Original: No Actionability Without actionability the code wants to be non-negative but separable.
Revised: No Actionability Without actionability the code wants to be non-negative but separable.

Original: Bumps are easily positivised and encode the space well, but there is no reason to arrange them into lattices, and it would actually be detrimental to co-ordinate the placement across neurons so that any hypothetical lattice axes align.
Revised: Bumps are easily positivised and encode the space well, but there is no reason to arrange them into lattices, and it would actually be detrimental to co-ordinate the placement across neurons so that any hypothetical lattice axes align.

Original: As such, it builds a random but bumpy looking pattern.
Revised: As such, it builds a random but bumpy looking pattern.

Original: This observation will be important for our discussion of the coding of 3-dimensional space, Appendix M.
Revised: This observation will be important for our discussion of the coding of 3-dimensional space, Appendix M.

Original: J L ATTICE S IZE :P EAK W IDTH R ATIO S CALES WITH N UMBER OF N EURONS
Revised: J L ATTICE S IZE :P EAK W IDTH R ATIO S CALES WITH N UMBER OF N EURONS

Original: Thanks to the actionability constraint, the number of neurons in a module controls the number of frequencies available to build the module’s response pattern.
Revised: Thanks to the actionability constraint, the number of neurons in a module controls the number of frequencies available to build the module’s response pattern.

Original: In our scheme the frequencies within one module organise into a lattice, and the more frequencies in the module, the higher the top frequency, and the correspondingly sharper the grid peaks.
Revised: In our scheme the frequencies within one module organise into a lattice, and the more frequencies in the module, the higher the top frequency, and the correspondingly sharper the grid peaks.

Original: This link allows us to extract the number of neurons from a single neuron’s tuning curve in the module.
Revised: This link allows us to extract the number of neurons from a single neuron’s tuning curve in the module.

Original: In this section we add slightly more justiﬁcation to this link.
Revised: In this section we add slightly more justiﬁcation to this link.

Original: We begin with a representation of 1D position, g ( x ) .
Revised: We begin with a representation of 1D position, g ( x ) .

Original: When there are enough neurons the griddy response can be approximated as,
Revised: When there are enough neurons the griddy response can be approximated as,

Original: [Equation]
Revised: [Equation]

Original: [Equation] where DC ( ν ) is a Dirac comb with lengthscale ν , ∗ is convolution, and µ is the grid peak width,
Revised: (91) where DC ( ν ) is a Dirac comb with lengthscale ν , ∗ is convolution, and µ is the grid peak width,

Original: Fig 16.
Revised: Fig 16.

Original: The fourier transform of this object is simple, Fig 16,
Revised: The fourier transform of this object is simple, Fig 16,

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: ν (cid:19) (92) the is approximately a dirac comb convolved with a Gaussian, and hence the frequency response is approximately a Gaussian multiplied by a Dirac Comb. Actionability implies sparsity in fourier space, which we approximate as a ﬁnite number of frequencies having power above some threshold τ , i.e. a sufﬁ ciently fast decay in frequency space, which corresponds to a sufﬁciently broad grid peak

Original: We can now extract the tradeoff exactly.
Revised: We can now extract the tradeoff exactly.

Original: In this continuous approximation the equivalent of actionability’s sparsity in frequency space is that the frequency spectra must decay sufﬁciently fast i.e. the number of frequencies with power above some small threshold must be less than half the number of neurons.
Revised: In this continuous approximation the equivalent of actionability’s sparsity in frequency space is that the frequency spectra must decay sufﬁciently fast i.e. the number of frequencies with power above some small threshold must be less than half the number of neurons.

Original: This number is controlled by the ratio of µ and ν .
Revised: This number is controlled by the ratio of µ and ν .

Original: We’ll set some arbitrary low scaled threshold, τ , which deﬁnes the power a frequency must have, relative to the constant component, to be considered as a member of the code.
Revised: We’ll set some arbitrary low scaled threshold, τ , which deﬁnes the power a frequency must have, relative to the constant component, to be considered as a member of the code.

Original: The actual value of τ is inconsequential, as we just want scaling rules.
Revised: The actual value of τ is inconsequential, as we just want scaling rules.

