots 10}\approx 2.1\dots 4.2$ and ${\sqrt{2}}^{5\dots 10}\approx 5.7\dots 32$, respectively (Mathis et al., 2012a, 2012b). Because aligned grid-cell lattices with perfectly periodic tuning curves imply that the posterior is periodic too (compare Equation 8), information from different scales would have to be combined to yield an unambiguous read-out. Whether the nested scales are indeed read out in this way in the brain remains to be seen (Mathis et al., 2012a, 2012b; Wei et al., 2013). An alternative hypothesis, as first suggested by Hafting et al., is that the slight, but apparently persistent irregularities in the firing fields across space (Hafting et al., 2005; Krupic et al., 2015; Stensola et al., 2015) are being used. Future experiments should tackle this key question.

We considered perfectly periodic structures (lattices) and asked which ones provide most resolution. However, the first recordings of grid cells already showed that the fields are not exactly hexagonally arranged and that different fields might have different peak firing rates (Hafting et al., 2005). More recently, deviations from hexagonal symmetry have gained considerable attention (Derdikman et al., 2009; Krupic et al., 2013, 2015; Stensola et al., 2015). Such ‘defects’ modulate the periodicity of the tuning and consequently affect the symmetry of the likelihood function. This might imply that a potential decoder might be able to distinguish different unit cells even given a single module, which is not possible for perfectly periodic tuning curves (compare Equation 8). The local resolution, on the other hand, is robust to small, incoherent variations as the FI is a statistical average over many tuning curves with different spatial phases. At a given location, Equation 9 becomes

where $\overline{{\text{Ω}}^{\mathcal{L}}}$ is the average of the variable tuning curves ${\text{Ω}}_i^{\mathcal{L}}$. Small variations in the peak rate and grid fields will therefore average out, unless these variations are coherent across grid cells. Thus, resolution bounded by the FI is robust with respect to minor differences in peak firing rates and hexagonality. Similar arguments hold in higher dimensions.

In this study, we focused on optimizing grid modules for an isotropic and homogeneous space, which means that the resolution should be equal everywhere and in each direction of space. From a mathematical point of view, this is the most general setting, but it is certainly not the only imaginable scenario; future studies should shed light on other geometries. Indeed, the topology of natural habitats, such as burrows or caves, can be highly complicated. Higher resolution might be required at spatial locations of behavioral relevance. Neural representations of 3D space may also be composed of multiple 1D and 2D patches (Jeffery et al., 2013). However, the mere fact that these habitats involve complicated low-dimensional geometries does not imply that an animal cannot acquire a general map for the environment. Poincaré already suggested that an isotropic and homogeneous representation for space can emerge out of non-Euclidean perceptual spaces, as one can move through physical space by learning the motion group (Poincaré, 1913). An isotropic and homogeneous representation of 3D space facilitates (mental) rotations in 3D and yields local coordinates that are independent of the environment's topology. On the other hand, the efficient-coding hypothesis (Barlow, 1959; Atick, 1992; Simoncelli and Olshausen, 2001) would argue that surface-bound animals might not need to dedicate their limited neuronal resources to acquiring a full representation of space, as flying animals might have to do, so that representations of 3D space will be species-specific (Las and Ulanovsky, 2014). Desert ants represent space only as a projection to flat space (Wohlgemuth et al., 2001; Grah et al., 2007). Likewise, experimental evidence suggests that rats do not encode 3D space in an isotropic manner (Hayman et al., 2011), but this might be a consequence of the specific anisotropic spatial navigation tasks these rats had to perform. Data from flying bats, on the other hand, demonstrate that, at least in this species, place cells represent 3D space in a uniform and nearly isotropic manner (Yartsev and Ulanovsky, 2013). The 3D, toroidal head-direction system in bats also suggests that they have access to the full motion group (Finkelstein et al., 2014). Our theoretical analysis assumes that the same is true for bat grid cells and that they have radially symmetric firing fields. From these assumptions, we showed the grid cells' firing fields should be arranged on an $\mathcal{F}\mathcal{C}\mathcal{C}$ lattice or packed as $\mathcal{HCP}$. Interestingly, such solutions also evolve dynamically in a self-organizing network model for 3D (Stella et al., 2013; Stella and Treves, 2015) that extends a previous 2D system which exhibits hexagonal grid patterns (Kropff and Treves, 2008). Experimentally, the effect of the arena's geometry on grid cells' tuning and anchoring has also been a question of great interest (Derdikman et al., 2009; Krupic et al., 2013, 2015; Stensola et al., 2015). First, let us note that even though the environment might be finite, the grid-cell representation need not be constrained by it. In particular, the firing fields are not required to be contained within the confines of the four walls of a box—experimental observations show that walls can intersect the firing fields (so that one measures only a part of the firing field). On the other hand, the borders clearly distort the hexagonal arrangement of nearby firing fields in 2D environments (Stensola et al., 2015), whereas central fields are more perfectly arranged. Deviations are also observed when only a few fields are present in the arena (Krupic et al., 2015). One might expect similar deviations in 3D, such as for bats flying in a confined space. Our mathematical results rely on symmetry arguments that do not cover non-periodic tuning curves. Given that the resolution is related to the packing ratio of a lattice, extensions of the theory to general packings might allow one to draw on the rich field of optimal finite packings (Böröczky, 2004; Toth et al., 2004), thereby providing new hypotheses to test.

