Population model

We consider a population ofN cells that respond in a stochastic fashion to a stimulus, corre-

sponding to 2D (planar) positionx = (x1, x2) ∈ B ⊂ R2. Thereby B is the environment of

the animal. The firing rate of each cell is given by its tuning curveΩi(x). We will assume that

the neuronal response is the spike countK = (k0,· · ·, kN) ∈NN, and that it is stochastically

dependent on the firing rate of each cell.

Generally, we try to formulate our results as universally as possible and therefore simply de- note the stochastic relationship betweenKandxby some functionP,

P(K_|x) =_P(K,Ωi(x)). (1)

More specifically, we consider independent Poisson firing, P(K,Ωi(x)) = Y i exp(₋Ωi(x)) Ωi(x)ki ki! . (2)

For the tuning curvesΩi(x)we mainly work with the general case of functions that are at least

twice continuously differentiable (C2_{). We often simply assume that these functions are uni-}

or multimodal. By unimodal we mean that the function has a single, global maximum on its definition domain. Multimodal on the other hand means that the function has multiple peaks. We will also deploy rotationally symmetric tuning curves, i.e.,

Ω(x) =fmaxT ·Φ kx−ck2 σ2 , (3)

with mean peak spike countfmaxT and a monotonically decreasing function Φ(s)with peak

Φ(0) = 1. The expression kvkdenotes the Euclidean distance, kvk = pv₁2+v₂2. The tuning

widthσscales the tuning curve, while leaving the shape unchanged. A Gaussian tuning curve

corresponds to the special caseΦ(s) = exp (₋s/2).

2.1.2 Decoding, Fisher information&Cramer-Rao inequality

The population of neurons encodes the position in a noisy manner. To investigate how precise the representation is, one looks at the performance of a suitable estimator. Due to the stochas- ticity of the firing, any estimator will make errors, when decoding the position; an estimatorbx

fixed positionx, for each realization the spike countKcan be different and hence the estimate

b

x(K(x)). This variability can be assessed by the properties of the posterior distributionp(_bx_|x),

being the probability distribution of estimates given a fixed positionx. The more precise a

population code is the more narrow this distribution will be.

For measuring the precision of the posterior we consider the second moment of the posterior, which is the mean square error [52, 7]:

χ2=_{E k}x₋xˆ_k2= X

K∈NN Z

Bk

x₋xˆ(K)_k2_·P(K_|x)dx, (4)

Theχ2error generally depends on the estimatorxˆ. For instance, given a particular population

responseK, the most likely stimulus that gave rise to it is ˆ

xMLE(K) =maxx∈BP(x|K), (5)

which is known as the maximum likelihood estimate (MLE). The corresponding mean maxi- mum likelihood estimate square error (MMLE) is defined asχ2

M LE =E

(x₋xˆM LE)2

. As the MMLE is numerically expensive to compute for large population sizes, we compare it against a reference that can be computed analytically. The maximum likelihood estimate is both statisti- cally efficient and consistent [35], which means thatχ2_{M LE}asymptotically approaches the mean

asymptotic square error (AE) for an increasing number of independent, identically distributed (i.i.d.) observations, which is given by

χ2_AE =E(1/J(x)), (6)

as shown in [7]. HereJ(x)stands for the Fisher information J(x) =E ∂

∂xln(P(K|x))

2!

. (7)

The Cramér-Rao bound gives a limit to the accuracy of any read-out mechanism and is there- fore useful for studying the accuracy of stochastic population codes. According to the Cramér- Rao inequality the Fisher information matrix bounds the covariance matrix of any unbiased estimator [35] at positionx:

E (x₋xˆM LE)2|x≥J(x)−1. (8)

For matrices such an inequality means that the difference is positive definite, which in this case, due to symmetry, is equivalent to the difference matrix possessing only positive eigenvalues. As we are dealing with navigation in arbitrary planar environments the error should be equal ineachdirection for an orthogonal basis, as trading accuracy in one direction for another one is not advantageous. From now on we will always pick the canonical basis, i.e.x = (x1, x2) =

x1(1,0) +x2(0,1). In these coordinates the mean square error is given by the sum over the

individual components:χ2 _=χ2

x1+χ 2

x2.

