Original: A CTIONABLE N EURAL R EPRESENTATIONS : G RID C ELLS FROM M INIMAL C ONSTRAINTS
Revised: A CTIONABLE N EURAL R EPRESENTATIONS : G RID C ELLS FROM M INIMAL C ONSTRAINTS

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: To afford ﬂexible behaviour, the brain must build internal representations that mirror the structure of variables in the external world.
Revised: To afford ﬂexible behaviour, the brain must build internal representations that mirror the structure of variables in the external world.

Original: For example, 2D space obeys rules: the same set of actions combine in the same way everywhere (step north, then south, and you won’t have moved, wherever you start).
Revised: For example, 2D space obeys rules: the same set of actions combine in the same way everywhere (step north, then south, and you won’t have moved, wherever you start).

Original: We suggest the brain must represent this consistent meaning of actions across space, as it allows you to ﬁnd new short-cuts and navigate in unfamiliar settings.
Revised: We suggest the brain must represent this consistent meaning of actions across space, as it allows you to ﬁnd new short-cuts and navigate in unfamiliar settings.

Original: We term this representation an ‘actionable representation’.
Revised: We term this representation an ‘actionable representation’.

Original: We formulate actionable representations using group and representation theory, and show that, when combined with biological and functional constraints - non-negative ﬁring, bounded neural activity, and precise coding - multiple modules of hexagonal grid cells are the optimal representation of 2D space.
Revised: We formulate actionable representations using group and representation theory, and show that, when combined with biological and functional constraints - non-negative ﬁring, bounded neural activity, and precise coding - multiple modules of hexagonal grid cells are the optimal representation of 2D space.

Original: We support this claim with intuition, analytic justiﬁcation, and simulations.
Revised: We support this claim with intuition, analytic justiﬁcation, and simulations.

Original: Our analytic results normatively explain a set of surprising grid cell phenomena, and make testable predictions for future experiments.
Revised: Our analytic results normatively explain a set of surprising grid cell phenomena, and make testable predictions for future experiments.

Original: Lastly, we highlight the generality of our approach beyond just understanding 2D space.
Revised: Lastly, we highlight the generality of our approach beyond just understanding 2D space.

Original: Our work characterises a new principle for understanding and designing ﬂexible internal representations: they should be actionable, allowing animals and machines to predict the consequences of their actions, rather than just encode.
Revised: Our work characterises a new principle for understanding and designing ﬂexible internal representations: they should be actionable, allowing animals and machines to predict the consequences of their actions, rather than just encode.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Animals and machines should build representations that afford ﬂexible behaviours.
Revised: Animals and machines should build representations that afford ﬂexible behaviours.

Original: However, different representation make some tasks easy and others hard; representing red versus white is good for understanding wines but less good for opening screw-top versus corked bottles.
Revised: However, different representation make some tasks easy and others hard; representing red versus white is good for understanding wines but less good for opening screw-top versus corked bottles.

Original: A central mystery in neuroscience is the relationship between tasks and their optimal representations.
Revised: A central mystery in neuroscience is the relationship between tasks and their optimal representations.

Original: Resolving this requires understanding the representational principles that permit ﬂexible behaviours such as zero-shot inference and short-cut planning.
Revised: Resolving this requires understanding the representational principles that permit ﬂexible behaviours such as zero-shot inference and short-cut planning.

Original: Here, we introduce actionable representations , a representation that permits ﬂexible behaviours.
Revised: Here, we introduce actionable representations , a representation that permits ﬂexible behaviours.

Original: Being actionable means encoding not only variables of interest, but also how the variable transforms.
Revised: Being actionable means encoding not only variables of interest, but also how the variable transforms.

Original: Actions cause many variables to transform in predictable ways.
Revised: Actions cause many variables to transform in predictable ways.

Original: For example, actions in 2D space obey rules; north, east, south, and west, have a universal meaning, and combine in the same way everywhere.
Revised: For example, actions in 2D space obey rules; north, east, south, and west, have a universal meaning, and combine in the same way everywhere.

Original: Embedding these rules into a representation of self-position permits deep inferences: having stepped north, then east, then south, an agent can infer that stepping west will lead home, having never taken that path - a zero-shot inference (Figure 1A).
Revised: Embedding these rules into a representation of self-position permits deep inferences: having stepped north, then east, then south, an agent can infer that stepping west will lead home, having never taken that path - a zero-shot inference (Figure 1A).

Original: Indeed biology represents 2D space in a structured manner.
Revised: Indeed biology represents 2D space in a structured manner.

Original: Grid cells in medial entorhinal cortex represent an abstracted ‘cognitive map’ of 2D space (Tolman, 1948).
Revised: Grid cells in medial entorhinal cortex represent an abstracted ‘cognitive map’ of 2D space (Tolman, 1948).

Original: These cells ﬁre in a hexagonal lattice of positions (Hafting et al., 2005), (Figure 1B), and are organised in modules; cells within one module have receptive ﬁelds that are translated versions of one another, and different modules have ﬁring lattices of different scales and orientations (Figure 1B), (Stensola et al., 2012).
Revised: These cells ﬁre in a hexagonal lattice of positions (Hafting et al., 2005), (Figure 1B), and are organised in modules; cells within one module have receptive ﬁelds that are translated versions of one another, and different modules have ﬁring lattices of different scales and orientations (Figure 1B), (Stensola et al., 2012).

Original: Biological representations must be more than just actionable - they must be functional , encoding the world efﬁciently, and obey biological constraints.
Revised: Biological representations must be more than just actionable - they must be functional , encoding the world efﬁciently, and obey biological constraints.

Original: We formalise these three ideas - actionable, functional, and biological - and analyse the resulting optimal representations.
Revised: We formalise these three ideas - actionable, functional, and biological - and analyse the resulting optimal representations.

Original: We deﬁne actionability using group and representation theory, as the requirement that each action has a corresponding matrix that linearly updates the representation; for example, the ‘step north’ matrix updates the representation to its value one step north.
Revised: We deﬁne actionability using group and representation theory, as the requirement that each action has a corresponding matrix that linearly updates the representation; for example, the ‘step north’ matrix updates the representation to its value one step north.

Original: Functionally , we want different points in space to be represented maximally differently, allowing inputs to be distinguished from one another.
Revised: Functionally , we want different points in space to be represented maximally differently, allowing inputs to be distinguished from one another.

Original: Biologically , we ensure all neurons have non-negative and bounded activity.
Revised: Biologically , we ensure all neurons have non-negative and bounded activity.

Original: From this constrained optimisation problem we derive optimal representations that resemble multiple modules of grid cells.
Revised: From this constrained optimisation problem we derive optimal representations that resemble multiple modules of grid cells.

Original: Our problem formulation allows analytic explanations for grid cell phenomena, matches experimental ﬁndings, such as the alignment of grids cells to room geometry (Stensola et al., 2015), and predicts some underappreciated aspects, such as the relative angle between modules.
Revised: Our problem formulation allows analytic explanations for grid cell phenomena, matches experimental ﬁndings, such as the alignment of grids cells to room geometry (Stensola et al., 2015), and predicts some underappreciated aspects, such as the relative angle between modules.

Original: In sum, we 1) propose actionable neural representations to support ﬂexible behaviours; 2) formalise the actionable constraint with group and representation theory; 3) mix actionability with biological and functional constraints to create a constrained optimisation problem; 4) analyse this problem and show that in 2D the optimal representation is a good model of grid cells, thus offering a mathematical understanding of why grid cells look the way they do; 5) provide several neural predictions; 6) highlight the generality of this normative method beyond 2D space.
Revised: In sum, we 1) propose actionable neural representations to support ﬂexible behaviours; 2) formalise the actionable constraint with group and representation theory; 3) mix actionability with biological and functional constraints to create a constrained optimisation problem; 4) analyse this problem and show that in 2D the optimal representation is a good model of grid cells, thus offering a mathematical understanding of why grid cells look the way they do; 5) provide several neural predictions; 6) highlight the generality of this normative method beyond 2D space.

Original: 1. R ELATED W ORK
Revised: 1. R ELATED W ORK

Original: Neuroscientists have long explained representations with normative principles like information maximisation (Attneave, 1954; Barlow et al., 1961), sparse (Olshausen & Field, 1996) or independent (Hyv ¨ arinen, 2010) latent encodings, often mixed with biological constraints such as non-negativity (Sengupta et al., 2018), energy budgets (Niven et al., 2007), or wiring minimisation (Hyv ¨ arinen et al., 2001).
Revised: Neuroscientists have long explained representations with normative principles like information maximisation (Attneave, 1954; Barlow et al., 1961), sparse (Olshausen & Field, 1996) or independent (Hyv ¨ arinen, 2010) latent encodings, often mixed with biological constraints such as non-negativity (Sengupta et al., 2018), energy budgets (Niven et al., 2007), or wiring minimisation (Hyv ¨ arinen et al., 2001).

Original: On the other hand, deep learning learns task optimised representations.
Revised: On the other hand, deep learning learns task optimised representations.

Original: A host of representation-learning principles have been considered (Bengio et al., 2013); but our work is most related to geometric deep learning (Bronstein et al., 2021) which emphasises input transformations, and building neural networks which respect (equivariant) or ignore (invariant) them.
Revised: A host of representation-learning principles have been considered (Bengio et al., 2013); but our work is most related to geometric deep learning (Bronstein et al., 2021) which emphasises input transformations, and building neural networks which respect (equivariant) or ignore (invariant) them.

Original: This is similar in spirit but not in detail to our approach, since equivariant networks do not build representations in which transformations are implementable through matrices.
Revised: This is similar in spirit but not in detail to our approach, since equivariant networks do not build representations in which transformations are implementable through matrices.

Original: There is much previous theory on grid cells, which can be categorised as relating to our actionable, functional, and biological constraints.
Revised: There is much previous theory on grid cells, which can be categorised as relating to our actionable, functional, and biological constraints.

Original: Functional:
Revised: Functional:

Original: Many works argue that grid cells provide an efﬁcient representation of position, that hexagons are optimal (Mathis et al., 2012a;b; Sreenivasan & Fiete, 2011; Wei et al., 2015) and make predictions for relative module lengthscales (Wei et al., 2015).
Revised: Many works argue that grid cells provide an efﬁcient representation of position, that hexagons are optimal (Mathis et al., 2012a;b; Sreenivasan & Fiete, 2011; Wei et al., 2015) and make predictions for relative module lengthscales (Wei et al., 2015).

Original: Since we use similar functional principles, we suspect that some of our novel results, such as grid-to-room alignment, could have been derived by these authors.
Revised: Since we use similar functional principles, we suspect that some of our novel results, such as grid-to-room alignment, could have been derived by these authors.

Original: However, in contrast to our work, these authors assume a grid-like tuning curve.
Revised: However, in contrast to our work, these authors assume a grid-like tuning curve.

Original: Instead we give a normative explanation of why be grid-like at all, explaining features like the alignment of grid axes within a module, which are detrimental from a pure decoding view (Stemmler et al., 2015).
Revised: Instead we give a normative explanation of why be grid-like at all, explaining features like the alignment of grid axes within a module, which are detrimental from a pure decoding view (Stemmler et al., 2015).

Original: Actionability: Grid cells have been classically understood as affording path-integration (integrating velocity to predict position) with units from both hand-tuned Burak & Fiete (2009) and trained recurrent neural network resembling grid cells
Revised: Actionability: Grid cells have been classically understood as affording path-integration (integrating velocity to predict position) with units from both hand-tuned Burak & Fiete (2009) and trained recurrent neural network resembling grid cells

Original: (Cueva & Wei, 2018; Banino et al., 2018; Sorscher et al., 2019).
Revised: (Cueva & Wei, 2018; Banino et al., 2018; Sorscher et al., 2019).

Original: Recently, these recurrent network approaches have been questioned for their parameter dependence (Schaeffer et al., 2022), or relying on decoding place cells with bespoke shapes that are not observed experimentally (Sorscher et al., 2019; Dordek et al., 2016).
Revised: Whether these results are overly parameter dependent, or rely on decoding place cells with bespoke shapes that are not observed experimentally, is currently a matter of debate (Schaeffer et al., 2022; Dordek et al., 2016; Sorscher et al., 2022).

Original: Our mathematical formalisation of path-integration, combined with biological and functional constraints, provides clarity on this issue.
Revised: Our mathematical formalisation of path-integration, combined with biological and functional constraints, provides clarity on this issue.

Original: Our approach is linear, in that actions update the representation linearly, which has previously been explored theoretically (Issa & Zhang, 2012), and numerically, in two works that learnt grid cells (Whittington et al., 2020; Gao et al., 2021).
Revised: Our approach is linear, in that actions update the representation linearly, which has previously been explored theoretically (Issa & Zhang, 2012), and numerically, in two works that learnt grid cells (Whittington et al., 2020; Gao et al., 2021).

Original: Our work could be seen as extracting the key ideas from these papers that make hexagonal grids optimal (see Appendix H), understanding why they produce grid cells, and extending them to multiple modules, something both papers had to hard code.
Revised: Our work could be seen as extracting the key ideas from these papers that make hexagonal grids optimal (see Appendix H), understanding why they produce grid cells, and extending them to multiple modules, something both papers had to hard code.

Original: Our dive into Representation
Revised: Our dive into Representation

Original: Theory builds on the group theoryideas in Gao et al.
Revised: Theory builds on the group theory in Gao et al.

Original: Biological: Lastly, both theoretically (Sorscher et al., 2019) and computationally (Dordek et al., 2016; Whittington et al., 2021), non-negativity has played a key role in normative derivations of hexagonal grid cells, as it will here.
Revised: Biological: Lastly, both theoretically (Sorscher et al., 2019) and computationally (Dordek et al., 2016; Whittington et al., 2021), non-negativity has played a key role in normative derivations of hexagonal grid cells, as it will here.