Many spatially modulated cells in rat medial entorhinal cortex have hexagonal tuning curves, but some have firing fields that are spatially periodic bands (Krupic et al., 2012). The orientation of these bands tends to coincide with one of the lattice vectors of the grid cells (as the lattices for different grid cells share a common orientation), so band cells might be a layout ‘defect’. In this context, we should point out that the lattice solutions are not globally optimal. For instance, in 2D, a higher resolution can result from two systems of nested 1D grid codes, which are aligned to the x and y axis, respectively, than from a lattice solution with the same number of neurons. The 1D cells would behave like band cells (Krupic et al., 2012). Similar counterexamples can be given in higher dimensions too. The anisotropy of the spatial tuning in grid cells of climbing rats when encoding 3D space (Hayman et al., 2011) might capitalize on this gain (Jeffery et al., 2013). Radial symmetry of the tuning curve may also be non-optimal. For example, two sets of elliptically tuned 2D unimodal cells, with orthogonal short axes, typically outperform unimodal cells with radially symmetric tuning curves (Wilke and Eurich, 2002). Why experimentally observed place fields and other tuning curves seem to be isotropically tuned is an open question (O'Keefe and Dostrovsky, 1971; Yartsev and Ulanovsky, 2013).

Grid cells which represent the position of an animal (Hafting et al., 2005) have been discovered only recently. By comparison, in technical systems, it has been known since the 1950s that the optimal quantizers for 2D signals rely on hexagonal lattices (Gray and Neuhoff, 1998). In this context, we note that lattice codes are also ideally suited to cover spaces that involve sensory or cognitive variables other than location. In higher-dimensional feature spaces, the potential gain could be enormous. For instance, the optimal eight-dimensional (8D) lattice is about 16 times denser than the orthogonal 8D lattice (Conway and Sloane, 1992) and would, therefore, dramatically increase the resolution of the corresponding population code. Advances in experimental techniques, which allow one to simultaneously record from large numbers of neurons (Ahrens et al., 2013; Deisseroth and Schnitzer, 2013) and to automate stimulus delivery for dense parametric mapping (Brincat and Connor, 2004), now pave the way to search for such representations in cortex. For instance, by parameterizing 19 metric features of cartoon faces, such as hair length, iris size, or eye size, Freiwald et al. showed that face-selective cells are broadly tuned to multiple feature dimensions (Freiwald et al., 2009). Especially in higher cortical areas, such joint feature spaces should be the norm rather than the exception (Rigotti et al., 2013). While no evidence for lattice codes was found in the specific case of face-selective cells, data sets like this one will be the test-bed for checking the hypothesis that other nested grid-like neural representations exist in cortex.

We study population codes of neurons encoding the D-dimensional space by considering the FI J as a measure for their resolution. The population coding model, the construction to periodify a tuning shape Ω onto a lattice $\mathcal{L}$ with center density ρ, as well as the definition of the FI, are given in the main text. In this section we give further background on the methods.

1. 1. Ahrens MB
   2. Orger MB
   3. Robson DN
   4. Li JM
   5. Keller PJ

   (2013) Whole-brain functional imaging at cellular resolution using light-sheet microscopy

   Nature Methods 10:413–420.

   https://doi.org/10.1038/nmeth.2434

   • Google Scholar
2. 1. Atick JJ

   (1992) Could information theory provide an ecological theory of sensory processing?