Original: Then we’ll derive the scaling of the frequency at which the amplitude of the gaussian drops below threshold, ω ∗ :
Revised: Then we’ll derive the scaling of the frequency at which the amplitude of the gaussian drops below threshold, ω ∗ :

Original: [Equation]
Revised: [Equation]

Original: Now we can ask how many frequencies there are in the code, by counting the number of peaks of the
Revised: Now we can ask how many frequencies there are in the code, by counting the number of peaks of the

Original: Dirac comb that occur before this threshold.
Revised: Dirac comb that occur before this threshold.

Original: Counting from 0, because in 1D negative frequencies are just scaled versions of positive frequencies:
Revised: Counting from 0, because in 1D negative frequencies are just scaled versions of positive frequencies:

Original: [Equation]
Revised: [Equation]

Original: Therefore, in 1D, when there are enough neurons, the number of frequencies, and therefore the number of the neurons scales with the lattice to peak width ratio:
Revised: Therefore, in 1D, when there are enough neurons, the number of frequencies, and therefore the number of the neurons scales with the lattice to peak width ratio:

Original: [Equation]
Revised: [Equation]

Original: A completely analogous argument can be made in 2D, Again the cutoff frequency is ω ∗ ∝ 1 µ .
Revised: A completely analogous argument can be made in 2D, Again the cutoff frequency is ω ∗ ∝ 1 µ .

Original: A different scaling emerges, however, because the number of frequencies in a dirac comb lower than some threshold scales with dimension.
Revised: A different scaling emerges, however, because the number of frequencies in a dirac comb lower than some threshold scales with dimension.

Original: The frequency density of the dirac comb is proportional toν 2 peaks per unit area with the speciﬁcs depending on the shape of the frequency lattice, so we’re asking how many peaks are there in a half circle of radius ω ∗ .
Revised: The frequency density of the dirac comb is proportional toν 2 peaks per unit area with the speciﬁcs depending on the shape of the frequency lattice, so we’re asking how many peaks are there in a half circle of radius ω ∗ .

Original: This is simple,
Revised: This is simple,

Original: [Equation]
Revised: [Equation]

Original: Hence we recover the stated scaling,
Revised: Hence we recover the stated scaling,

Original: [Equation]

Original: [Equation]
Revised: [Equation]

Original: K O PTIMAL R ELATIVE A NGLE B ETWEEN M ODULES
Revised: K O PTIMAL R ELATIVE A NGLE B ETWEEN M ODULES

Original: Our analysis of the full loss, Appendix E, suggested that optimal representations should contain multiple modules, and each module should be chosen such that its frequencies are optimally nonharmonically related to the frequencies in other modules.
Revised: Our analysis of the full loss, Appendix E, suggested that optimal representations should contain multiple modules, and each module should be chosen such that its frequencies are optimally nonharmonically related to the frequencies in other modules.

Original: This is a useful principle, but the full implications are hard to work through exactly.
Revised: This is a useful principle, but the full implications are hard to work through exactly.

Original: In this section we use rough approximations and coarse calculations to make estimates of one aspect of the relationships between modules: the optimal relative angle between lattice axes.
Revised: In this section we use rough approximations and coarse calculations to make estimates of one aspect of the relationships between modules: the optimal relative angle between lattice axes.

Original: Our scheme begins by calculating a smoothed frequency lattice density as a function of angle, ρ ( ψ ) .
Revised: Our scheme begins by calculating a smoothed frequency lattice density as a function of angle, ρ ( ψ ) .

Original: It then aligns two of these smoothed densities, representing two modules, at different angular offsets, and ﬁnds the angle which minimises the overlap.
Revised: It then aligns two of these smoothed densities, representing two modules, at different angular offsets, and ﬁnds the angle which minimises the overlap.

Original: This angle is the predicted optimal angle.
Revised: This angle is the predicted optimal angle.

Original: We then extend this to multiple modules by searching over all doubles or triples of alignment angles for the minimal overlap of three or four densities.
Revised: We then extend this to multiple modules by searching over all doubles or triples of alignment angles for the minimal overlap of three or four densities.