Thei, j-th component of the Fisher information matrix is given by: J(ij)(x) =E ∂ ∂xi lnP(K_|x)_· ∂ ∂xj lnP(K_|x) . (9)

The goalχx1 =χx2 translates intoJ(x)being a diagonal matrix with equal entries.

As a concrete example, we consider a module of tuning curves with Poisson noise, centersci,

tuning widthσ, and spatial periodλ, as given by Ωi(x) =fmax·exp 1 σ2 2 X α=1 {cos [2π(xα−ci,α)/λ]−1} ! , (10)

similar to [44]. If the number of neurons per periodλisM 1and the centers are uniformly

distributed, the module’s average Fisher information is given by:

J(ii)= M4π 2_f maxT ·exp(−2σ−2) λ2_σ2 ·I1(σ− 2_)·I 0(σ−2), (11)

withi = 1,2, whereIn(s)is thenth order modified Bessel function of the first kind. The non-

diagonal values of the Fisher information vanish [39]. 2.1.3 Periodification and planar lattices

For studying the lattice-type dependence let us consider the two equidistant planar lattices: the hexagonal and the square lattice (Figure 1). A planar lattice is a discrete subgroup of the plane

R2_{, and can therefore be generated by linear combinations with integer coefficients of two basis}

vectors.

These lattices can be used to construct periodic tilings of the plane and therefore qualify to underlie a grid code. In the following we outline a basic construction for the periodification of a unimodaltuning curve given a certain lattice and will later show that under certain assumptions a grid code based on the symmetries of a hexagonal lattice has a higher population Fisher information and therefore offers more spatial resolution.

LetΓ_⊂R2be a planar point lattice [34]: Γ =

2

X

α=1

kαvα for kα∈Z; vα ∈R2, (12)

such that(vα)_1≤α≤2 is a basis forR2. For the quadratic and hexagonal lattice it holds that the

length of the basis vectors is equalkv1k2=kv2k2and that the angle between]v1, v2respectively

|v | = |v |, φ = 90°_{1 2} |v | = |v |, φ = 120°_{1 2} φ v1 v2 φ v1 v2 D D A B 6

Figure 1: Equidistant planar lattices. A: Hexagonal latticeΓ6 generated by basis vectors v1

andv2 of equal length and angle120◦. The shaded area labeled byD6 depicts a fundamental

domain for this lattice. B: Quadratic latticeΓ generated by basis vectors v1 and v2 of equal

length and angle90◦. The shaded area labeled by D depicts a fundamental domain for this

lattice.

The orbit of a pointxinR2_{is the set of elements ofR}2_{to whichxcan be moved by the elements}

ofΓand is denoted byΓx={g+x|g∈Γ}. A fundamental domain of a latticeΓis a connected

subsetD ⊂ R2 _{that has the property that the orbits of D under the lattice operation fulfill}

ΓD= _R2_{and that it contains exactly one point from each orbit, i.e. Γx∩D = xfor allx∈D.}

LetΓ6stand for the hexagonal lattice with fundamental domain being the regular hexagonD6

(Fig. 1). For the quadratic lattice we pick a square centered around a node as fundamental domain.

LetDbe a fundamental domain of latticeΓand letΩ(x)be a tuning curve. With the canonical

inclusionι:D_→R2_{one can define the periodic extension ofΩas}

ΩΓ:R2→R+, x7→Ω◦ι(xmodΓ). (13)

HeresmodΓmeans congruent modulo the latticeΓ, so an element x _∈R2_{is mapped to the}

unique element of the same orbit inD. The definition is illustrated in Fig. 2 and is analogous

to the construction of wallpapers, where each motif is placed at any lattice point to define a pattern. Here the restriction of the tuning curve toDis themotif.1

Having defined the periodic functionΩΓ, we want to focus our attention on the Fisher informa-

tion for a population of such neurons. A family ofshifted, periodic tuning curvesΩΓ(x−ci) =

1_{This construction yields a function, which everywhere but possible at the boundary of the fundamental domain,}

has the same differentiability class (Ck_{) as the initial tuning curveΩ. At the boundary it is at least continuous and}

can also be smoothed. As a one-dimensional subset ofD, the boundary can be neglected when computing the

Fisher information.