   Network Computation in Neural Systems 3:213–251.

   https://doi.org/10.1088/0954-898X/3/2/009

   • Google Scholar
3. 1. Barlow W

   (1883) Probable nature of the internal symmetry of crystals

   Nature 29:205–207.

   https://doi.org/10.1038/029205a0

   • Google Scholar
4. 1. Barlow HB

   (1959)

   NPL Symposium on the Mechanization of Thought Process. No. 10

   535–539, NPL Symposium on the Mechanization of Thought Process. No. 10, London, Her Majesty's Stationary Office.

   • Google Scholar
5. 1. Berens P
   2. Ecker AS
   3. Gerwinn S
   4. Tolias AS
   5. Bethge M

   (2011) Reassessing optimal neural population codes with neurometric functions

   Proceedings of the National Academy of Sciences of USA 108:4423–4428.

   https://doi.org/10.1073/pnas.1015904108

   • Google Scholar
6. 1. Bethge M
   2. Rotermund D
   3. Pawelzik K

   (2002) Optimal short-term population coding: when Fisher information fails

   Neural Computation 14:2317–2351.

   https://doi.org/10.1162/08997660260293247

   • Google Scholar
7. Book

   1. Böröczky K

   (2004)

   Finite packing and covering (2nd edition)

   Cambridge University Press.

   • Google Scholar
8. 1. Brincat SL
   2. Connor CE

   (2004) Underlying principles of visual shape selectivity in posterior inferotemporal cortex

   Nature Neuroscience 7:880–886.

   https://doi.org/10.1038/nn1278

   • Google Scholar
9. 1. Brown WM
   2. Bäcker A

   (2006) Optimal neuronal tuning for finite stimulus spaces

   Neural Computation 18:511–1526.

   https://doi.org/10.1162/neco.2006.18.7.1511

   • Google Scholar
10. 1. Brun VH
    2. Solstad T
    3. Kjelstrup KB
    4. Fyhn M
    5. Witter MP
    6. Moser EI
    7. Moser MB

    (2008) Progressive increase in grid scale from dorsal to ventral medial entorhinal cortex

    Hippocampus 18:1200–1212.

    https://doi.org/10.1002/hipo.20504

    • Google Scholar
11. Book

    1. Conway JH
    2. Sloane NJA

    (1992)

    Sphere packings, lattices and groups (2nd edition)

    New York: Springer-Verlag.

    • Google Scholar
12. 1. Deisseroth K
    2. Schnitzer MJ

    (2013) Engineering approaches to illuminating brain structure and dynamics

    Neuron 80:568–577.

    https://doi.org/10.1016/j.neuron.2013.10.032

    • Google Scholar
13. 1. Derdikman D
    2. Whitlock JR
    3. Tsao A
    4. Fyhn M
    5. Hafting T
    6. Moser M-B
    7. Moser EI

    (2009) Fragmentation of grid cell maps in a multicompartment environment

    Nature Neuroscience 12:1325–1332.

    https://doi.org/10.1038/nn.2396

    • Google Scholar
14. 1. Eurich W
    2. Wilke SD

    (2000) Multidimensional Encoding Strategy of Spiking Neurons

    Neural Computation 12:1519–1529.

    https://doi.org/10.1162/089976600300015240

    • Google Scholar
15. 1. Fiete IR
    2. Burak Y
    3. Brookings T

    (2008) What grid cells convey about rat location

    Journal of Neuroscience 28:6858–6871.

    https://doi.org/10.1523/JNEUROSCI.5684-07.2008

    • Google Scholar
16. 1. Finkelstein A
    2. Derdikman D
    3. Rubin A
    4. Foerster JN
    5. Las L
    6. Ulanovsky N

    (2014) Three-dimensional head-direction coding in the bat brain

    Nature 517:159–164.

    https://doi.org/10.1038/nature14031

    • Google Scholar
17. 1. Freiwald WA
    2. Tsao DY
    3. Livingstone MS

    (2009) A face feature space in the macaque temporal lobe

    Nature Neuroscience 12:1187–1196.

    https://doi.org/10.1038/nn.2363

    • Google Scholar
18. 1. Fyhn M
    2. Hafting T
    3. Witter MP
    4. Moser EI
    5. Moser M-B

    (2008) Grid cells in mice

    Hippocampus 18:1230–1238.

    https://doi.org/10.1002/hipo.20472

    • Google Scholar
19. 1. Gauss CF

    (1831)

    Recension der ’Untersuchungen über die Eigenschaften der positiven ternären quadratischen Formen von Ludwig August Seeber‘

    Göttingsche Gelehrte Anzeigen, July 9, pp. 1065; reprinted in J. Reine Angew. Math. 20 (1840)312–320.