Original:  A CTIONABLE N EURAL R EPRESENTATIONS : A N O BJECTIVE
Revised:  A CTIONABLE N EURAL R EPRESENTATIONS : A N O BJECTIVE

Original: We seek a representation g ( x ) ∈ R N of 2D position x ∈ T 2 , where N is the number of neurons.
Revised: We seek a representation g ( x ) ∈ R N of 2D position x ∈ T 2 , where N is the number of neurons.

Original: Our representation is built using three ideas: functional, biological, and actionable; whose combination will lead to multiple modules of grid cells, and which we’ll now formalise.
Revised: Our representation is built using three ideas: functional, biological, and actionable; whose combination will lead to multiple modules of grid cells, and which we’ll now formalise.

Original: Functional:
Revised: Functional:

Original: To be useful, the representation must encode different positions differently.
Revised: To be useful, the representation must encode different positions differently.

Original: However, it is more important to distinguish positions 1km apart than 1mm, and frequently visited positions should be separated the most.
Revised: However, it is more important to distinguish positions 1km apart than 1mm, and frequently visited positions should be separated the most.

Original: To account for these, we ask our representation to minimise
Revised: To account for these, we ask our representation to minimise

Original: [Equation]
Revised: [Equation]

Original: [Equation] The red term measures the representational similarity of x and x (cid:48) ; it is large if their representations are nearer than some distance σ in neural space and small otherwise.
Revised: The red term measures the representational similarity of x and x (cid:48) ; it is large if their representations are nearer than some distance σ in neural space and small otherwise.

Original: By integrating over all pairs x and x (cid:48) , L measures the total representational similarity, which we seek to minimise.
Revised: By integrating over all pairs x and x (cid:48) , L measures the total representational similarity, which we seek to minimise.

Original: The green term is the agent’s position occupancy distribution, which ensures only visited points contribute to the loss, for now simply a Gaussian of lengthscale L .
Revised: The green term is the agent’s position occupancy distribution, which ensures only visited points contribute to the loss, for now simply a Gaussian of lengthscale L .

Original: Finally, the blue term weights the importance of separating each pair, encouraging separation of points more distant than a lengthscale, l .
Revised: Finally, the blue term weights the importance of separating each pair, encouraging separation of points more distant than a lengthscale, l .

Original: [Equation]
Revised: [Equation]

Original: Biological: Neurons have non-negative ﬁring rates, so we constrain g
Revised: Biological: Neurons have non-negative ﬁring rates, so we constrain g

Original: i ( x ) ≥ 0 .
Revised: i ( x ) ≥ 0 .

Original: Further, neurons can’t ﬁre arbitrarily fast, and ﬁring is energetically costly, so we constrain each neuron’s response g n ( x ) via (cid:82) g 2 n ( x ) p ( x ) d x =
Revised: Further, neurons can’t ﬁre arbitrarily fast, and ﬁring is energetically costly, so we constrain each neuron’s response g n ( x ) via (cid:82) g 2 n ( x ) p ( x ) d x =

Original: Actionable: Our ﬁnal constraint requires that the representation is actionable.
Revised: Actionable: Our ﬁnal constraint requires that the representation is actionable.

Original: This means each transformations of the input must have its own transformation in neural space, independent of position.
Revised: This means each transformations of the input must have its own transformation in neural space, independent of position.

Original: For mathematical convenience we enact the neural transformation using a matrix.
Revised: For mathematical convenience we enact the neural transformation using a matrix.

Original: Labelling this matrix T (∆ x ) ∈ R N × N , for transformation ∆ x , this means that for all positions x ,
Revised: Labelling this matrix T (∆ x ) ∈ R N × N , for transformation ∆ x , this means that for all positions x ,

Original: [Equation]
Revised: [Equation]

Original: For intuition into how this constrains the neural code g ( x ) , we consider a simple example of two neurons representing an angle θ ∈ [0 , 2 π ) .
Revised: For intuition into how this constrains the neural code g ( x ) , we consider a simple example of two neurons representing an angle θ ∈ [0 , 2 π ) .

Original: Replacing x with θ in equation 

Revised: Replacing x with θ in equation

Original: we get the equivalent constraint: g ( θ + ∆ θ )
Revised: we get the equivalent constraint: g ( θ + ∆ θ )

Original: =
Revised: =

Original: T (∆ θ ) g ( θ ) .
Revised: T (∆ θ ) g ( θ ) .

Original: Here the matrix T performs a rotation, and the solution (up to a linear transform) is for T to be the standard 2 × 2 rotation matrix, with frequency n .
Revised: Here the matrix T performs a rotation, and the solution (up to a linear transform) is for T to be the standard 2 × 2 rotation matrix, with frequency n .

Original: [Equation]
Revised: [Equation]

Original: Thus as θ rotates by 2 π the neural representation traces out a circle an integer number, n , times.
Revised: Thus as θ rotates by 2 π the neural representation traces out a circle an integer number, n , times.

Original: Thanks to the problem’s linearity, extending to more neurons is easy.
Revised: Thanks to the problem’s linearity, extending to more neurons is easy.

Original: Adding two more neurons lets the population contain another sine and cosine at some frequency, just like the two neurons in equation 4.
Revised: Adding two more neurons lets the population contain another sine and cosine at some frequency, just like the two neurons in equation 4.

Original: Extrapolating this we get our actionability constraint: the neural response must be constructed from some invertible linear mixing of the sines and cosines of D < N 2 frequencies,
Revised: Extrapolating this we get our actionability constraint: the neural response must be constructed from some invertible linear mixing of the sines and cosines of D < N 2 frequencies,

Original: [Equation]
Revised: [Equation]

Original: The vectors { a d , b d } Dd =1 ∈ R N are coefﬁcient vectors that mix together the sines and cosines, of which there are D .
Revised: The vectors { a d , b d } Dd =1 ∈ R N are coefﬁcient vectors that mix together the sines and cosines, of which there are D .

Original: a 0 is the coefﬁcient vector for a frequency that cycles 0 times.
Revised: a 0 is the coefﬁcient vector for a frequency that cycles 0 times.

Original: This argument comes from an area of maths called Representation Theory (a different meaning of representation!)
Revised: This argument comes from an area of maths called Representation Theory (a different meaning of representation!)

Original: that places constraints on the matrices T for variables whose transformations form a mathematical object called a group.
Revised: that places constraints on the matrices T for variables whose transformations form a mathematical object called a group.

Original: This includes many of interest, such as position on a circle, torus, or sphere.
Revised: This includes many of interest, such as position on a circle, torus, or sphere.

Original: These constraints on matrices can be translated into constraints on an actionable neural code just like we did for g ( θ ) (see Appendix A).
Revised: These constraints on matrices can be translated into constraints on an actionable neural code just like we did for g ( θ ) (see Appendix A).

Original: When generalising the above example to 2D space (a torus), we must consider a few things: First, the space is two-dimensional, so compared to our previous equation 5, the frequencies, denoted k d , are now two dimensional.
Revised: When generalising the above example to 2D space (a torus), we must consider a few things: First, the space is two-dimensional, so compared to our previous equation 5, the frequencies, denoted k d , are now two dimensional.

Original: Second, to approximate a ﬁnite region of ﬂat 2D space, we consider a similarly sized region of a torus.
Revised: Second, to approximate a ﬁnite region of ﬂat 2D space, we consider a similarly sized region of a torus.

Original: As the radius of the torus grows this approximation becomes arbitrarily good (see Appendix A.4 for discussion).
Revised: As the radius of the torus grows this approximation becomes arbitrarily good (see Appendix A.4 for discussion).

Original: Periodicity constrains the frequencies in equation 5 to be nR for integer n and ring radius
Revised: Periodicity constrains the frequencies in equation 5 to be nR for integer n and ring radius

Original: R .
Revised: R .

Original: As the loop (torus in 2D) becomes very large these permitted frequencies become arbitrarily close, so we drop the integer constraint,
Revised: As the loop (torus in 2D) becomes very large these permitted frequencies become arbitrarily close, so we drop the integer constraint,

Original: [Equation]
Revised: [Equation]

Original: Our constrained optimisation problem is complete.
Revised: Our constrained optimisation problem is complete.

Original: Equation 6 speciﬁes the set of actionable representations.
Revised: Equation 6 speciﬁes the set of actionable representations.

Original: Without additional constraints these codes are meaningless: random combinations of sines and cosines produce random neural responses (Figure 2A).
Revised: Without additional constraints these codes are meaningless: random combinations of sines and cosines produce random neural responses (Figure 2A).

Original: We will choose from amongst the set of actionable codes by optimising the parameters a 0 , { a d , b d , k d } Dd =1 to minimise L , subject to biological (non-negative and bounded ﬁring rates) constraints.
Revised: We will choose from amongst the set of actionable codes by optimising the parameters a 0 , { a d , b d , k d } Dd =1 to minimise L , subject to biological (non-negative and bounded ﬁring rates) constraints.

Original:  O PTIMAL R EPRESENTATIONS
Revised:  O PTIMAL R EPRESENTATIONS

Original: Optimising over the set of actionable codes to minimise L with biological constraints gives multiple modules of grid cells (Figure 2B).
Revised: Optimising over the set of actionable codes to minimise L with biological constraints gives multiple modules of grid cells (Figure 2B).

Original: This section will, using intuition and analytics, explain why.
Revised: This section will, using intuition and analytics, explain why.

Original: (Figure 6 shows that the population belong to one of these three
Revised: (Figure 6 shows that the population belong to one of these three

Original: 3. N ON - NEGATIVITY L EADS TO A M ODULE OF L ATTICE C ELLS
Revised: 3. N ON - NEGATIVITY L EADS TO A M ODULE OF L ATTICE C ELLS

Original: To understand how non-negativity produces modules of lattice responses we will study the following simpliﬁed loss, which maximises the Euclidean distance between representations of angle, g ( θ ) ,
Revised: To understand how non-negativity produces modules of lattice responses we will study the following simpliﬁed loss, which maximises the Euclidean distance between representations of angle, g ( θ ) ,

Original: [Equation]
Revised: [Equation]

Original: This is equivalent to the full loss (equation 1) for uniform p ( θ ) , χ ( θ, θ (cid:48) )
Revised: This is equivalent to the full loss (equation 1) for uniform p ( θ ) , χ ( θ, θ (cid:48) )

Original: = 1 , and σ very large.
Revised: = 1 , and σ very large.

Original: Make no mistake, this is a bad loss.
Revised: Make no mistake, this is a bad loss.

Original: For contrast, the full loss encouraged the representations of different positions to be separated by more than σ , enabling discrimination 1 .
Revised: For contrast, the full loss encouraged the representations of different positions to be separated by more than σ , enabling discrimination 1 .

Original: Therefore, sensibly, the representation is most rewarded for separating points that would otherwise be closer than σ .
Revised: Therefore, sensibly, the representation is most rewarded for separating points that would otherwise be closer than σ .

Original: Ldoes the opposite!
Revised: Ldoes the opposite!

Original: It grows quadratically with separation, so g ( θ ) is most rewarded for pushing apart already well-separated points, a terrible representational principle!
Revised: It grows quadratically with separation, so g ( θ ) is most rewarded for pushing apart already well-separated points, a terrible representational principle!

Original: Nonetheless, L 0 will give us key insights
Revised: Nonetheless, L 0 will give us key insights

Original: Since actionability gives us a parameterised form of the representations (equation 5), we can compute the integrals to obtain the following constrained optimisation problem (details: Appendix C)
Revised: Since actionability gives us a parameterised form of the representations (equation 5), we can compute the integrals to obtain the following constrained optimisation problem (details: Appendix C)

Original: [Equation]
Revised: [Equation]

Original: with
Revised: with

Original: Non
Revised: Non

Original: [Equation]
Revised: [Equation]

Original: negativity
Revised: negativity

Original: [Equation]
Revised: [Equation]

Original: [Figure]
Revised: [Figure]

Original: [Equation]
Revised: [Equation]

Original: Where N is the number of neurons.
Revised: Where N is the number of neurons.

Original: This is now something we can understand.
Revised: This is now something we can understand.

Original: First, reminding ourselves that the neural code, g ( θ ) , is made from a constant vector, a 0 , and θ − dependent parts (equation 5; Figure 3A), we can see that L 0 separates representations by encouraging the size of each varying part, (cid:107) a d (cid:107) 

Revised: First, reminding ourselves that the neural code, g ( θ ) , is made from a constant vector, a 0 , and θ − dependent parts (equation 5; Figure 3A), we can see that L 0 separates representations by encouraging the size of each varying part, (cid:107) a d (cid:107)

Original: + (cid:107) b d (cid:107) 2 , to be maximised.
Revised: + (cid:107) b d (cid:107) 2 , to be maximised.

Original: This effect is limited by the ﬁring rate bound, (cid:107) a 0 (cid:107) 2 − 12 L 0 = N .
Revised: This effect is limited by the ﬁring rate bound, (cid:107) a 0 (cid:107) 2 − 12 L 0 = N .

Original: Thus, to minimise L 0 we must minimise the constant vector, a 0 .
Revised: Thus, to minimise L 0 we must minimise the constant vector, a 0 .

Original: This would be easy without non-negativity (when any code with (cid:107) a 0 (cid:107) = 0 is optimal), but no sum of sines and cosines can be non-negative for all θ without an offset.
Revised: This would be easy without non-negativity (when any code with (cid:107) a 0 (cid:107) = 0 is optimal), but no sum of sines and cosines can be non-negative for all θ without an offset.

Original: Thus the game is simple; choose frequencies and coefﬁcients so the ﬁring rates are non-negative, but using the smallest possible constant vector.
Revised: Thus the game is simple; choose frequencies and coefﬁcients so the ﬁring rates are non-negative, but using the smallest possible constant vector.

Original: loops.
Revised: loops.

Original: B adding harmonic frequencies increases the allowing code made
Revised: B adding harmonic frequencies increases the allowing code made

Original: One lattice cell .
Revised: One lattice cell .