Original: We will talk through the details of this method, and arrive at the results that are quoted in section 4.2.
Revised: We will talk through the details of this method, and arrive at the results that are quoted in section 4.2.

Original: If you imagine standing at the origin in frequency space, corresponding to the constant component, and looking to inﬁnity at an angle ψ to the k x axis, the density function, ρ ( ψ ) , measures, as a function of ψ , the number of smoothed frequencies that you see, ﬁg 17A.
Revised: If you imagine standing at the origin in frequency space, corresponding to the constant component, and looking to inﬁnity at an angle ψ to the k x axis, the density function, ρ ( ψ ) , measures, as a function of ψ , the number of smoothed frequencies that you see, ﬁg 17A.

Original: We assume there are many neurons in this module, and hence many frequencies in the frequency lattice.
Revised: We assume there are many neurons in this module, and hence many frequencies in the frequency lattice.

Original: Then we make the same approximation as in eqns.
Revised: Then we make the same approximation as in eqns.

Original: 91 and 92, that the frequency response is a hexagonal grid multiplied by a decaying Gaussian.
Revised: 91 and 92, that the frequency response is a hexagonal grid multiplied by a decaying Gaussian.

Original: The lengthscale of this Guassian is the ﬁrst parameter that will effect our results, and it scales linearly with the number of neurons in the module.
Revised: The lengthscale of this Guassian is the ﬁrst parameter that will effect our results, and it scales linearly with the number of neurons in the module.

Original: Then we try and capture the repulsive effects of the lattice.
Revised: Then we try and capture the repulsive effects of the lattice.

Original: Each lattice point repels frequencies over a nearby region, as calculated in Section E.4.
Revised: Each lattice point repels frequencies over a nearby region, as calculated in Section E.4.

Original: This region is of size ∼ 12 L , where L is the exploration lengthscale of the animal.
Revised: This region is of size ∼ 12 L , where L is the exploration lengthscale of the animal.

Original: We therefore smooth the grid peaks in our density function by convolving the lattice with a Gaussian.
Revised: We therefore smooth the grid peaks in our density function by convolving the lattice with a Gaussian.

Original: We choose our units such that L = 1 , then the width of this smoothing Gaussian is also order 1 .
Revised: We choose our units such that L = 1 , then the width of this smoothing Gaussian is also order 1 .

Original: The second important parameter is then the grid lattice lengthscale.
Revised: The second important parameter is then the grid lattice lengthscale.

Original: The larger the grid cell lengthscale the more overlapping the repulsive effects of the frequencies will be, and the smoother the repulsive density, ρ ( ψ ) .
Revised: The larger the grid cell lengthscale the more overlapping the repulsive effects of the frequencies will be, and the smoother the repulsive density, ρ ( ψ ) .

Original: In maths,
Revised: In maths,

Original: [Equation]
Revised: [Equation]

Original: where k r is the radial part of the frequency.
Revised: where k r is the radial part of the frequency.

Original: We calculate the obvious numerical approximation to this integral, Fig 17B, and in these calculations we ignore the effect of higher harmonics in equation E.4.
Revised: We calculate the obvious numerical approximation to this integral, Fig 17B, and in these calculations we ignore the effect of higher harmonics in equation E.4.

Original: This isn’t as bad as is sounds, since higher harmonics contribute density at the same angle, ψ , part of the reason we chose to calculate this angular density.
Revised: This isn’t as bad as is sounds, since higher harmonics contribute density at the same angle, ψ , part of the reason we chose to calculate this angular density.

Original: The next stage is ﬁnding the optimal angle between modules, starting with two identical modules.
Revised: The next stage is ﬁnding the optimal angle between modules, starting with two identical modules.

Original: We do this by overlapping copies of ρ ( ψ (cid:107) µ, ν ) .
Revised: We do this by overlapping copies of ρ ( ψ (cid:107) µ, ν ) .