    • Google Scholar
20. 1. Giocomo LM
    2. Hussaini SA
    3. Zheng F
    4. Kandel ER
    5. Moser M-B
    6. Moser EI

    (2011) Grid cells use HCN1 channels for spatial scaling

    Cell 147:1159–1170.

    https://doi.org/10.1016/j.cell.2011.08.051

    • Google Scholar
21. 1. Grah G
    2. Wehner R
    3. Ronacher B

    (2007) Desert ants do not acquire and use a three-dimensional global vector

    Frontiers in Zoology 4:12.

    https://doi.org/10.1186/1742-9994-4-12

    • Google Scholar
22. 1. Gray RM
    2. Neuhoff DL

    (1998) Quantization

    IEEE Transactions on Information Theory 44:2325–2383.

    https://doi.org/10.1109/18.720541

    • Google Scholar
23. 1. Gruber PM

    (2004) Optimum quantization and its applications

    Advances in Mathematics 186:456–497.

    https://doi.org/10.1016/j.aim.2003.07.017

    • Google Scholar
24. 1. Guanella A
    2. Verschure PF

    (2007) Prediction of the position of an animal based on populations of grid and place cells: a comparative simulation study

    Journal of Integrative Neuroscience 6:433–446.

    https://doi.org/10.1142/S0219635207001556

    • Google Scholar
25. 1. Hafting T
    2. Fyhn M
    3. Molden S
    4. Moser MB
    5. Moser EI

    (2005) Microstructure of a spatial map in the entorhinal cortex

    Nature 436:801–806.

    https://doi.org/10.1038/nature03721

    • Google Scholar
26. 1. Hales T

    (2005) A proof of the Kepler conjecture

    Annals of Mathematics 162:1065–1185.

    https://doi.org/10.4007/annals.2005.162.1065

    • Google Scholar
27. Book

    1. Hales T

    (2012)

    Dense sphere packings: a blueprint for formal proofs

    Cambridge: Cambridge University Press.

    • Google Scholar
28. 1. Hayman R
    2. Verriotis M
    3. Jovalekic A
    4. Fenton AA
    5. Jeffery KJ

    (2011) Anisotropic encoding of three-dimensional space by place cells and grid cells

    Nature Neuroscience 14:1182–1188.

    https://doi.org/10.1038/nn.2892

    • Google Scholar
29. 1. Jacobs J
    2. Weidemann CT
    3. Miller JF
    4. Solway A
    5. Burke JF
    6. Wei X-X
    7. Suthana N
    8. Sperling MR
    9. Sharan AD
    10. Fried I
    11. Kahana MJ

    (2013) Direct recordings of grid-like neuronal activity in human spatial navigation

    Nature Neuroscience 16:1188–1190.

    https://doi.org/10.1038/nn.3466

    • Google Scholar
30. 1. Jeffery KJ
    2. Jovalekic A
    3. Verriotis M
    4. Hayman R

    (2013) Navigating in a three-dimensional world

    The Behavioral and Brain Sciences 36:523–543.

    https://doi.org/10.1017/S0140525X12002476

    • Google Scholar
31. 1. Kropff E
    2. Treves A

    (2008) The emergence of grid cells: intelligent design or just adaptation?

    Hippocampus 18:1256–1269.

    https://doi.org/10.1002/hipo.20520

    • Google Scholar
32. 1. Krupic J
    2. Burgess N
    3. O'Keefe J

    (2012) Neural representations of location composed of spatially periodic bands

    Science 337:853–857.

    https://doi.org/10.1126/science.1222403

    • Google Scholar
33. 1. Krupic J
    2. Bauza M
    3. Burton S
    4. Barry C
    5. O'Keefe J

    (2015) Grid cell symmetry is shaped by environmental geometry

    Nature 518:232–235.

    https://doi.org/10.1038/nature14153

    • Google Scholar
34. 1. Krupic J
    2. Bauza M
    3. Burton S
    4. Lever C
    5. O'Keefe J

    (2013) How environment geometry affects grid cell symmetry and what we can learn from it

    Philosophical Transactions of the Royal Society of London B Biological Sciences 369:20130188.

    https://doi.org/10.1098/rstb.2013.0188

    • Google Scholar
35. 1. Lagrange JL

    (1773)

    Recherches d’arithmétique

    693–758, Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Berlin, Années, 3, Reprinted in Oeuvres.