Original: We now heuristically argue, and conﬁrm in simulation, that the optimal solution for a single neuron is a lattice tuning curve (see Appendix G for why this problem is non-convex).
Revised: We now heuristically argue, and conﬁrm in simulation, that the optimal solution for a single neuron is a lattice tuning curve (see Appendix G for why this problem is non-convex).

Original: Starting with a single frequency component, e.g. sin( θ ) , achieving non-negativity requires adding a constant offset, sin( θ ) + 1 (Figure 3B).
Revised: Starting with a single frequency component, e.g. sin( θ ) , achieving non-negativity requires adding a constant offset, sin( θ ) + 1 (Figure 3B).

Original: However, we could also have just added another frequency.
Revised: However, we could also have just added another frequency.

Original: In particular adding harmonics of the base frequency (with appropriate phase shifts) pushes up the minima (Figure 3B).
Revised: In particular adding harmonics of the base frequency (with appropriate phase shifts) pushes up the minima (Figure 3B).

Original: Extending this argument, we suggest non-negativity, for a single cell, can be achieved by including a grid of frequencies.
Revised: Extending this argument, we suggest non-negativity, for a single cell, can be achieved by including a grid of frequencies.

Original: This gives a lattice tuning curve (Figure 3B right).
Revised: This gives a lattice tuning curve (Figure 3B right).

Original: Module of lattice cells .
Revised: Module of lattice cells .

Original: Achieving non-negativity for this cell used up many frequencies.
Revised: Achieving non-negativity for this cell used up many frequencies.

Original: But as discussed (Section 2), actionability only allows a limited number frequencies ( < N 2 since each frequency uses 2 neurons (sine and cosine)), thus how can we make lots of neurons non-negative with limited frequencies?
Revised: But as discussed (Section 2), actionability only allows a limited number frequencies ( < N 2 since each frequency uses 2 neurons (sine and cosine)), thus how can we make lots of neurons non-negative with limited frequencies?

Original: Fortunately, we can do so by making all neuron’s tuning curves translated versions of each other, as translated curves contain the same frequencies but with different phases.
Revised: Fortunately, we can do so by making all neuron’s tuning curves translated versions of each other, as translated curves contain the same frequencies but with different phases.

Original: This is a module of lattice cells.
Revised: This is a module of lattice cells.

Original: We validate our arguments by numerically optimising the coefﬁcients a 0 , { a d , b d } Dd =1 and frequencies { n d } Dd =1 to minimise L 0 subject to constraints, producing a module of lattices (Figure 3C; details in Appendix B).
Revised: We validate our arguments by numerically optimising the coefﬁcients a 0 , { a d , b d } Dd =1 and frequencies { n d } Dd =1 to minimise L 0 subject to constraints, producing a module of lattices (Figure 3C; details in Appendix B).

Original: These arguments equally apply to representations of a periodic 2D space (a torus; Figure 3D).
Revised: These arguments equally apply to representations of a periodic 2D space (a torus; Figure 3D).

Original: Studying L 0 has told us why lattice response curves are good.
Revised: Studying L 0 has told us why lattice response curves are good.

Original: But surprisingly, all lattices are equally good, even at inﬁnitely high frequency.
Revised: But surprisingly, all lattices are equally good, even at inﬁnitely high frequency.

Original: Returning to the full loss will break this degeneracy.
Revised: Returning to the full loss will break this degeneracy.

Original: 3. P RIORITISING I MPORTANT P AIRS OF P OSITIONS P RODUCES H EXAGONAL G RID C ELLS
Revised: 3. P RIORITISING I MPORTANT P AIRS OF P OSITIONS P RODUCES H EXAGONAL G RID C ELLS

Original: Now we return to the full loss and understand its impact in two steps, beginning with the reintroduction of χ and p , which break the lattice degeneracy, forming hexagonal grid cells.
Revised: Now we return to the full loss and understand its impact in two steps, beginning with the reintroduction of χ and p , which break the lattice degeneracy, forming hexagonal grid cells.

Original: [Equation]
Revised: [Equation]

Original: χ prefers low frequencies: recall that χ = 1 − e
Revised: χ prefers low frequencies: recall that χ = 1 − e

Original: [Equation]
Revised: [Equation]

Original: ensures very distant inputs have different representations, while allowing similar inputs to have similar representations, up to a resolution, l .
Revised: ensures very distant inputs have different representations, while allowing similar inputs to have similar representations, up to a resolution, l .

Original: This encourages low frequencies ( (cid:107) k d (cid:107) <
Revised: This encourages low frequencies ( (cid:107) k d (cid:107) <

Original: 1 l ), which separate distant points but produce similar representations for pairs closer than l (Analytics: Appendix D.1).
Revised: 1 l ), which separate distant points but produce similar representations for pairs closer than l (Analytics: Appendix D.1).

Original: At this stage, for periodic 2D space, the lowest frequency lattices, place cells, are optimal (see Appendix F; Sengupta et al.
Revised: At this stage, for periodic 2D space, the lowest frequency lattices, place cells, are optimal (see Appendix F; Sengupta et al.

Original: p ( x ) prefers high frequencies: However, the occupancy distribution of the animal, p ( x ) , counters χ .
Revised: p ( x ) prefers high frequencies: However, the occupancy distribution of the animal, p ( x ) , counters χ .

Original: On an inﬁnite 2D plane animals must focus on representing a limited area, of lengthscale L .
Revised: On an inﬁnite 2D plane animals must focus on representing a limited area, of lengthscale L .

Original: This encourages high frequencies ( (cid:107) k d (cid:107) > 1 L ), whose response varies among the visited points (Analytics: Appendix D.2).
Revised: This encourages high frequencies ( (cid:107) k d (cid:107) > 1 L ), whose response varies among the visited points (Analytics: Appendix D.2).

Original: More complex p ( x ) induce more complex frequency biases, but, to ﬁrst order, the effect is always a high frequency bias (Figure 5F-G, Appendix L).
Revised: More complex p ( x ) induce more complex frequency biases, but, to ﬁrst order, the effect is always a high frequency bias (Figure 5F-G, Appendix L).

Original: Since, of all lattices, hexagons ﬁt the most frequencies within the annulus, they are preferred, and hexagonal frequency lattices lead to hexagonal grid cells.
Revised: Since, of all lattices, hexagons ﬁt the most frequencies within the annulus, they are preferred, and hexagonal frequency lattices lead to hexagonal grid cells.

Original: B Simulations conﬁrm.
Revised: B Simulations conﬁrm.

Original: C Harmonically related frequencies co-repeat more often than non-harmonic, meaning that, as a pair, harmonically related frequencies are worse at encoding, since they encode many points in the same way.
Revised: C Harmonically related frequencies co-repeat more often than non-harmonic, meaning that, as a pair, harmonically related frequencies are worse at encoding, since they encode many points in the same way.

Original: Combination → Hexagons: Satisfying non-negativity and functionality required a lattice of many frequencies, but now p and χ bias our frequency choice, preferring those beyond 1 L (to separate points the animal visits) but smaller than 1 l (to separate distant visited points).
Revised: Combination → Hexagons: Satisfying non-negativity and functionality required a lattice of many frequencies, but now p and χ bias our frequency choice, preferring those beyond 1 L (to separate points the animal visits) but smaller than 1 l (to separate distant visited points).

Original: Thus to get as many of these preferred frequencies as possible, we want the lattice with the densest packing within a
Revised: Thus to get as many of these preferred frequencies as possible, we want the lattice with the densest packing within a

Original: Goldilocks annulus in frequency space (Figure 4A).
Revised: Goldilocks annulus in frequency space (Figure 4A).

Original: This is a hexagonal lattice in frequency space which leads to a hexagonal grid cell.
Revised: This is a hexagonal lattice in frequency space which leads to a hexagonal grid cell.

Original: Simulations with few neurons agree, giving a module of hexagonal grid cells (Figure 4B).
Revised: Simulations with few neurons agree, giving a module of hexagonal grid cells (Figure 4B).

Original: 3. A H ARMONIC T USSLE P RODUCES M ULTIPLE M ODULES
Revised: 3. A H ARMONIC T USSLE P RODUCES M ULTIPLE M ODULES

Original: Finally, we will study the neural lengthscale σ , and understand how it produces multiple modules.
Revised: Finally, we will study the neural lengthscale σ , and understand how it produces multiple modules.

Original: [Equation]
Revised: [Equation]

Original: As discussed, L prioritises the separation of poorly distinguished points, those whose representations are closer than σ .
Revised: As discussed, L prioritises the separation of poorly distinguished points, those whose representations are closer than σ .

Original: This causes certain frequencies to be desired in the overall population , in particular those unrelated to existing frequencies by simple harmonic ratios, i.e. not ω 1 = 32 ω 2 (Figure 4C; see Appendix E for a perturbative derivation of this effect).
Revised: This causes certain frequencies to be desired in the overall population , in particular those unrelated to existing frequencies by simple harmonic ratios, i.e. not ω 1 = 32 ω 2 (Figure 4C; see Appendix E for a perturbative derivation of this effect).

Original: This is because pairs of harmonically related frequencies represent more positions identically than a non-harmonically related pair, so are worse for separation (similar to arguments made in Wei et al.
Revised: This is because pairs of harmonically related frequencies represent more positions identically than a non-harmonically related pair, so are worse for separation (similar to arguments made in Wei et al.

Original: This, however, sets up a ‘harmonic tussle’ between what the population wants - non-harmonically related frequencies for L - and what single neurons want - harmonically related frequency lattices for non-negativity (Section 3.1).
Revised: This, however, sets up a ‘harmonic tussle’ between what the population wants - non-harmonically related frequencies for L - and what single neurons want - harmonically related frequency lattices for non-negativity (Section 3.1).

Original: Modules of grid cells resolve this tension: harmonic frequencies exist within modules to give non-negativity, and non-harmonically related modules allow for separation, explaining the earlier simulation results (Figure 2B; further details in Appendix E.3).
Revised: Modules of grid cells resolve this tension: harmonic frequencies exist within modules to give non-negativity, and non-harmonically related modules allow for separation, explaining the earlier simulation results (Figure 2B; further details in Appendix E.3).

Original: This concludes our main result.
Revised: This concludes our main result.

Original: We have shown three constraints on neural populations - actionable, functional, and biological - lead to multiple modules of hexagonal grid cells, and we have understood why.
Revised: We have shown three constraints on neural populations - actionable, functional, and biological - lead to multiple modules of hexagonal grid cells, and we have understood why.

Original: We posit this is the minimal set of requirements for grid cells (see Appendix I for ablation simulations and discussion).
Revised: We posit this is the minimal set of requirements for grid cells (see Appendix I for ablation simulations and discussion).

Original:  P REDICTIONS
Revised:  P REDICTIONS

Original: Our theory makes testable predictions about the structure of optimal actionable codes for 2D space.
Revised: Our theory makes testable predictions about the structure of optimal actionable codes for 2D space.

Original: We describe three here: tuning curve sharpness scales with the number of neurons in a module; the optimal angle between modules; and the optimal grid alignment to room geometry.
Revised: We describe three here: tuning curve sharpness scales with the number of neurons in a module; the optimal angle between modules; and the optimal grid alignment to room geometry.

Original: 4. L ATTICE S IZE :F IELD W IDTH R ATIO SCALES WITH N UMBER OF N EURONS IN M ODULE
Revised: 4. L ATTICE S IZE :F IELD W IDTH R ATIO SCALES WITH N UMBER OF N EURONS IN M ODULE

Original: In our framework the number of neurons controls the number of frequencies in the representation (equation 6).
Revised: In our framework the number of neurons controls the number of frequencies in the representation (equation 6).

Original: A neuron within a module only contains frequencies from that module’s frequency lattice, since other modules have non-harmonically related frequencies.
Revised: A neuron within a module only contains frequencies from that module’s frequency lattice, since other modules have non-harmonically related frequencies.

Original: More neurons in a module, means more and higher frequencies in the lattice, which sharpen grid peaks (Figure 5A).
Revised: More neurons in a module, means more and higher frequencies in the lattice, which sharpen grid peaks (Figure 5A).

Original: We formalise this (Appendix J) and predict that the number of neurons within a module scales with the square of the lattice lengthscale, ν , to ﬁeld width, µ , ratio, N ∝ (cid:0) νµ (cid:1) 2 .
Revised: We formalise this (Appendix J) and predict that the number of neurons within a module scales with the square of the lattice lengthscale, ν , to ﬁeld width, µ , ratio, N ∝ (cid:0) νµ (cid:1) 2 .

Original: This matches the intuition that the sharper a module’s peak, the more neurons you need to tile the entire space.
Revised: This matches the intuition that the sharper a module’s peak, the more neurons you need to tile the entire space.

Original: In a rudimentary analysis, our predictions compare favourably to data from Stensola et al.
Revised: In a rudimentary analysis, our predictions compare favourably to data from Stensola et al.

Original: (2012) assuming uniform sampling of grid cells across modules (Figure 5B).
Revised: (2012) assuming uniform sampling of grid cells across modules (Figure 5B).

Original: We are eager to test these claims quantitatively.
Revised: We are eager to test these claims quantitatively.

Original: 4. M ODULES ARE O PTIMALLY O RIENTED AT S MALL O FFSETS ( ∼ 4 ◦ )
Revised: 4. M ODULES ARE O PTIMALLY O RIENTED AT S MALL O FFSETS ( ∼ 4 ◦ )

Original: In section 3.3 we saw how frequencies of different modules are maximally non-harmonically related in order to separate the representation of as many points as possible.
Revised: In section 3.3 we saw how frequencies of different modules are maximally non-harmonically related in order to separate the representation of as many points as possible.

Original: To maximise non-harmonicity between two modules, the second module’s frequency lattice can be both stretched and rotated relative to the ﬁrst. 0 or 30 ◦ relative orientations are particularly bad coding choices as they align the high density axes of the two lattices (Figure 5C).
Revised: To maximise non-harmonicity between two modules, the second module’s frequency lattice can be both stretched and rotated relative to the ﬁrst. 0 or 30 ◦ relative orientations are particularly bad coding choices as they align the high density axes of the two lattices (Figure 5C).