Original: For simplicity, we start with two identical lattices and calculate the overlap as a function of angular offset:
Revised: For simplicity, we start with two identical lattices and calculate the overlap as a function of angular offset:

Original: [Equation]
Revised: [Equation]

Original: Then the predicted angular offset is,
Revised: Then the predicted angular offset is,

Original: [Equation]
Revised: [Equation]

Original: Figure 17C shows this is a small angular offset around 4 degrees, though it can vary between 3 and 8 degrees depending on µ and ν .
Revised: Figure 17C shows this is a small angular offset around 4 degrees, though it can vary between 3 and 8 degrees depending on µ and ν .

Original: An identical procedure can be applied to multiple modules.
Revised: An identical procedure can be applied to multiple modules.

Original: For example, for the arrangement of three modules we deﬁne,
Revised: For example, for the arrangement of three modules we deﬁne,

Original: [Equation]
Revised: [Equation]

Original: and ﬁnd the pair of offsets that minimise the overlap.
Revised: and ﬁnd the pair of offsets that minimise the overlap.

Original: For identical modules this also predicts small offsets between modules, of around 4 degrees between each module, Figure 17D, and similarly three identical modules should be oriented at multiples of 4 degrees from a reference lattice (i.e. offsets of 4, 8, and 12).
Revised: For identical modules this also predicts small offsets between modules, of around 4 degrees between each module, Figure 17D, and similarly three identical modules should be oriented at multiples of 4 degrees from a reference lattice (i.e. offsets of 4, 8, and 12).

Original: As shown in Figure 5E, these small offsets appear to be present in data, validating our prediction.
Revised: As shown in Figure 5E, these small offsets appear to be present in data, validating our prediction.

Original: There are many natural next steps for this kind of analysis.
Revised: There are many natural next steps for this kind of analysis.

Original: In future we will explore the effect of different lattice lengthscales and try to predict the optimal module arrangement.
Revised: In future we will explore the effect of different lattice lengthscales and try to predict the optimal module arrangement.

Original: Another simple analysis you can do is explore how changing the grid lengthscale, ν , effects the optimal relative angle.
Revised: Another simple analysis you can do is explore how changing the grid lengthscale, ν , effects the optimal relative angle.

Original: The larger the grid lattice lengthscale, the smaller its frequency lattice, and the more smooth the resulting repulsion density is.
Revised: The larger the grid lattice lengthscale, the smaller its frequency lattice, and the more smooth the resulting repulsion density is.

Original: As a result, the optimal relative angle between large modules is larger than between small modules, a prediction we would like to test in data.
Revised: As a result, the optimal relative angle between large modules is larger than between small modules, a prediction we would like to test in data.

Original: It would be interesting to apply more careful numerical comparisons, for example matching the ν and µ to those measured in data, and comparing the real and optimal angular offsets.
Revised: It would be interesting to apply more careful numerical comparisons, for example matching the ν and µ to those measured in data, and comparing the real and optimal angular offsets.

Original: L R OOM G EOMETRY A NALYSIS
Revised: L R OOM G EOMETRY A NALYSIS

Original: The functional objective depends on the points the animal wants to encode, which we assume are the points the animal visits.
Revised: The functional objective depends on the points the animal wants to encode, which we assume are the points the animal visits.

Original: As such, the exploration pattern of the animal determines the optimal arrangement of grids, through the appearance of p ( x ) in the functional objective.
Revised: As such, the exploration pattern of the animal determines the optimal arrangement of grids, through the appearance of p ( x ) in the functional objective.

Original: Different constraints to the exploration of the animal, such as different sized environments, will lead to different optimal representations.
Revised: Different constraints to the exploration of the animal, such as different sized environments, will lead to different optimal representations.

Original: In this section we analyse the functional objective for a series of room shapes, and show they lead to the intricate frequency biases suggested in Section 4.3.
Revised: In this section we analyse the functional objective for a series of room shapes, and show they lead to the intricate frequency biases suggested in Section 4.3.

Original: L. 1-D IMENSONAL B OX
Revised: L. 1-D IMENSONAL B OX

Original: To begin and to set the pattern for this section, we will study the representation of 1-dimensional position in a ﬁnite box.
Revised: To begin and to set the pattern for this section [Equation] we will study the representation of 1-dimensional position in a ﬁnite box.