    • Google Scholar
36. Book

    1. Las L
    2. Ulanovsky N

    (2014)

    Hippocampal neurophysiology across species

    In: Derdikman D, Knierim JJ, editors. Space, time and memory in the hippocampal formation. Springer Vienna. pp. 431–461.

    • Google Scholar
37. Book

    1. Lehmann EL

    (1998)

    Theory of Point estimation (2nd edition)

    NY, USA: Springer-Verlag.

    • Google Scholar
38. 1. Mathis A

    (2012)

    The representation of space in mammals: resolution of stochastic place and grid codes

    PhD thesis, Ludwig-Maximilians-Univeristät München.

    • Google Scholar
39. 1. Mathis A
    2. Herz AV
    3. Stemmler MB

    (2012a) Optimal population codes for space: grid cells outperform place cells

    Neural Computation 24:2280–2317.

    https://doi.org/10.1162/NECO_a_00319

    • Google Scholar
40. 1. Mathis A
    2. Herz AV
    3. Stemmler MB

    (2012b) Resolution of nested neuronal representations can be exponential in the number of neurons

    Physical Review Letters 109:018103.

    https://doi.org/10.1103/PhysRevLett.109.018103

    • Google Scholar
41. 1. Mathis A
    2. Herz AV
    3. Stemmler MB

    (2013) Multiscale codes in the nervous system: the problem of noise correlations and the ambiguity of periodic scales

    Physical Review E, Statistical, Nonlinear, and Soft Matter Physics 88:022713.

    https://doi.org/10.1103/PhysRevE.88.022713

    • Google Scholar
42. 1. Montemurro MA
    2. Panzeri S

    (2006) Optimal tuning widths in population coding of periodic variables

    Neural Computation 18:1555–1576.

    https://doi.org/10.1162/neco.2006.18.7.1555

    • Google Scholar
43. Book

    1. Nelson DR

    (2002)

    Defects and geometry in condensed matter physics

    Cambridge University Press.

    • Google Scholar
44. 1. O'Keefe J
    2. Dostrovsky J

    (1971) The hippocampus as a spatial map: preliminary evidence from unit activity in the freely-moving rat

    Brain Research 34:171–175.

    https://doi.org/10.1016/0006-8993(71)90358-1

    • Google Scholar
45. 1. Paradiso MA

    (1988) A theory for the use of visual orientation information which exploits the columnar structure of striate cortex

    Biological Cybernetics 58:35–49.

    https://doi.org/10.1007/BF00363954

    • Google Scholar
46. Book

    1. Poincaré H

    (1913)

    The foundations of science: science and hypothesis, the value of science, science and method

    Garrison, NY: The Science Press.

    • Google Scholar
47. 1. Pouget S
    2. Deneve S
    3. Ducom JC
    4. Latham PE

    (1999) Narrow versus wide tuning curves: what's best for a population code?

    Neural Computation 11:85–90.

    https://doi.org/10.1162/089976699300016818

    • Google Scholar
48. 1. Ray S
    2. Naumann R
    3. Burgalossi A
    4. Tang Q
    5. Schmidt H
    6. Brecht M

    (2014) Grid-layout and theta-modulation of layer 2 pyramidal neurons in medial entorhinal cortex

    Science 13:987–994.

    https://doi.org/10.1126/science.1243028

    • Google Scholar
49. 1. Rigotti M
    2. Barak O
    3. Warden MR
    4. Wang X-J
    5. Daw ND
    6. Miller EK
    7. Fusi S

    (2013) The importance of mixed selectivity in complex cognitive tasks

    Nature 497:585–590.

    https://doi.org/10.1038/nature12160

    • Google Scholar
50. 1. Seung HS
    2. Sompolinsky H

    (1993) Simple models for reading neuronal population codes

    Proceedings of the National Academy of Sciences of USA 90:10749–10753.

    https://doi.org/10.1073/pnas.90.22.10749

    • Google Scholar
51. 1. Shannon CE

    (1948)

    A mathematical theory of communication

    The Bell System Technical Journal XXVII:379–423.