Original: The optimal angular offset of two modules, calculated via a frequency overlap metric (Appendix K), is small (Figure 5D); the value depends on the grid peak and lattice lengthscales, µ and ν , but varies between 3 ◦ and 8 ◦ degrees.
Revised: The optimal angular offset of two modules, calculated via a frequency overlap metric (Appendix K), is small (Figure 5D); the value depends on the grid peak and lattice lengthscales, µ and ν , but varies between 3 ◦ and 8 ◦ degrees.

Original: Multiple modules should orient at a sequence of small angles (Appendix K).
Revised: Multiple modules should orient at a sequence of small angles (Appendix K).

Original: In a rudimentary analysis, our predictions compare favourably to the observations of Stensola et al.
Revised: In a rudimentary analysis, our predictions compare favourably to the observations of Stensola et al.

Original: (2012) (Figure 5E).
Revised: (2012) (Figure 5E).

Original: (2012) the most sharply peaked grids were recorded most often (2nd left), and the broadest the least rightmost).
Revised: (2012) the most sharply peaked grids were recorded most often (2nd left), and the broadest the least rightmost).

Original: C Aligning the high-density axes of two lattices creates high overlap, while small offsets minimise it.
Revised: C Aligning the high-density axes of two lattices creates high overlap, while small offsets minimise it.

Original: D
Revised: D

Original: We quantiﬁed the overlap as a function of offset angle (Appendix K) and found the minima occured at ∼ 4 ◦ from aligned (aligned = 6 maxima, 4 ◦ offsets are the 12 minima).
Revised: We quantiﬁed the overlap as a function of offset angle (Appendix K) and found the minima occured at ∼ 4 ◦ from aligned (aligned = 6 maxima, 4 ◦ offsets are the 12 minima).

Original: E The orientation and spacing of all grids in animals with multiple modules recorded by Stensola et al.
Revised: E The orientation and spacing of all grids in animals with multiple modules recorded by Stensola et al.

Original: Many modules are misaligned by small offsets, matching our prediction.
Revised: Many modules are misaligned by small offsets, matching our prediction.

Original: F Square or
Revised: F Square or

Original: G circular rooms create complex frequency biases, L ( k ) is the loss for a one frequency code of ﬁxed amplitude.
Revised: G circular rooms create complex frequency biases, L ( k ) is the loss for a one frequency code of ﬁxed amplitude.

Original: H
Revised: H

Original: The optimal frequencies along one axis of a box occur at 2 πnL for integer n : Squishing a room makes the optimal frequencies expand.
Revised: The optimal frequencies along one axis of a box occur at 2 πnL for integer n : Squishing a room makes the optimal frequencies expand.

Original: Grids should change to ﬁt the optimal patterns in the recorded environment, unless they happen to be optimal for both, as 6 πL is.
Revised: Grids should change to ﬁt the optimal patterns in the recorded environment, unless they happen to be optimal for both, as 6 πL is.

Original: I Most grid cells scale with the room, but, when one side is squashed by a factor of 2/3, those at 6 πL are stable.
Revised: I Most grid cells scale with the room, but, when one side is squashed by a factor of 2/3, those at 6 πL are stable.

Original: 4. O PTIMAL G RIDS M ORPH TO R OOM G EOMETRY
Revised: 4. O PTIMAL G RIDS M ORPH TO R OOM G EOMETRY

Original: In Section 3.2 (and Appendix D) we showed that p ( x ) , the animal’s occupancy distribution, introduced a high frequency bias - grid cells must peak often enough to encode visited points.
Revised: In Section 3.2 (and Appendix D) we showed that p ( x ) , the animal’s occupancy distribution, introduced a high frequency bias - grid cells must peak often enough to encode visited points.

Original: However, changing p ( x ) changes the shape of this high frequency bias (Appendix L).
Revised: However, changing p ( x ) changes the shape of this high frequency bias (Appendix L).

Original: In particular, we examine an animal’s encoding of square, circular, or rectangular environments, Appendix L, with the assumption that p ( x ) is uniform over that space.
Revised: In particular, we examine an animal’s encoding of square, circular, or rectangular environments, Appendix L, with the assumption that p ( x ) is uniform over that space.

Original: In each case the bias is coarsely towards high frequencies, but has additional intricacies: in square and rectangular rooms optimal frequencies lie on a lattice, with peaks at integer multiples of 2 πL along one of the cardinal axes, for room width/height
Revised: In each case the bias is coarsely towards high frequencies, but has additional intricacies: in square and rectangular rooms optimal frequencies lie on a lattice, with peaks at integer multiples of 2 πL along one of the cardinal axes, for room width/height

Original: L (Figure 5F); whereas in circular rooms optima are at the zeros of a Bessel function (Figure 5G).
Revised: L (Figure 5F); whereas in circular rooms optima are at the zeros of a Bessel function (Figure 5G).

Original: These ideas make several predictions.
Revised: These ideas make several predictions.

Original: For example, grid modules in circular rooms should have lengthscales set by the optimal radii in Figure 5G, but they should still remain hexagonal since the
Revised: For example, grid modules in circular rooms should have lengthscales set by the optimal radii in Figure 5G, but they should still remain hexagonal since the

Original: Bessel function is circularly symmetric.
Revised: Bessel function is circularly symmetric.

Original: However, the optimal representation in square rooms should not be perfectly hexagonal since p ( x ) induces a bias inconsistent with a hexagonal lattice (this effect is negligible for high frequency grids).
Revised: However, the optimal representation in square rooms should not be perfectly hexagonal since p ( x ) induces a bias inconsistent with a hexagonal lattice (this effect is negligible for high frequency grids).

Original: Intriguingly, shearing towards squarer lattices is observed in square rooms (Stensola et al., 2015), and it would be interesting to test its grid-size dependence.
Revised: Intriguingly, shearing towards squarer lattices is observed in square rooms (Stensola et al., 2015), and it would be interesting to test its grid-size dependence.

Original: Lastly, these effects make predictions about how grid cells morph when the environment geometry changes.
Revised: Lastly, these effects make predictions about how grid cells morph when the environment geometry changes.

Original: A grid cell that is optimal in both geometries can remain the same, however sub-optimal grid cells should change.
Revised: A grid cell that is optimal in both geometries can remain the same, however sub-optimal grid cells should change.

Original: For example turning a square into a squashed square (i.e. a rectangle), stretches the optimal frequencies along the squashed dimension.
Revised: For example turning a square into a squashed square (i.e. a rectangle), stretches the optimal frequencies along the squashed dimension.

Original: Thus, some cells are optimal in both rooms and should stay stable, will others will change, presumably to nearby optimal frequencies (Figure 5H).
Revised: Thus, some cells are optimal in both rooms and should stay stable, will others will change, presumably to nearby optimal frequencies (Figure 5H).

Original: Indeed Stensola et al.
Revised: Indeed Stensola et al.

Original: (2012) recorded the same grid cells in a square and rectangular environment (Figure 5I), and observed exactly these phenomena.
Revised: (2012) recorded the same grid cells in a square and rectangular environment (Figure 5I), and observed exactly these phenomena.

Original:  D ISCUSSION & C ONCLUSIONS
Revised:  D ISCUSSION & C ONCLUSIONS

Original: We have proposed actionability as a fundamental representational principle to afford ﬂexible behaviours.
Revised: We have proposed actionability as a fundamental representational principle to afford ﬂexible behaviours.

Original: We have shown in simulation and with analytic justiﬁcation that the optimal actionable representations of 2D space are, when constrained to be both biological and functional, multiple modules of hexagonal grid cells, thus offering a mathematical understanding of grid cells.
Revised: We have shown in simulation and with analytic justiﬁcation that the optimal actionable representations of 2D space are, when constrained to be both biological and functional, multiple modules of hexagonal grid cells, thus offering a mathematical understanding of grid cells.

Original: We then used this theory to make three novel grid cell predictions that match data on early inspection.
Revised: We then used this theory to make three novel grid cell predictions that match data on early inspection.

Original: While this is promising for our theory, there remain some grid cell phenomena that, as it stands, it will never predict.
Revised: While this is promising for our theory, there remain some grid cell phenomena that, as it stands, it will never predict.

Original: For example, grid cell peaks vary in intensity (Dunn et al., 2017), and grid lattices bend in trapezoidal environments (Krupic et al., 2015).
Revised: For example, grid cell peaks vary in intensity (Dunn et al., 2017), and grid lattices bend in trapezoidal environments (Krupic et al., 2015).

Original: These effects may be due to incorporation of sensory information - something we have not included in our theory - to better infer position.
Revised: These effects may be due to incorporation of sensory information - something we have not included in our theory - to better infer position.

Original: Including such effects may recapitulate these ﬁndings, as in Ocko et al.
Revised: Including such effects may recapitulate these ﬁndings, as in Ocko et al.

Original: Our theory is normative and abstracted from implementation.
Revised: Our theory is normative and abstracted from implementation.

Original: However, both modelling (Burak &
Revised: However, both modelling (Burak &

Original: Fiete, 2009) and experimental (Gardner et al., 2022; Kim et al., 2017) work suggests that continuous attractor networks (CANs) implement path integrating circuits.
Revised: Fiete, 2009) and experimental (Gardner et al., 2022; Kim et al., 2017) work suggests that continuous attractor networks (CANs) implement path integrating circuits.

Original: Actionability and CANs imply seemingly different representation update equations; future work could usefully compare the two.
Revised: Actionability and CANs imply seemingly different representation update equations; future work could usefully compare the two.

Original: While we focused on understanding the optimal representations of 2D space and their relationship to grid cells, our theory is more general.
Revised: While we focused on understanding the optimal representations of 2D space and their relationship to grid cells, our theory is more general.

Original: Most simply, it can be applied to behaviour in other, non 2D, spaces.
Revised: Most simply, it can be applied to behaviour in other, non 2D, spaces.

Original: In fact many variables whose transformations form a group are relatively easily analysed.
Revised: In fact many variables whose transformations form a group are relatively easily analysed.

Original: The brain represents many such variables, e.g. heading directions, (Finkelstein et al., 2015), object orientations, (Logothetis et al., 1995), the‘4-loop task’ of Sun et al.
Revised: The brain represents many such variables, e.g. heading directions, (Finkelstein et al., 2015), object orientations, (Logothetis et al., 1995), the‘4-loop task’ of Sun et al.

Original: (2020) or 3-dimensional space (Grieves et al., 2021;
Revised: (2020) or 3-dimensional space (Grieves et al., 2021;

Original: Ginosar et al., 2021).
Revised: Ginosar et al., 2021).

Original: Interestingly, our theory predicts 3D representations with regular order (Figure 19E in Appendix M), unlike those found in the brain (Grieves et al., 2021; Ginosar et al., 2021) suggesting the animal’s 3D navigation is sub-optimal.
Revised: Interestingly, our theory predicts 3D representations with regular order (Figure 19E in Appendix M), unlike those found in the brain (Grieves et al., 2021; Ginosar et al., 2021) suggesting the animal’s 3D navigation is sub-optimal.

Original: But, most expansively, these principles express a view that representations must be more than just passive encodings of the world; they must embed the consequences of predictable actions, allowing planning and inferences in never-before-seen situations.
Revised: But, most expansively, these principles express a view that representations must be more than just passive encodings of the world; they must embed the consequences of predictable actions, allowing planning and inferences in never-before-seen situations.

Original: We codiﬁed these ideas using Group and Representation theory, and demonstrated their utility in understanding grid cells.
Revised: We codiﬁed these ideas using Group and Representation theory, and demonstrated their utility in understanding grid cells.

Original: However, the underlying principle is broader than the crystalline nature of group structures: the world and your actions within it have endlessly repeating structures whose understanding permits creative analogising and ﬂexible behaviours.
Revised: However, the underlying principle is broader than the crystalline nature of group structures: the world and your actions within it have endlessly repeating structures whose understanding permits creative analogising and ﬂexible behaviours.

Original: A well-designed representation should reﬂect this.
Revised: A well-designed representation should reﬂect this.

Original: A C ONSTRAINING R EPRESENTATIONS WITH R EPRESENTATION T HEORY
Revised: A C ONSTRAINING R EPRESENTATIONS WITH R EPRESENTATION T HEORY

Original: Having an actionable code means the representational effect of every transformation of the variable, ∆ x , can be implemented by a matrix:
Revised: Having an actionable code means the representational effect of every transformation of the variable, ∆ x , can be implemented by a matrix:

Original: [Equation]
Revised: [Equation]

Original: In Section 2 we outlined a rough argument for how this equation leads to the following constraint on actionable representations of 2D position, that was vital for our work:
Revised: In Section 2 we outlined a rough argument for how this equation leads to the following constraint on actionable representations of 2D position, that was vital for our work:

Original: [Equation]
Revised: [Equation]

Original: In this section, we’ll repeat this argument more robustly using Group and Representation Theory.
Revised: In this section, we’ll repeat this argument more robustly using Group and Representation Theory.

Original: In doing so, it’ll become clear how broadly our version of actionability can be used to derive clean analytic representational constraints; namely, the arguments presented here can be applied to the representation of any variable whose transformations form a group whose representations are well understood.
Revised: In doing so, it’ll become clear how broadly our version of actionability can be used to derive clean analytic representational constraints; namely, the arguments presented here can be applied to the representation of any variable whose transformations form a group whose representations are well understood.

Original: Sections A.1 and A.2 contain a review of the Group and Representation theory used.
Revised: Sections A.1 and A.2 contain a review of the Group and Representation theory used.

Original: Section A. applies it to our problem.
Revised: Section A. applies it to our problem.

Original: A. G ROUP T HEORY
Revised: A. G ROUP T HEORY

Original: A mathematical group is a collection of things (like the set of integers), and a way to combine two members of the group that makes a third (like addition of two integers, that always creates another integer), in a way that satisﬁes a few rules:
Revised: A mathematical group is a collection of things (like the set of integers), and a way to combine two members of the group that makes a third (like addition of two integers, that always creates another integer), in a way that satisﬁes a few rules:

Original: There is an identity element, which is a member of the group that when combined with any other element doesn’t change it.
Revised: There is an identity element, which is a member of the group that when combined with any other element doesn’t change it.