Original: Ignoring the effects of χ for now, our loss is,
Revised: Ignoring the effects of χ for now, our loss is,

Original: [Equation]
Revised: [Equation]

Original: Making the symmetry assumption as in Section E our loss is,
Revised: Making the symmetry assumption as in Section E our loss is,

Original: [Equation]
Revised: [Equation]

Original: Changing variables to α = x − x (cid:48) , β [Equation] x + x (cid:48) , Figure 18A,
Revised: Changing variables to α = x − x (cid:48) , β = x + x (cid:48) , Figure 18A,

Original: [Equation]
Revised: [Equation]

Original: Performing the expansion as in Section E we reach:
Revised: Performing the expansion as in Section E we reach:

Original: [Equation]
Revised: [Equation]

Original: Term A behaves the same as before, so we again study the series expension of term B, the analogue of equation 77.
Revised: Term A behaves the same as before, so we again study the series expension of term B, the analogue of equation 77.

Original: The 0 th order term is a constant with respect to the code, but the ﬁrst order term provides the main frequency biasing effect, as in Section E.4.
Revised: The 0 th order term is a constant with respect to the code, but the ﬁrst order term provides the main frequency biasing effect, as in Section E.4.

Original: [Equation]
Revised: [Equation]

Original: Hence the 1-dimensional box coarsly creates a high frequency bias ( > 1 L ), but also encourages frequencies that ﬁt whole numbers of wavelengths within the length of the box, Figure 18B.
Revised: Hence the 1-dimensional box coarsely creates a high frequency bias ( > 1 L ), but also encourages frequencies that ﬁt whole numbers of wavelengths within the length of the box, Figure 18B.

Original: Before we move on we will study the second order term.
Revised: Before we move on we will study the second order term.

Original: We won’t repeat this for future ﬁnite rooms as it shows basically the same effect as in Section E.4: coarsly frequencies repel each other up to a lengthscale 1 L .
Revised: We won’t repeat this for future ﬁnite rooms as it shows basically the same effect as in Section E.4: coarsly frequencies repel each other up to a lengthscale 1 L .

Original: However, as for the ﬁrst order term, there is additional structure, i.e. subsidiary optimal zeros in the loss, that could be of interest:
Revised: However, as for the ﬁrst order term, there is additional structure, i.e. subsidiary optimal zeros in the loss, that could be of interest:

Original: [Equation]
Revised: [Equation]

Original: This is fundamentally a repulsion between frequencies closer than 1 L from one another, ﬁg 18C.
Revised: This is fundamentally a repulsion between frequencies closer than 1 L from one another, ﬁg 18C.

Original: Additionally, the repulsion is zero when the difference between frequencies is exactly 2 πL times an integer.
Revised: Additionally, the repulsion is zero when the difference between frequencies is exactly 2 πL times an integer.

Original: L. 2-D IMENSIONAL B OX
Revised: L. 2-D IMENSIONAL B OX

Original: A very similar analysis can be applied to the representation of a 2-dimensional box, and used to derive the frequency bias plotted in Figure 6X.
Revised: A very similar analysis can be applied to the representation of a 2-dimensional box, and used to derive the frequency bias plotted in Figure 5F.

Original: We reach an exactly analogous point to equation 106,
Revised: We reach an exactly analogous point to equation 106,

Original: [Equation]
Revised: [Equation]

Original: Again we’ll study the ﬁrst order term of the series expansion,
Revised: Again we’ll study the ﬁrst order term of the series expansion,

Original: [Equation]
Revised: [Equation]

Original: This is exactly the bias plotted in Figure 5F.
Revised: This is exactly the bias plotted in Figure 5F.

Original: Changing to rectangles is easy, it is just a rescaling of one axis, which just inversely scales the corresponding frequency axis.
Revised: Changing to rectangles is easy, it is just a rescaling of one axis, which just inversely scales the corresponding frequency axis.

Original: L. C IRCULAR R OOM
Revised: L. C IRCULAR R OOM

Original: The ﬁnal analysis we do of this type is the hardest.
Revised: The ﬁnal analysis we do of this type is the hardest, a circular environment.

Original: It follows the same pattern, make the symmetry assumption, expand the series.
Revised: It follows the same pattern, make the symmetry assumption, expand the series.