    • Google Scholar
52. 1. Simoncelli EP
    2. Olshausen BA

    (2001) Natural image statistics and neural representation

    Annual Review of Neuroscience 24:1193–1216.

    https://doi.org/10.1146/annurev.neuro.24.1.1193

    • Google Scholar
53. 1. Stella F
    2. Si B
    3. Kropff E
    4. Treves A

    (2013) Grid maps for spaceflight, anyone? They are for free!

    Behavioral and Brain Sciences 36:566–567.

    https://doi.org/10.1017/S0140525X13000575

    • Google Scholar
54. 1. Stella F
    2. Treves A

    (2015) The self-organization of grid cells in 3D

    eLife 3:e05913.

    https://doi.org/10.7554/eLife.05913

    • Google Scholar
55. 1. Stensola H
    2. Stensola T
    3. Froland K
    4. Moser M-B
    5. Moser EI

    (2012) The entorhinal grid map is discretized

    Nature 492:72–78.

    https://doi.org/10.1038/nature11649

    • Google Scholar
56. 1. Stensola T
    2. Stensola H
    3. Moser M-B
    4. Moser EI

    (2015) Shearing-induced asymmetry in entorhinal grid cells

    Nature 518:207–212.

    https://doi.org/10.1038/nature14151

    • Google Scholar
57. 1. Thue A

    (1910)

    Über die dichteste Zusammenstellung von kongruenten Kreisen in einer Ebene

    Norske Videnskabs-Selskabets Skrifter 1:1–9.

    • Google Scholar
58. 1. Toth CD
    2. O'Rourke J
    3. Goodman JE

    (2004)

    Discrete and combinatorial mathematics series

    Discrete and combinatorial mathematics series, 2nd edition, CRC Press.

    • Google Scholar
59. 1. Wei X-X
    2. Prentice J
    3. Balasubramanian V

    (2013)

    The sense of place: grid cells in the brain and the transcendental number e

    arXiv:1304.0031v1:20.

    • Google Scholar
60. Book

    1. Whittaker EJ

    (1981)

    Crystallography: an introduction for earth (and other solid state) students (1st edition)

    Pergamon Press.

    • Google Scholar
61. 1. Wohlgemuth S
    2. Ronacher B
    3. Wehner R

    (2001) Ant odometry in the third dimension

    Nature 411:795–798.

    https://doi.org/10.1038/35081069

    • Google Scholar
62. 1. Yaeli S
    2. Meir R

    (2010) Error-based analysis of optimal tuning functions explains phenomena observed in sensory neurons

    Frontiers in Computational Neuroscience 4:130.

    https://doi.org/10.3389/fncom.2010.00130

    • Google Scholar
63. 1. Yartsev MM
    2. Ulanovsky N

    (2013) Representation of three-dimensional space in the hippocampus of flying bats

    Science 340:367–372.

    https://doi.org/10.1126/science.1235338

    • Google Scholar
64. 1. Yartsev MM
    2. Witter MP
    3. Ulanovsky N

    (2011) Grid cells without theta oscillations in the entorhinal cortex of bats

    Nature 479:103–107.

    https://doi.org/10.1038/nature10583

    • Google Scholar
65. 1. Zhang K
    2. Sejnowski TJ

    (1999) Neuronal tuning: to sharpen or broaden?

    Neural Computation 11:75–84.

    https://doi.org/10.1162/089976699300016809

    • Google Scholar

Author details

1. Alexander Mathis

   1. Department of Molecular and Cellular Biology, Harvard University, Cambridge, United States
   2. Center for Brain Science, Harvard University, Cambridge, United States

   Contribution

   AM, Conception and design, Analysis and interpretation of data, Drafting or revising the article

   For correspondence

   amathis@fas.harvard.edu

   Competing interests

   The authors declare that no competing interests exist.

   "This ORCID iD identifies the author of this article:" 0000-0002-3777-2202

2. Martin B Stemmler

   1. Bernstein Center for Computational Neuroscience, Germany
   2. Fakultät für Biologie, Ludwig-Maximilians-Universität München, Germany

   Contribution

   MBS, Conception and design, Drafting or revising the article

   Competing interests

   The authors declare that no competing interests exist.

3. Andreas VM Herz

   1. Bernstein Center for Computational Neuroscience, Germany
   2. Fakultät für Biologie, Ludwig-Maximilians-Universität München, Germany

   Contribution

   AVMH, Conception and design, Drafting or revising the article

   Competing interests

   The authors declare that no competing interests exist.

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