Original: For adding integers the identity element is 0 , since a + 0 = a for all a . 2.
Revised: For adding integers the identity element is 0 , since a + 0 = a for all a . 2.

Original: Every member of the group has an inverse, deﬁned by its property that combining an element with its inverse produces the identity element.
Revised: Every member of the group has an inverse, deﬁned by its property that combining an element with its inverse produces the identity element.

Original: In our integer-addition example the inverse of any integer a is − a , since − a + a = 0 , and 0 is the identity element.
Revised: In our integer-addition example the inverse of any integer a is − a , since − a + a = 0 , and 0 is the identity element.

Original: Associativity applies, which just means the order in which you perform operations doesn’t matter: ( a + b )
Revised: Associativity applies, which just means the order in which you perform operations doesn’t matter: ( a + b )

Original: + c = a + ( b + c )
Revised: + c = a + ( b + c )

Original: Groups are ubiquitous.
Revised: Groups are ubiquitous.

Original: We mention them here because they will be our route to understanding actionable representations - representations in which transformations are also encoded consistently.
Revised: We mention them here because they will be our route to understanding actionable representations - representations in which transformations are also encoded consistently.

Original: The set of transformations of many variables of interest are groups.
Revised: The set of transformations of many variables of interest are groups.

Original: For example, the set of transformations of 2D position, i.e. 2D translations - ∆ x , is a group if you deﬁne the combination of two translations via simple vector addition, ∆ x 1+2 = ∆ x 1 + ∆ x 2 .
Revised: For example, the set of transformations of 2D position, i.e. 2D translations - ∆ x , is a group if you deﬁne the combination of two translations via simple vector addition, ∆ x 1+2 = ∆ x 1 + ∆ x 2 .

Original: We can easily check that they satisfy all three of the group rules:
Revised: We can easily check that they satisfy all three of the group rules:

Original: There is an identity translation: simply add 0 . 2.
Revised: There is an identity translation: simply add 0 . 2.

Original: Each 2D translation has its inverse: − ∆ x + ∆ x =3.
Revised: Each 2D translation has its inverse: − ∆ x + ∆ x =3.

Original: Associativity: (∆ a + ∆ b ) + ∆ c = ∆ a + (∆ b + ∆ c )
Revised: Associativity: (∆ a + ∆ b ) + ∆ c = ∆ a + (∆ b + ∆ c )

Original: The same applies for the transformations of an angle, θ , or of positions on a torus or sphere, and much else besides.
Revised: The same applies for the transformations of an angle, θ , or of positions on a torus or sphere, and much else besides.

Original: This is a nice observation, but in order to put it to work we need a second area of mathematics, Representation Theory.
Revised: This is a nice observation, but in order to put it to work we need a second area of mathematics, Representation Theory.

Original: A. R EPRESENTATION T HEORY
Revised: A. R EPRESENTATION T HEORY

Original: Groups are abstract, in the sense that the behaviour of many different mathematical objects can embody the same group.
Revised: Groups are abstract, in the sense that the behaviour of many different mathematical objects can embody the same group.

Original: For example, the integers modulo P with an addition combination rule form a group, called C P .
Revised: For example, the integers modulo P with an addition combination rule form a group, called C P .

Original: But equally, the P roots of unity ( { e 2 πnP } P − 1 n =0 ) with a multiplication combination rule obey all the same rules: you can create a 1-1 correspondence between the integers 0 through P − 1 and the P roots of unity by labelling all the roots with the integer n that appears in the exponent e 2 πnP , and, under their respective combination rules, all additions of integers will exactly match onto multiplication of complex roots (i.e. e 2 πn 1 P
Revised: But equally, the P roots of unity ( { e 2 πnP } P − 1 n =0 ) with a multiplication combination rule obey all the same rules: you can create a 1-1 correspondence between the integers 0 through P − 1 and the P roots of unity by labelling all the roots with the integer n that appears in the exponent e 2 πnP , and, under their respective combination rules, all additions of integers will exactly match onto multiplication of complex roots (i.e. e 2 πn 1 P

Original: [Equation]
Revised: [Equation]

Original: In this work we’ll be interested in a particular instantiation of our groups, deﬁned by sets of matrices,
Revised: In this work we’ll be interested in a particular instantiation of our groups, deﬁned by sets of matrices,

Original: T (∆ x ) , combined using matrix multiplication.
Revised: T (∆ x ) , combined using matrix multiplication.

Original: For example, a matrix version of the group C P would be the set of 2-by-2 rotation matrices that rotate by 2 πP increments:
Revised: For example, a matrix version of the group C P would be the set of 2-by-2 rotation matrices that rotate by 2 πP increments:

Original: [Equation]
Revised: [Equation]

Original: such that T ( n 1 ) T ( n 2 ) =
Revised: such that T ( n 1 ) T ( n 2 ) =

Original: T (mod P ( n 1 + n 2 ))
Revised: T (mod P ( n 1 + n 2 ))

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: Combining these matrices using matrix multiplication follows all the same patterns that adding the integers modulo P, or multiplying the P roots of unity, followed; they all embody the same group.
Revised: Combining these matrices using matrix multiplication follows all the same patterns that adding the integers modulo P, or multiplying the P roots of unity, followed; they all embody the same group.

Original: Representation theory is the branch of mathematics that speciﬁes the structure of sets of matrices that, when combined using matrix multiplication, follow the rules of a particular group.
Revised: Representation theory is the branch of mathematics that speciﬁes the structure of sets of matrices that, when combined using matrix multiplication, follow the rules of a particular group.

Original: Such sets of matrices are called a representation of the group; to avoid confusion arising from this unfortunate though understandable convergent terminology, we distinguish group representations from neural representations by henceforth denoting representations of a group in italics.
Revised: Such sets of matrices are called a representation of the group; to avoid confusion arising from this unfortunate though understandable convergent terminology, we distinguish group representations from neural representations by henceforth denoting representations of a group in italics.

Original: We will make use of one big result from Representation Theory, hinted at in Section 2: the Peter-Weyl theorem (Knapp, 2002).
Revised: We will make use of one big result from Representation Theory, hinted at in Section 2: the Peter-Weyl theorem (Knapp, 2002).

Original: For compact topological groups (which include transformations of a point on a circle, torus, and sphere) any matrix that is a representation of a group can be composed from the direct product of a set of blocks, called irreducible representations , or irreps for short, up to a linear transformation.
Revised: For compact topological groups (which include transformations of a point on a circle, torus, and sphere) any matrix that is a representation of a group can be composed from the direct product of a set of blocks, called irreducible representations , or irreps for short, up to a linear transformation.

Original: i.e. if T (∆ a ) is a representation of the group of transformations of some variable a :
Revised: i.e. if T (∆ a ) is a representation of the group of transformations of some variable a :

Original: [Equation]
Revised: [Equation]

Original: where I d (∆ a ) are the irreps of the group in question, which are square matrices not necessarily of the same size as d varies, and S is some invertible square matrix.
Revised: where I d (∆ a ) are the irreps of the group in question, which are square matrices not necessarily of the same size as d varies, and S is some invertible square matrix.

Original: To motivate our current rabbit hole further, this is exactly the result we hinted towards in Section 2.
Revised: To motivate our current rabbit hole further, this is exactly the result we hinted towards in Section 2.

Original: We discussed T (∆ θ ) , and, in 2-dimensions, argued it was, up to a linear transform, the 2-dimensional rotation matrix.
Revised: We discussed T (∆ θ ) , and, in 2-dimensions, argued it was, up to a linear transform, the 2-dimensional rotation matrix.

Original: Further, we discussed how every extra two neurons allowed you to add another frequency, i.e. another rotation matrix.
Revised: Further, we discussed how every extra two neurons allowed you to add another frequency, i.e. another rotation matrix.

Original: This was an attempt to motivate the plausibility of the Peter-Weyl theorem: the rotation matrices are the irreps of the group of transformations of an angle, and adding two neurons allows you to create a larger T (∆ θ ) by stacking rotation matrices on top of one another.
Revised: This was an attempt to motivate the plausibility of the Peter-Weyl theorem: the rotation matrices are the irreps of the group of transformations of an angle, and adding two neurons allows you to create a larger T (∆ θ ) by stacking rotation matrices on top of one another.

Original: Now including the invertible linear transform, S , we can state the 4-dimensional version of equation 4:
Revised: Now including the invertible linear transform, S , we can state the 4-dimensional version of equation 4:

Original: [Equation]
Revised: [Equation]

Original: In performing this decomposition the role of the rotation matrices as irreps is clear.
Revised: In performing this decomposition the role of the rotation matrices as irreps is clear.

Original: There are inﬁnitely many types, each speciﬁed by an integer frequency, n d , and their direct product produces any representation of the rotation group.
Revised: There are inﬁnitely many types, each speciﬁed by an integer frequency, n d , and their direct product produces any representation of the rotation group.

Original: We can now return to actionable neural representations of periodic 2D space, a torus, where this theorem comes in very useful.
Revised: We can now return to actionable neural representations of periodic 2D space, a torus, where this theorem comes in very useful.

Original: We sought codes, g ( x ) , that could be manipulated via matrices:
Revised: We sought codes, g ( x ) , that could be manipulated via matrices:

Original: [Equation]
Revised: [Equation]

Original: Now we recognise that we’re asking T (∆ x ) to be a representation of the transformation group of x , and we know the shape T (∆ x ) must take to ﬁt this criteria.
Revised: Now we recognise that we’re asking T (∆ x ) to be a representation of the transformation group of x , and we know the shape T (∆ x ) must take to ﬁt this criteria.

Original: To specify T (∆ x ) from this constrained class we must simply choose a set of irreps that ﬁll up the dimensionality of the neural space, stack them one on top of each other, and rotate and scale them using the matrix S .
Revised: To specify T (∆ x ) from this constrained class we must simply choose a set of irreps that ﬁll up the dimensionality of the neural space, stack them one on top of each other, and rotate and scale them using the matrix S .

Original: A ﬁnal detail is that there is a trivial way to make a representation of any group: choose T ( θ )
Revised: A ﬁnal detail is that there is a trivial way to make a representation of any group: choose T ( θ )

Original: =
Revised: =

Original: I N , the N-by-N identity matrix.
Revised: I N , the N-by-N identity matrix.

Original: This also ﬁts our previous discussion, but corresponds to a representation made from the direct product of N copies of something called the trivial irrep , a 1-dimensional irrep , I trivial (∆ a ) = 1 .
Revised: This also ﬁts our previous discussion, but corresponds to a representation made from the direct product of N copies of something called the trivial irrep , a 1-dimensional irrep , I trivial (∆ a ) = 1 .

Original: Armed with this knowledge, the following subsection will show how this theory can be used to constrain the neural code, g ( x ) .
Revised: Armed with this knowledge, the following subsection will show how this theory can be used to constrain the neural code, g ( x ) .

Original: To ﬁnish this review subsection, we list the non-trivial irreps of the groups used in this work.
Revised: To ﬁnish this review subsection, we list the non-trivial irreps of the groups used in this work.

Original: To be speciﬁc, in all our work we use only the real-valued irreps , i.e all elements of T are real numbers, since ﬁring rates are real numbers.
Revised: To be speciﬁc, in all our work we use only the real-valued irreps , i.e all elements of T are real numbers, since ﬁring rates are real numbers.

Original: These are less common than complex irreps , which is often what is implied by the simple name irreps .
Revised: These are less common than complex irreps , which is often what is implied by the simple name irreps .

Original: [Table]
Revised: [Table]

Original: Non
Revised: Non

Original: [Table]
Revised: [Table]

Original: Irreps
Revised: Irreps

Original: An angle on a unit circle, θ
Revised: An angle on a unit circle, θ

Original: [Equation]
Revised: [Equation]

Original: n
Revised: n

Original: [Equation]
Revised: [Equation]

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: n
Revised: n

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: n
Revised: n

Original: [Equation]
Revised: [Equation]

Original: ) (cid:19) for n ∈
Revised: ) (cid:19) for n ∈

Original: Z
Revised: Z

Original: Position on a very big circle ≈ a line, x
Revised: Position on a very big circle ≈ a line, x

Original: (cid:18)
Revised: (cid:18)

Original: [Equation]
Revised: [Equation]

Original: ∆
Revised: ∆

Original: [Equation]
Revised: [Equation]

Original: ) − sin
Revised: ) − sin

Original: [Equation]
Revised: [Equation]

Original: ) sin(
Revised: ) sin(

Original: [Equation]
Revised: [Equation]

Original: θ ) cos( ω ∆ θ ) (cid:19) for ω ∈ R
Revised: θ ) cos( ω ∆ θ ) (cid:19) for ω ∈ R

Original: Angles on a unit torus, θ
Revised: Angles on a unit torus, θ

Original: (cid:18) cos( a · ∆ θ ) − sin( a · ∆ θ ) sin( a · ∆ θ ) cos( a · ∆ θ ) (cid:19) for a
Revised: (cid:18) cos( a · ∆ θ ) − sin( a · ∆ θ ) sin( a · ∆ θ ) cos( a · ∆ θ ) (cid:19) for a

Original: ∈ ZPosition on a very big torus ≈ plane, x (cid:18) cos( k · ∆ x ) − sin( k · ∆ x ) sin( k · ∆ x ) cos( k · ∆ x ) (cid:19) for k ∈ RPosition on a unit sphere, φ Real Wigner-D Matrices
Revised: ∈ ZPosition on a very big torus ≈ plane, x (cid:18) cos( k · ∆ x ) − sin( k · ∆ x ) sin( k · ∆ x ) cos( k · ∆ x ) (cid:19) for k ∈ RPosition on a unit sphere, φ Real Wigner-D Matrices

Original: Proofs and discussions for deriving these irreps of the circle, torus, and sphere transformation groups can be found in any textbook on the representation theory of compact Lie Groups (e.g. Knapp (2002)).
Revised: Proofs and discussions for deriving these irreps of the circle, torus, and sphere transformation groups can be found in any textbook on the representation theory of compact Lie Groups (e.g. Knapp (2002)).