Original: Doing so we arrive at the following ﬁrst order term:
Revised: Doing so we arrive at the following ﬁrst order term:

Original: [Equation]
Revised: [Equation]

Original: 113) where ( r, ψ ) is a radial co-ordinate system, x = (cid:18) r cos ψ r sin ψ (cid:19) .
Revised: (113) where ( r, ψ ) is a radial co-ordinate system, x = (cid:18) [Equation] r sin ψ (cid:19) .

Original: Using the double angle formula to spread the cosine, and rotating the 0 of each ψ variable to align with k d , we get: 
Revised: Using the double angle formula to spread the cosine, and rotating the 0 of each ψ variable to align with k d , we get: ∆ [Equation] [Equation]

Revised: (cid:34)(cid:18)

Original: [Equation]
Revised: [Equation]

Revised: R

Original: [Equation]
Revised: [Equation]

Original: The second integral is 0, because sine is odd and periodic.
Revised: The second integral is 0, because sine is odd and periodic.

Original: We can perform the ﬁrst integral using the following Bessel function properties,
Revised: We can perform the ﬁrst integral using the following Bessel function properties,

Original: [Equation]
Revised: [Equation]

Original: Hence,
Revised: Hence,

Original: [Equation]
Revised: [Equation]

Original: Which can be solved using another Bessel function property,
Revised: Which can be solved using another Bessel function property,

Original: [Equation]
Revised: [Equation]

Original: Therefore,
Revised: Therefore,

Original: [Equation]
Revised: [Equation]

Original: This is exactly the circular bias plotted in
Revised: This is exactly the circular bias plotted in

Original: Figure 5G.
Revised: Figure 5G.

Original: Placing the base frequencies of grid module on the zeros of this bessel function would be optimal, i.e. calling a zero ξ , then the wavelengths of the grid modules should optimally align with,
Revised: Placing the base frequencies of grid module on the zeros of this bessel function would be optimal, i [Equation] e. calling a zero ξ , then the wavelengths of the grid modules should optimally align with,

Original: [Equation]
Revised: [Equation]

Original: This corresponds to grids with wavelengths that are the following multiples of the circle diameter: 0 .
Revised: This corresponds to grids with wavelengths that are the following multiples of the circle diameter: 0 .

Original: Though, as with the other predictions, the major effect is the polynomial decay with frequency, so these effects will all be of larger importance for the larger grid modules.
Revised: Though, as with the other predictions, the major effect is the polynomial decay with frequency, so these effects will all be of larger importance for the larger grid modules.

Original: As such, we might expect only the ﬁrst two grid modules to ﬁt this pattern, the others should, in our framework, focus on choosing their lattice parameters to be optimally non-harmonic.
Revised: As such, we might expect only the ﬁrst two grid modules to ﬁt this pattern, the others should, in our framework, focus on choosing their lattice parameters to be optimally non-harmonic.

Original: M R EPRESENTATIONS OF C IRCLES , S PHERES , AND 3- DIMENSIONS
Revised: M R EPRESENTATIONS OF C IRCLES , S PHERES , AND 3- DIMENSIONS

Original: We believe our approach is relevant to the representation of many variables beyond 2-dimensional space.
Revised: We believe our approach is relevant to the representation of many variables beyond 2-dimensional space.

Original: Most simply we can directly apply the tools we’ve developed to the representation of any variable whose transition structure is a group, Section A.1.
Revised: Most simply we can directly apply the tools we’ve developed to the representation of any variable whose transition structure is a group, Section A.1.

Original: This is actually true for both numerical and analytic approaches, but here we focus mainly on the numerical approach.
Revised: This is actually true for both numerical and analytic approaches, but here we focus mainly on the numerical approach.

Original: We discuss the representation of an angle on a ring, a point on a sphere, and 3 dimensional space.
Revised: We discuss the representation of an angle on a ring, a point on a sphere, and 3 dimensional space.

Original: Additional interesting variables might include a position in hyperbolic space (since tree-like category structures, such as the taxonomic hierarchy of species, are best described by hyperbolic spaces), or the full orientation of an object in 3-dimensional space.
Revised: Additional interesting variables might include a position in hyperbolic space (since tree-like category structures, such as the taxonomic hierarchy of species, are best described by hyperbolic spaces), or the full orientation of an object in 3-dimensional space.