Figure 2: A unimodal tuning curve Ω, shown on the left, can be rescaled and periodically

extended using Eqs. (12)-(13). The periodic tuning curvesΩΓ in the middle and right panels

are based on a rectangular latticeΓspanned byv1=λ·(1,0)0andv2=λ·(0,1)0. For the middle

panel the spatial period isλ= 1/2, whereas for the right panelλ= 1/4. Ω|(x−ci)modΓ|2

σ2

on the latticeΓwith different centersci constitutes a module and is associ-

ated with a Fisher informationJM

Ω,Γ. As we assume that all neurons are statistically independent

the Fisher information is given by summing over the contributions from all individual cells,

JM Ω,Γ(x) = M X i=1 JΩΓ(x−ci). (14)

In the limit of large population with uniformly distributed centersc∈Dthis finite sum fulfills

volMD ·J M Ω,Γ(x)− Z D JΩΓ(x−ϕ)dϕ →0 (15)

forM _{→ ∞}. As this difference is already tight for decent neuron numbers, we will from now

on consider the Fisher information of theaveragecell, which is defined as

JΩ,Γ(x) :=

Z

D

JΩΓ(x−ϕ)dϕ (16)

For the following theorem let us assume that the initial unimodal tuning curve is radially sym- metric. As a first step to compute the Fisher information, we note that

∂ ∂xilnP(K|x) = ∂ ∂slnP(K, s) s=ΩΓ(x)·Φ 0_(x)·f maxT ·2(xi−ci) σ2 . (17)

Together with the definition (9) of the Fisher information this yields J_ΩΓ(ij)(x) =f_max2 T2· 4(xi−ci)(xj−cj) σ4 ·Φ0(x)2 ·X K ∂ ∂slnP(K, s) s=ΩΓ(x) 2 · P(K,ΩΓ(x)) | {z } =:N(kx−ck2₎ . (18)

Note that for i ₆= j this function is odd in c around x. As Eq. (16) shows, the population

Fisher information is given by averaging these individual contributions over the fundamental domain,

J_Ω(ij_,Γ)(x) =

Z

D

J_ΩΓ(ij)(x₋ϕ)dϕ. (19)

Therefore, as the fundamental domains are symmetric one gets thatJ_Ω(ij_,Γ)(x) = 0fori ₆=j. The

diagonal entries are all identical and the trace of the Fisher information matrix becomes trJΩΓ(x) =X i J_ΩΓ(ii)(x) = Z D f_max2 T24 P i(xi−ϕi)2 σ4 Φ0_Γ kx₋ϕ_k2 σ2 2 · N(_kx₋ϕ_k2) | {z } =:F(kx−ϕk2₎ dϕ (20) We will use the notationBR(c) ={x ∈R2| kx−ck2 ≤R}for the closed ball of radiusRwith

centerc.

3 Results

3.1 Theory

To analyze the Fisher information of a grid code, we start by focusing on its basic building block: the module, a population of grid cells that share the same lattice, but are spatially shifted and represent different (spatial) phases. For a single module we show that the key parameter for improving the resolution is the spatial period of the lattice. In order to harness a small spatial period for a module, the ambiguities of an individual module have to be excluded by the other modules. We will show that this is best done in a nested fashion, which suggests that there should be a progression of grid sizes across modules. Such a code with multiple scales qualitatively changes the scaling behavior of the Fisher information.

Finally, we derive that for a wide range of tuning shapes the Fisher information is maximized by arranging the peaks on a hexagonal lattice rather than on a quadratic lattice. Furthermore, we study the optimal tuning width for such a module. All the model predictions will then be compared to measured characteristics of grid cells.