Original: The step from complex to real irreps can be done using the Frobenius-Schur indicator, which gives a recipe for mapping from the complex irreps of a compact group to the real irreps (Fulton & Harris, 1991).
Revised: The step from complex to real irreps can be done using the Frobenius-Schur indicator, which gives a recipe for mapping from the complex irreps of a compact group to the real irreps (Fulton & Harris, 1991).

Original: The real Wigner D-Matrices were calculated recursively as detailed in
Revised: The real Wigner D-Matrices were calculated recursively as detailed in

Original: Ivanic & Ruedenberg (1996; 1998), by translating matlab code from Politis et al.
Revised: Ivanic & Ruedenberg (1996; 1998), by translating matlab code from Politis et al.

Original: (2016) into python.
Revised: (2016) into python.

Original: Our derivations of the irreps on the very large circle and torus are simple.
Revised: Our derivations of the irreps on the very large circle and torus are simple.

Original: In 1D the constraint that the frequencies n be integers is only because θ lives on the unit circle.
Revised: In 1D the constraint that the frequencies n be integers is only because θ lives on the unit circle.

Original: Then the frequencies are constrained such that after rotating by 2 π the function is identical.
Revised: Then the frequencies are constrained such that after rotating by 2 π the function is identical.

Original: If you change the radius of the circle to R this constraint becomes ω d = n d R where n d is any integer.
Revised: If you change the radius of the circle to R this constraint becomes ω d = n d R where n d is any integer.

Original: As you take the circle’s radius to inﬁnity, in the process making ﬁnite sections of the circle a better and better approximation of ﬁnite patches of ﬂat 1D space, the lattice of permitted frequencies ω d becomes arbitrarily close together.
Revised: As you take the circle’s radius to inﬁnity, in the process making ﬁnite sections of the circle a better and better approximation of ﬁnite patches of ﬂat 1D space, the lattice of permitted frequencies ω d becomes arbitrarily close together.

Original: Eventually they are separated by machine precision, and so we can simply implement the code as if they were continuous, hence ω d ∈ R .
Revised: Eventually they are separated by machine precision, and so we can simply implement the code as if they were continuous, hence ω d ∈ R .

Original: The same argument applies analogously for 2D frequencies on a very very large torus.
Revised: The same argument applies analogously for 2D frequencies on a very very large torus.

Original: A. R EPRESENTATIONAL C ONSTRAINTS
Revised: A. R EPRESENTATIONAL C ONSTRAINTS

Original: Finally, we will translate these constraints on transformation matrices into constraints on the neural code.
Revised: Finally, we will translate these constraints on transformation matrices into constraints on the neural code.

Original: This is, thankfully, not too difﬁcult.
Revised: This is, thankfully, not too difﬁcult.

Original: Consider the representation of an angle for simplicity, and take some arbitrary origin in the input space, θ 0 = 0 .
Revised: Consider the representation of an angle for simplicity, and take some arbitrary origin in the input space, θ 0 = 0 .

Original: The representation of all other angles can be derived via:
Revised: The representation of all other angles can be derived via:

Original: [Equation]
Revised: [Equation]

Original: ( θ ) =
Revised: ( θ ) =

Original: T ( θ ) g (0) =
Revised: T ( θ ) g (0) =

Original: S  
Revised: S  

Original: I 1 ( θ )  . . .
Revised: I 1 ( θ )  . . .

Original: I 2 ( θ ) . . .
Revised: I 2 ( θ ) . . .

Original: I 3 ( θ ) . . .
Revised: I 3 ( θ ) . . .

Original: I D ( θ ) 
Revised: I D ( θ ) 

Original:  S − 1 g (0) (17)
Revised:  S − 1 g (0) (17)

Original: In the case of an angle, each irrep , I d ( θ ) , is just a 2-by-2 rotation matrix at frequency d , table A.2, and for an N -dimensional T we can ﬁt maximally N 2 (for N even) different frequencies, where N is the number of neurons.
Revised: In the case of an angle, each irrep , I d ( θ ) , is just a 2-by-2 rotation matrix at frequency d , table A.2, and for an N -dimensional T we can ﬁt maximally N 2 (for N even) different frequencies, where N is the number of neurons.

Original: Hence the representation, g ( θ ) , is just a linear combination of the sine and cosine of these different frequencies, exactly as quoted previously:
Revised: Hence the representation, g ( θ ) , is just a linear combination of the sine and cosine of these different frequencies, exactly as quoted previously:

Original: [Equation]
Revised: [Equation]

Original: a 0 corresponds to the trivial irrep , and we include it since we know it must be in the code to make the ﬁring rates non-negative for all θ .
Revised: a 0 corresponds to the trivial irrep , and we include it since we know it must be in the code to make the ﬁring rates non-negative for all θ .

Original: It also cleans up the constraint on the number of allowable frequencies, for even or odd N we require D < N 2 .
Revised: It also cleans up the constraint on the number of allowable frequencies, for even or odd N we require D < N 2 .

Original: This is because if N is odd one dimension is used for the trivial irrep , the rest can be shared up amongst N − 1frequencies, so D must be an integer smaller than N 2 .
Revised: This is because if N is odd one dimension is used for the trivial irrep , the rest can be shared up amongst N − 1frequencies, so D must be an integer smaller than N 2 .

Original: If N is even, one dimension must still be used by the trivial irrep , so D can still maximally be only the largest integer smaller than N 2 .
Revised: If N is even, one dimension must still be used by the trivial irrep , so D can still maximally be only the largest integer smaller than N 2 .

Original: Extending this to other groups is relatively simple, the representation is just a linear combination of the functions that appear in the irrep of the group in question, see table A.2.
Revised: Extending this to other groups is relatively simple, the representation is just a linear combination of the functions that appear in the irrep of the group in question, see table A.2.

Original: For position on a circle, line (very large circle), torus, or plane (very large torus), they all take the relatively simple form as in equation 18, but requiring appropriate changes from 1 to 2 dimensions, or integer to real frequencies.
Revised: For position on a circle, line (very large circle), torus, or plane (very large torus), they all take the relatively simple form as in equation 18, but requiring appropriate changes from 1 to 2 dimensions, or integer to real frequencies.

Original: Representations of position on a sphere are slightly more complex, instead being constructed from linear combinations of sets of spherical harmonics.
Revised: Representations of position on a sphere are slightly more complex, instead being constructed from linear combinations of sets of spherical harmonics.

Original: A. P ERIODIC VS I NFINITE S PACES
Revised: A. P ERIODIC VS I NFINITE S PACES

Original: We now describe a key step in our argument for representations of ﬂat 2D space.
Revised: We now describe a key step in our argument for representations of ﬂat 2D space.

Original: We approximate a ﬁnite region of the inﬁnite 2D plane by a ﬁnite region of a very large periodic 2D space, the torus.
Revised: We approximate a ﬁnite region of the inﬁnite 2D plane by a ﬁnite region of a very large periodic 2D space, the torus.

Original: We do this because the representation theory of the group of 2D translations is not as well characterised (to the best our knowledge there is no equivalent of the Peter-Weyl theorem that could be used to provide a target for optimisation).
Revised: We do this because the representation theory of the group of 2D translations is not as well characterised (to the best our knowledge there is no equivalent of the Peter-Weyl theorem that could be used to provide a target for optimisation).

Original: Fortunately, any animal only cares about a ﬁnite region of 2D space (encoded in equation 2 via the lengthscale L ).
Revised: Fortunately, any animal only cares about a ﬁnite region of 2D space (encoded in equation 2 via the lengthscale L ).

Original: We therefore approximate this ﬁnite region of ﬂat 2D space with an equivalently sized region of a torus.
Revised: We therefore approximate this ﬁnite region of ﬂat 2D space with an equivalently sized region of a torus.

Original: This enables us to use the fully characterised representation theory of the set of translations of periodic space.
Revised: This enables us to use the fully characterised representation theory of the set of translations of periodic space.

Original: Now, there are legitimate concerns over whether this is a reasonable approximation.
Revised: Now, there are legitimate concerns over whether this is a reasonable approximation.

Original: Fortunately, we are free to choose the radii of the torus as we wish, and we make use of this freedom to make the approximation of the ﬁnite ﬂat 2D space arbitrarily good.
Revised: Fortunately, we are free to choose the radii of the torus as we wish, and we make use of this freedom to make the approximation of the ﬁnite ﬂat 2D space arbitrarily good.

Original: As the radii increase to inﬁnity the small region of torus approximates the ﬂat 2D space arbitrarily well.
Revised: As the radii increase to inﬁnity the small region of torus approximates the ﬂat 2D space arbitrarily well.

Original: This use of periodic space does exclude some representations of the full set of 2D translations, for example:
Revised: This use of periodic space does exclude some representations of the full set of 2D translations, for example:

Original: [Equation]
Revised: [Equation]

Original: There are two reasons not to be concerned by this.
Revised: There are two reasons not to be concerned by this.

Original: First, this solution would never have been allowed, as including it ensures that your representation, g ( x ) , cannot have non-negative or bounded ﬁring rates (we suspect this would be the case for all such additional representations of ﬂat 2D translations).
Revised: First, this solution would never have been allowed, as including it ensures that your representation, g ( x ) , cannot have non-negative or bounded ﬁring rates (we suspect this would be the case for all such additional representations of ﬂat 2D translations).

Original: Second, any result which was dependent on these kind of boundary effects at ∞ would be highly suspect.
Revised: Second, any result which was dependent on these kind of boundary effects at ∞ would be highly suspect.

Original: After all, animals are not truly representing ﬂat 2D space, rather they are representing an approximately ﬂat section of the surface of a sphere (the Earth).
Revised: After all, animals are not truly representing ﬂat 2D space, rather they are representing an approximately ﬂat section of the surface of a sphere (the Earth).

Original: B N UMERICAL O PTIMISATION D ETAILS
Revised: B N UMERICAL O PTIMISATION D ETAILS

Original: In order to test our claims, we numerically optimise the parameters that deﬁne the representation to minimise the loss, subject to constraints.
Revised: In order to test our claims, we numerically optimise the parameters that deﬁne the representation to minimise the loss, subject to constraints.

Original: Despite sharing many commonalities, our optimisation procedures are different depending on whether the variable being represented lives in a very large periodic space (approximations to the line, plane, or volume) or ﬁnite (circle, torus, sphere).
Revised: Despite sharing many commonalities, our optimisation procedures are different depending on whether the variable being represented lives in a very large periodic space (approximations to the line, plane, or volume) or ﬁnite (circle, torus, sphere).

Original: We describe each of these schemes in turn, beginning with the very large spaces.
Revised: We describe each of these schemes in turn, beginning with the very large spaces.

Original: All code will be made available on github.
Revised: All code will be made available on github.

Original: B. N UMERICAL O PTIMISATION FOR V ERY L ARGE S PACES
Revised: B. N UMERICAL O PTIMISATION FOR V ERY L ARGE S PACES

Original: We will use the representation of a point on a line for explanation, planes or volumes are a simple extension.
Revised: We will use the representation of a point on a line for explanation, planes or volumes are a simple extension.

Original: g ( x ) is parameterised as follows:
Revised: g ( x ) is parameterised as follows:

Original: [Equation]
Revised: [Equation]

Original: Our loss is made from three terms: the ﬁrst is the functional objective, the last two enforce the non-negativity and boundedness constraint respectively:
Revised: Our loss is made from three terms: the ﬁrst is the functional objective, the last two enforce the non-negativity and boundedness constraint respectively:

Original: [Equation]
Revised: [Equation]

Original: To evaluate the functional component of our loss we sample a set of points { x i } Mi =1 from p ( x ) and calculate their representations { g ( x i ) }
Revised: To evaluate the functional component of our loss we sample a set of points { x i } Mi =1 from p ( x ) and calculate their representations { g ( x i ) }

Original: Mi =1 .
Revised: Mi =1 .

Original: To make the bounded constraint approximately satisﬁed (there’s still some work to be done, taken care of by L bounded ) we calculate the following neuron-wise norm and use it to normalise each neuron’s ﬁring:
Revised: To make the bounded constraint approximately satisﬁed (there’s still some work to be done, taken care of by L bounded ) we calculate the following neuron-wise norm and use it to normalise each neuron’s ﬁring:

Original: [Equation]
Revised: [Equation]

Original: The functional form of L functional varies, as discussed in the main paper.
Revised: The functional form of L functional varies, as discussed in the main paper.

Original: For example, for the full loss we compute the following using the normalised ﬁring, a discrete approximation to equation 1:
Revised: For example, for the full loss we compute the following using the normalised ﬁring, a discrete approximation to equation 1:

Original: [Equation]
Revised: [Equation]

Original: Now we come to our two constraints, which enforce that the representation is non-negative and bounded.
Revised: Now we come to our two constraints, which enforce that the representation is non-negative and bounded.

Original: We would like our representation to be reasonable (i.e. non-negative and bounded) for all values of x .
Revised: We would like our representation to be reasonable (i.e. non-negative and bounded) for all values of x .

Original: If we do not enforce this then the optimiser ﬁnds various trick solutions in which the representation is non-negative and bounded only in small region, but negative and growing explosively outside of this, which is completely unbiological, and uninteresting.
Revised: If we do not enforce this then the optimiser ﬁnds various trick solutions in which the representation is non-negative and bounded only in small region, but negative and growing explosively outside of this, which is completely unbiological, and uninteresting.

Original: Of course, x is inﬁnite, so we cannot numerically ensure these constraints are satisﬁed for all x .
Revised: Of course, x is inﬁnite, so we cannot numerically ensure these constraints are satisﬁed for all x .