Original: For starters, we have frequently referenced the representation of an angle, g ( θ ) .
Revised: For starters, we have frequently referenced the representation of an angle, g ( θ ) .

Original: This is also something measured in brain, in the famous head direction circuit, perhaps the most beautiful circuit in the brain (Kim et al., 2017).
Revised: This is also something measured in brain, in the famous head direction circuit, perhaps the most beautiful circuit in the brain (Kim et al., 2017).

Original: There you measure neurons that have unimodal tuning to heading direction.
Revised: There, you measure neurons that have unimodal tuning to heading direction.

Original: We match this ﬁnding, Figure 19A. This is understandable since, as discussed, on a ﬁnite space you still want a lattice module, but χ , encourages you to be low frequency and there is no high frequency bias, since the space is ﬁnite and p ( θ ) is uniform.
Revised: We match this ﬁnding, Figure 19A. This is understandable since, as discussed, on a ﬁnite space you still want a lattice module, but χ , encourages you to be low frequency and there is no high frequency bias, since the space is ﬁnite and p ( θ ) is uniform.

Original: Thus, place cells are optimal.
Revised: Thus, place cells are optimal.

Original: If we included more neurons it would likely form multiple modules of cells with higher frequency response properties.
Revised: If we included more neurons it would likely form multiple modules of cells with higher frequency response properties.

Original: We can also apply our work to the representation of a point on a sphere.
Revised: We can also apply our work to the representation of a point on a sphere.

Original: Again, with the low frequency bias place cells are optimal, Figure 19B. Without it, or conceivably with enough neurons that multiple modules form, you can get many interesting patterns, Figure 19C-D. These could be relevant to the representation of object orientations, or in virtual reality experiments where animals must truly explore spherical spaces.
Revised: Again, with the low frequency bias place cells are optimal, Figure 19B. Without it, or conceivably with enough neurons that multiple modules form, you can get many interesting patterns, Figure 19C-D. These could be relevant to the representation of object orientations, or in virtual reality experiments where animals must truly explore spherical spaces.

Original: Finally, we can equally apply our framework to the representation of a point in three dimensional space.
Revised: Finally, we can equally apply our framework to the representation of a point in three dimensional space.

Original: Grid cells have been measured while an animal explores in 3D and shown to have irregular ﬁring ﬁelds in both bats (Ginosar et al., 2021), and mice (Grieves et al., 2021).
Revised: Grid cells have been measured while an animal explores in 3D and shown to have irregular ﬁring ﬁelds in both bats (Ginosar et al., 2021), and mice (Grieves et al., 2021).

Original: Analogously to the griddy modules in one and two dimensions, our framework predicts griddy three dimensional responses that are similarly densely packing, which is veriﬁed in 19E. However, in Appendix I we performed ablation studies and found that removing the actionable constraint led to multimodal tuning curves without structure, and we ﬁnd the same for 3D, Figure 19F, perhaps matching the multimodal but unstructured representation found in the brain (e.g. Figure 19G from Ginosar et al.
Revised: Analogously to the griddy modules in one and two dimensions, our framework predicts griddy three dimensional responses that are similarly densely packing, which is veriﬁed in 19E. However, in Appendix I we performed ablation studies and found that removing the actionable constraint led to multimodal tuning curves without structure, and we ﬁnd the same for 3D, Figure 19F, perhaps matching the multimodal but unstructured representation found in the brain (e.g. Figure 19G from Ginosar et al.

Original: As such, it seems that in three dimensions the animals are building a non-negative and discriminative representation without the constraint to path integrate.
Revised: As such, it seems that in three dimensions the animals are building a non-negative and discriminative representation without the constraint to path integrate.

Original: Therefore, we suggest that the animals have sacriﬁced the ability to manipulate their internal representations in 3D and instead focus on efﬁciently encoding the space.
Revised: Therefore, we suggest that the animals have sacriﬁced the ability to manipulate their internal representations in 3D and instead focus on efﬁciently encoding the space.