Original: However, ensuring they are true in a region local to the animal’s explorations (which are represented by p ( x ) ) sufﬁces to remove most of these trick solutions.
Revised: However, ensuring they are true in a region local to the animal’s explorations (which are represented by p ( x ) ) sufﬁces to remove most of these trick solutions.

Original: As such, we sample a second set of M S ‘shift positions’, { x m } M S m =1 , from a scaled up version of p ( x ) , using a scale factor S .
Revised: As such, we sample a second set of M S ‘shift positions’, { x m } M S m =1 , from a scaled up version of p ( x ) , using a scale factor S .

Original: We then create a much larger set of positions, by shifting the original set by each of the ‘shift positions’, creating a dataset of size M ∗ M S , and use these to calculate our two constraint losses.
Revised: We then create a much larger set of positions, by shifting the original set by each of the ‘shift positions’, creating a dataset of size M ∗ M S , and use these to calculate our two constraint losses.

Original: Our non-negativity loss penalises negative ﬁring rates:
Revised: Our non-negativity loss penalises negative ﬁring rates:

Original: [Equation]
Revised: [Equation]

Original: Where I is the indicator function, 1 if the ﬁring rate is negative, 0 else, i.e. we just average the magnitude of the negative portion of the ﬁring rates.
Revised: Where I is the indicator function, 1 if the ﬁring rate is negative, 0 else, i.e. we just average the magnitude of the negative portion of the ﬁring rates.

Original: The bounded loss penalises the deviation of each neuron’s norm from 1, in each of the shifted rooms:
Revised: The bounded loss penalises the deviation of each neuron’s norm from 1, in each of the shifted rooms:

Original: [Equation]
Revised: [Equation]

Original: That completes our speciﬁcation of the losses.
Revised: That completes our speciﬁcation of the losses.

Original: We minimise the full loss over the parameters ( a 0 , { a d , b d , ω d } Dd =1 ) using a gradient-based algorithm, ADAM (Kingma & Ba, 2014).
Revised: We minimise the full loss over the parameters ( a 0 , { a d , b d , ω d } Dd =1 ) using a gradient-based algorithm, ADAM (Kingma & Ba, 2014).

Original: We initialise these parameters by sampling from independent zero-mean gaussians, with variances as in table B.3.
Revised: We initialise these parameters by sampling from independent zero-mean gaussians, with variances as in table B.3.

Original: The ﬁnal detail of our approach is the setting of λ p and λ b , which control the relative weight of the constraints vs. the functional objective.
Revised: The ﬁnal detail of our approach is the setting of λ p and λ b , which control the relative weight of the constraints vs. the functional objective.

Original: We don’t want the optimiser to strike a tradeoff between the objective and constraints, since the constraints must actually be satisﬁed.
Revised: We don’t want the optimiser to strike a tradeoff between the objective and constraints, since the constraints must actually be satisﬁed.

Original: But we also don’t want to make λ p so large that the objective is badly minimised in the pursuit of only constraint satisfaction.
Revised: But we also don’t want to make λ p so large that the objective is badly minimised in the pursuit of only constraint satisfaction.

Original: To balance these demands we use GECO (Rezende & Viola, 2018), which speciﬁes a set of stepwise updates for λ p and λ b .
Revised: To balance these demands we use GECO (Rezende & Viola, 2018), which speciﬁes a set of stepwise updates for λ p and λ b .

Original: These ensure that if the constraint is not satisﬁed their coefﬁcients increase, else they decrease allowing the optimiser to focus on the loss.
Revised: These ensure that if the constraint is not satisﬁed their coefﬁcients increase, else they decrease allowing the optimiser to focus on the loss.

Original: The dynamics that implement this are as follows, for a timestep, t .
Revised: The dynamics that implement this are as follows, for a timestep, t .

Original: GECO deﬁnes a log-space measure of how well the constraint is being satisﬁed:
Revised: GECO deﬁnes a log-space measure of how well the constraint is being satisﬁed:

Original: [Equation]
Revised: [Equation]

Original: (26) (or small variations on this), where k is a log-threshold that sets the target log-size of the constraint loss in question, and L t is the value of the loss at timestep t .
Revised: (26) (or small variations on this), where k is a log-threshold that sets the target log-size of the constraint loss in question, and L t is the value of the loss at timestep t .

Original: L t is then smoothed:
Revised: L t is then smoothed:

Original: [Equation]
Revised: [Equation]

Original: And this smoothed measure of how well the constraint is being satisﬁed controls the behaviour of the coefﬁcient:
Revised: And this smoothed measure of how well the constraint is being satisﬁed controls the behaviour of the coefﬁcient:

Original: [Equation]
Revised: [Equation]

Original: This speciﬁes the full optimisation, parameters are listed in table B.3.
Revised: This speciﬁes the full optimisation, parameters are listed in table B.3.

Original: B. N UMERICAL O PTIMISATION FOR F INITE S PACES
Revised: B. N UMERICAL O PTIMISATION FOR F INITE S PACES

Original: Optimising the representation of ﬁnite variables has one added difﬁculty, and one simpliﬁcation.
Revised: Optimising the representation of ﬁnite variables has one added difﬁculty, and one simpliﬁcation.

Original: Again, we have a parameterised form, e.g. for an angle θ :
Revised: Again, we have a parameterised form, e.g. for an angle θ :

Original: [Equation]
Revised: [Equation]

Original: The key difﬁculty is that each frequency, n d , has to be an integer, so we cannot optimise it by gradient descent, and similar problems arise for positions on a torus, or on a sphere.
Revised: The key difﬁculty is that each frequency, n d , has to be an integer, so we cannot optimise it by gradient descent, and similar problems arise for positions on a torus, or on a sphere.

Original: We shall spend the rest of this section outlining how we circumvent this problem.
Revised: We shall spend the rest of this section outlining how we circumvent this problem.

Original: However, we do also have one major simpliﬁcation.
Revised: However, we do also have one major simpliﬁcation.

Original: Because the variable is ﬁnite, we can easily ensure the representation is nonnegative and bounded across the whole space, avoiding the need for an additional normalisation constraint.
Revised: Because the variable is ﬁnite, we can easily ensure the representation is nonnegative and bounded across the whole space, avoiding the need for an additional normalisation constraint.

Original: Further, for the uniform occupancy distributions we consider in ﬁnite spaces, we can analytically calculate the neuron norm, and scale the parameters appropriately to ensure it is always 1:
Revised: Further, for the uniform occupancy distributions we consider in ﬁnite spaces, we can analytically calculate the neuron norm, and scale the parameters appropriately to ensure it is always 1:

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: As such, we simply sample a set of angles { θ m } Mm =1 , and their normalised representations { ˜ g ( θ m ) } Mm =1 , and compute the appropriate functional and non-negativity loss, as in equations& 24.
Revised: As such, we simply sample a set of angles { θ m } Mm =1 , and their normalised representations { ˜ g ( θ m ) } Mm =1 , and compute the appropriate functional and non-negativity loss, as in equations& 24.

Original: The ﬁnal thing we must clear up is how to learn which frequencies, n d , to include in the code.
Revised: The ﬁnal thing we must clear up is how to learn which frequencies, n d , to include in the code.

Original: To do this, we make a code that contains many many frequencies, up to some cutoff D max , where D max is bigger than N :
Revised: To do this, we make a code that contains many many frequencies, up to some cutoff D max , where D max is bigger than N :

Original: [Equation]
Revised: [Equation]

Original: We then add a term to the loss that forces the code to choose only D of these D max frequencies, setting all other coefﬁcient vectors to zero, and hence making the code actionable again but allowing the optimiser to do so in a way that minimises the functional loss.
Revised: We then add a term to the loss that forces the code to choose only D of these D max frequencies, setting all other coefﬁcient vectors to zero, and hence making the code actionable again but allowing the optimiser to do so in a way that minimises the functional loss.

Original: The term that we add to achieve this is inspired by the representation theory we used to write these constraints, Section A.2.
Revised: The term that we add to achieve this is inspired by the representation theory we used to write these constraints, Section A.2.

Original: We create ﬁrst a 2 D max + 1 -dimesional vector, I ( θ ) , that contains all the irreps i.e. each of the D max sines and cosines and a constant.
Revised: We create ﬁrst a 2 D max + 1 -dimesional vector, I ( θ ) , that contains all the irreps i.e. each of the D max sines and cosines and a constant.

Original: We then create a (2 D max + 1) ×
Revised: We then create a (2 D max + 1) ×

Original: (2 D max + 1) -dimensional transition matrix, G I (∆ θ ) that is a representation of the rotation group in this space: G I (∆ θ ) I ( θ ) =
Revised: (2 D max + 1) -dimensional transition matrix, G I (∆ θ ) that is a representation of the rotation group in this space: G I (∆ θ ) I ( θ ) =

Original: I ( θ + ∆ θ ) .
Revised: I ( θ + ∆ θ ) .

Original: G I can be simply made by stacking 2-by-2 rotation matrices.
Revised: G I can be simply made by stacking 2-by-2 rotation matrices.

Original: Then we create the neural code by projecting the frequency basis through a rectangular weight matrix, W : g ( θ )
Revised: Then we create the neural code by projecting the frequency basis through a rectangular weight matrix, W : g ( θ )

Original: =
Revised: =

Original: W I ( θ ) .
Revised: W I ( θ ) .

Original: Finally, we create the best possible representation of the group in the neural space:
Revised: Finally, we create the best possible representation of the group in the neural space:

Original: [Equation]
Revised: [Equation]

Original: Where W ∗ denotes the Moore-Penrose pseudoinverse.
Revised: Where W ∗ denotes the Moore-Penrose pseudoinverse.

Original: We then learn W , which is equivalent to learning a 0 and { a d , b d } D max d =1 .
Revised: We then learn W , which is equivalent to learning a 0 and { a d , b d } D max d =1 .

Original: Representation theory tells us this will not do a perfect job at rotating the neural representation, unless the optimiser chooses W to cut out all but D of the frequencies.
Revised: Representation theory tells us this will not do a perfect job at rotating the neural representation, unless the optimiser chooses W to cut out all but D of the frequencies.

Original: As such, we sample a set of shifts { θ m s } Mm s =1 , and measure the following discrepancy:
Revised: As such, we sample a set of shifts { θ m s } Mm s =1 , and measure the following discrepancy:

Original: [Equation]
Revised: [Equation]

Original: Minimising it as we minimised the other constraint terms will force the code to choose D frequencies and hence make the code actionable.
Revised: Minimising it as we minimised the other constraint terms will force the code to choose D frequencies and hence make the code actionable.

Original: Since calculating the pseudoinverse is expensive, we replace W ∗ with a matrix B that we also learn by gradient descent.
Revised: Since calculating the pseudoinverse is expensive, we replace W ∗ with a matrix B that we also learn by gradient descent.

Original: B quickly learns to approximate the pseudoinverse, speeding our computations.
Revised: B quickly learns to approximate the pseudoinverse, speeding our computations.

Original: That wraps up our numerical description.
Revised: That wraps up our numerical description.

Original: We are again left with three loss terms, two of which enforce constraints.
Revised: We are again left with three loss terms, two of which enforce constraints.

Original: This can be applied to any group, even if you can’t do gradient descent through their irreps , at the cost of limiting the optimisation to a subset of irreps below a certain frequency,
Revised: This can be applied to any group, even if you can’t do gradient descent through their irreps , at the cost of limiting the optimisation to a subset of irreps below a certain frequency,

Original: [Equation]
Revised: [Equation]

Original: max
Revised: max

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: P ARAMETERS V ALUES
Revised: P ARAMETERS V ALUES

Original: We list the parameter values used to generate the grid cells in Figure 2B, and show the full population of neurons in ﬁgure 6.
Revised: We list the parameter values used to generate the grid cells in Figure 2B, and show the full population of neurons in ﬁgure 6.

Original: Parameter
Revised: Parameter

Original: Meaning
Revised: Meaning

Original: Value
Revised: Value

Original: [Table]
Revised: [Table]

Original: Neural Lengthscale
Revised: Neural Lengthscale

Original: [Table]
Revised: [Table]

Original: [Table] [Table]
Revised: [Table] [Table]

Original: Number
Revised: Number

Original: [Table]
Revised: [Table]

Original: steps
Revised: steps

Original: [Table]
Revised: [Table]

Original: Number of neurons
Revised: Number of neurons

Original: [Table]
Revised: [Table]

Original: number of sampled points every n resample steps
Revised: number of sampled points every n resample steps

Original: [Table]
Revised: [Table]

Original: number of room shifts sampled every n resample steps
Revised: number of room shifts sampled every n resample steps

Original: [Table]
Revised: [Table]

Original: standard deviation of normal for shift sampling
Revised: standard deviation of normal for shift sampling

Original: [Table]
Revised: [Table]

Original: number of steps per resample of points
Revised: number of steps per resample of points

Original: [Table]
Revised: [Table]

Original: positivity weighting coefﬁcient 0.
Revised: positivity weighting coefﬁcient 0.

Original: [Table]
Revised: [Table]

Original: log positivity target -
Revised: log positivity target -

Original: [Table]
Revised: [Table]

Original: smoothing 0.
Revised: smoothing 0.

Original: [Table]
Revised: [Table]

Original: coefﬁcient
Revised: coefﬁcient

Original: [Table] [Table]
Revised: [Table] [Table]

Original: GECO
Revised: GECO

Original: [Table]
Revised: [Table]

Original: norm constraint
Revised: norm constraint

Original: [Table] [Table]
Revised: [Table] [Table]

Original: [Table] [Table]
Revised: [Table] [Table]

Original: ditto
Revised: ditto

Original: [Table] [Table]
Revised: [Table] [Table]

Original: coefﬁcient gradient step size 0.
Revised: coefﬁcient gradient step size 0.

Original: [Table]
Revised: [Table]

Original: frequency gradient step size 0.
Revised: frequency gradient step size 0.

Original: [Table]
Revised: [Table]

Original: moment 0.
Revised: moment 0.

Original: [Table] [Table]
Revised: [Table] [Table]

Original: ADAM non-exploding term 1 × 10 −
Revised: ADAM non-exploding term 1 × 10 −

Original: B. R OBUSTNESS TO P ARAMETER V ALUES
Revised: B. R OBUSTNESS TO P ARAMETER V ALUES

Original: In this section we explore the parameter-dependence of our numerical solutions.
Revised: In this section we explore the parameter-dependence of our numerical solutions.

Original: Most of the parameters in table B.3 control the behaviour of the optimisation scheme (for example setting ADAM (Kingma & Ba, 2014) or GECO (Rezende & Viola, 2018) parameters).
Revised: Most of the parameters in table B.3 control the behaviour of the optimisation scheme (for example setting ADAM (Kingma & Ba, 2014) or GECO (Rezende & Viola, 2018) parameters).

Original: These were tuned so that the optimiser found the best solutions possible; any parameter dependence in these does not seem deeply worrying, reﬂecting the optimisers used rather than the core problem we’re solving.
Revised: These were tuned so that the optimiser found the best solutions possible; any parameter dependence in these does not seem deeply worrying, reﬂecting the optimisers used rather than the core problem we’re solving.

Original: We therefore focus our explorations on the parameters that deﬁne key parts of the loss function: the neural lengthscale, σ , the two spatial lengthscales, L and l , and the number of neurons, N .
Revised: We therefore focus our explorations on the parameters that deﬁne key parts of the loss function: the neural lengthscale, σ , the two spatial lengthscales, L and l , and the number of neurons, N .

Original: We choose the units of the input space such that L = 1 , leaving us with three parameters to explore.
Revised: We choose the units of the input space such that L = 1 , leaving us with three parameters to explore.

Original: (The neural space units are not so arbitrary, due to the ﬁring rate constraint)
Revised: (The neural space units are not so arbitrary, due to the ﬁring rate constraint)

Original: Our theory actually makes predictions for how these parameters should change the representation.
Revised: Our theory actually makes predictions for how these parameters should change the representation.

Original: As discussed in appendix E.3, if the ratio of N and σ is small there should be one module of neurons, and increasing it should increase the number of modules in the optimal solutions.
Revised: As discussed in appendix E.3, if the ratio of N and σ is small there should be one module of neurons, and increasing it should increase the number of modules in the optimal solutions.

Original: l enforces the push to hexagons, so, assuming there’s only one module for simplicity, if l is sufﬁciently large the optimal solution should be hexagonal grid cells.
Revised: l enforces the push to hexagons, so, assuming there’s only one module for simplicity, if l is sufﬁciently large the optimal solution should be hexagonal grid cells.

Original: If l is very small then it will have no effect, so all sufﬁciently high frequency grids should be equal and their shape should matter less.
Revised: If l is very small then it will have no effect, so all sufﬁciently high frequency grids should be equal and their shape should matter less.

Original: We verify these claims in a series of numerical experiments, and we show that there are reasonably large parameter regimes in which our suggested qualitative solutions emerge (one module for high σ , as in ﬁgure 4, multiple modules for low σ , as in ﬁgure 2).
Revised: We verify these claims in a series of numerical experiments, and we show that there are reasonably large parameter regimes in which our suggested qualitative solutions emerge (one module for high σ , as in ﬁgure 4, multiple modules for low σ , as in ﬁgure 2).

Original: In these experiments most other parameters are kept at the values in table B.3, barring the number of steps which was varied with the number of neurons, and λ p 0 and λ n 0 which should be scaled with the approximate size of the loss, that varies as σ and l vary.
Revised: In these experiments most other parameters are kept at the values in table B.3, barring the number of steps which was varied with the number of neurons, and λ p 0 and λ n 0 which should be scaled with the approximate size of the loss, that varies as σ and l vary.

Original: B.4. E FFECT OF V ARYING l
Revised: B.4. E FFECT OF V ARYING l

Original: The top 6 panels of ﬁgure 8 show that for a ﬁxed number of neurons one module of hexagonal grids is optimal for a range of l .
Revised: The top 6 panels of ﬁgure 8 show that for a ﬁxed number of neurons one module of hexagonal grids is optimal for a range of l .

Original: We start at l = 1 , as that corresponds to the rough lengthscale of the environment.
Revised: We start at l = 1 , as that corresponds to the rough lengthscale of the environment.

Original: Eventually for small l the optimiser chooses other, non-hexagonal grids.
Revised: Eventually for small l the optimiser chooses other, non-hexagonal grids.

Original: While hexagons are optimal for a reasonable range of l , we might wonder why our arguments do not hold for even smaller l ?
Revised: While hexagons are optimal for a reasonable range of l , we might wonder why our arguments do not hold for even smaller l ?

Original: We believe this is due to the small number of neurons we are using.
Revised: We believe this is due to the small number of neurons we are using.

Original: We argued hexagons were optimal because they packed the most frequencies into a Goldilocks annulus in frequency space, ﬁgure 4.
Revised: We argued hexagons were optimal because they packed the most frequencies into a Goldilocks annulus in frequency space, ﬁgure 4.

Original: As l decreases the outer ring of this annulus moves further away, providing a larger Goldilocks region.
Revised: As l decreases the outer ring of this annulus moves further away, providing a larger Goldilocks region.

Original: When the number of neurons in a module, i.e. the number of frequencies, is small this permits many different lattices to pack frequencies within the Goldilocks annulus equally well, ﬁgure 7.
Revised: When the number of neurons in a module, i.e. the number of frequencies, is small this permits many different lattices to pack frequencies within the Goldilocks annulus equally well, ﬁgure 7.

Original: So, for a given number of neurons, there is a l -threshold, beyond which there is no longer a push towards hexagons, and this threshold can be decreased by increasing the number of neurons.
Revised: So, for a given number of neurons, there is a l -threshold, beyond which there is no longer a push towards hexagons, and this threshold can be decreased by increasing the number of neurons.

Original: We verify this in the last panel of ﬁgure 8, where we show that at l = 0 . 1 , where 64 neuron modules were not hexagonal, 100 neuron modules were.
Revised: We verify this in the last panel of ﬁgure 8, where we show that at l = 0 . 1 , where 64 neuron modules were not hexagonal, 100 neuron modules were.

Original: Therefore, we expect for modules containing many neurons (like those in the brain) hexagons are optimal for a much larger range of l .
Revised: Therefore, we expect for modules containing many neurons (like those in the brain) hexagons are optimal for a much larger range of l .

Original: B.4. E FFECT OF V ARYING σ
Revised: B.4. E FFECT OF V ARYING σ

Original: As discussed in Appendix E.3, varying σ varies the push towards non-harmonicity, and hence how many modules we would expect in the optimal solution.
Revised: As discussed in Appendix E.3, varying σ varies the push towards non-harmonicity, and hence how many modules we would expect in the optimal solution.

Original: We conﬁrm this in Figure 9.
Revised: We conﬁrm this in Figure 9.

Original: For large σ (up to σ = 1 ) we get one hexagonal module, decreasing it leads to solutions with more modules.
Revised: For large σ (up to σ = 1 ) we get one hexagonal module, decreasing it leads to solutions with more modules.

Original: The more modules the less hexagonal they are.
Revised: The more modules the less hexagonal they are.

Original: We suspect this is again partly a ﬁnite neuron effect, as in Appendix B.4.1.
Revised: We suspect this is again partly a ﬁnite neuron effect, as in Appendix B.4.1.

Original: Future work could usefully explore whether more constraints are needed to robustly generate many hexagonal modules, or whether more neurons is enough as it was in ﬁgure 8.
Revised: Future work could usefully explore whether more constraints are needed to robustly generate many hexagonal modules, or whether more neurons is enough as it was in ﬁgure 8.

Original: B.4. V ARYING N UMBER OF N EURONS
Revised: B.4. V ARYING N UMBER OF N EURONS

Original: Figures 10 and 11 show that as we vary the number of neurons (with ﬁxed l ), we can ﬁnd values of σ for which the population produces either one module of hexagonal grids (Figure 10) or multiple modules (Figure 11).
Revised: Figures 10 and 11 show that as we vary the number of neurons (with ﬁxed l ), we can ﬁnd values of σ for which the population produces either one module of hexagonal grids (Figure 10) or multiple modules (Figure 11).

Original: Hence, our qualitative solution types are found across a range of population sizes.
Revised: Hence, our qualitative solution types are found across a range of population sizes.

Original: C A NALYSIS OF OUR S IMPLEST L OSS THAT L EADS TO O NE L ATTICE M ODULE
Revised: C A NALYSIS OF OUR S IMPLEST L OSS THAT L EADS TO O NE L ATTICE M ODULE

Original: In this section we analytically study the simpliﬁed loss suggested in section 3.
Revised: In this section we analytically study the simpliﬁed loss suggested in section 3.

Original: [Equation]
Revised: [Equation]

Original: and we show how to derive the equations 8.
Revised: and we show how to derive the equations 8.

Original: The actionability constraint tells us that:
Revised: The actionability constraint tells us that:

Original: [Equation]
Revised: [Equation]

Original: Where D is smaller than half the number of neurons (Appendix A)
Revised: Where D is smaller than half the number of neurons (Appendix A)

Original: [Equation]
Revised: [Equation]

Original: We ensure n d (cid:54) =
Revised: We ensure n d (cid:54) =

Original: n d
Revised: n d

Original: [Equation]
Revised: [Equation]

Original: if d (cid:54) =
Revised: if d (cid:54) =

Original: d (cid:48) by combining like terms.
Revised: d (cid:48) by combining like terms.

Original: Our loss is:
Revised: Our loss is:

Original: [Equation]
Revised: [Equation]

Original: Developing this and using trig identities for the sum of sines and cosines:
Revised: Developing this and using trig identities for the sum of sines and cosines:

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: We now change variables to α = θ − θ (cid:48)and β
Revised: We now change variables to α = θ − θ (cid:48)and β

Original: = θ + θ (cid:48) 2 .
Revised: = θ + θ (cid:48) 2 .

Original: This introduces a factor of two from the
Revised: This introduces a factor of two from the

Original: Jacobian ( 12 ), which we cancel by doubling the range of the integral while keeping the same limits, despite the change of variable, Figure 12.
Revised: Jacobian ( 12 ), which we cancel by doubling the range of the integral while keeping the same limits, despite the change of variable, Figure 12.

Original: This gives us:
Revised: This gives us:

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: Performing these integrals is easy using the fourier orthogonality relations; the two cross terms with mixed sin and cos of β are instantly zero, the other two terms are non-zero only if n d = n d (cid:48) , i.e. d = d (cid:48) .
Revised: Performing these integrals is easy using the fourier orthogonality relations; the two cross terms with mixed sin and cos of β are instantly zero, the other two terms are non-zero only if n d = n d (cid:48) , i.e. d = d (cid:48) .

Original: When d = d (cid:48)
Revised: When d = d (cid:48)

Original: these integrals evaluate to π
Revised: these integrals evaluate to π

Original: [Equation]
Revised: [Equation]

Original: giving:
Revised: giving:

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: Exactly as quoted in equation 7.
Revised: Exactly as quoted in equation 7.

Original: We can do the same development for the ﬁring rate constraint:
Revised: We can do the same development for the ﬁring rate constraint:

Original: [Equation]
Revised: [Equation]

Original: Where we have again simply used the orthogonality relation of sines and cosines.
Revised: Where we have again simply used the orthogonality relation of sines and cosines.

Original: These are the constraints that must be enforced.
Revised: These are the constraints that must be enforced.

Original: For illustration it is useful to see one slice of the constraints made by summing over the neuron index:
Revised: For illustration it is useful to see one slice of the constraints made by summing over the neuron index:

Original: [Equation]
Revised: [Equation]

Original: This is the constraint shown in equation 8, and is strongly intuition guiding.
Revised: This is the constraint shown in equation 8, and is strongly intuition guiding.

Original: But remember it is really a stand-in for the N constraints, one per neuron, that must each be satisﬁed!
Revised: But remember it is really a stand-in for the N constraints, one per neuron, that must each be satisﬁed!

Original: This analysis can be generalised to derive exactly the same result in 2D by integrating over pairs of points on the torus.
Revised: This analysis can be generalised to derive exactly the same result in 2D by integrating over pairs of points on the torus.

Original: It yields no extra insight, and is no different, so we just quote the result:
Revised: It yields no extra insight, and is no different, so we just quote the result:

Original: [Equation]
Revised: [Equation]

Original: D A NALYSIS OF P ARTIALLY S IMPLIFIED L OSS THAT L EADS TO A M ODULE OF H EXAGONAL G RIDS
Revised: D A NALYSIS OF P ARTIALLY S IMPLIFIED L OSS THAT L EADS TO A M ODULE OF H EXAGONAL G RIDS

Original: In this section we will study an intermediate loss: it will compute the euclidean distance, like in
Revised: In this section we will study an intermediate loss: it will compute the euclidean distance, like in

Original: Appendix C, but we’ll include χ and p .
Revised: Appendix C, but we’ll include χ and p .

Original: We’ll show that they induce a frequency bias, χ for low frequencies, and p for high, which together lead to hexagonal lattices.
Revised: We’ll show that they induce a frequency bias, χ for low frequencies, and p for high, which together lead to hexagonal lattices.

Original: We’ll study χ ﬁrst for representations of a circular variable, then we’ll study p for representations of a line, then we’ll combine them.
Revised: We’ll study χ ﬁrst for representations of a circular variable, then we’ll study p for representations of a line, then we’ll combine them.

Original: At each stage generalising to 2-dimensions is a conceptually trivial but algebraically nasty operation with no utility, so we do not detail here.
Revised: At each stage generalising to 2-dimensions is a conceptually trivial but algebraically nasty operation with no utility, so we do not detail here.

Original: In Appendix L we will return to the differences between 1D and 2D, and show instances where they become important.
Revised: In Appendix L we will return to the differences between 1D and 2D, and show instances where they become important.

Original: D. L OW F REQUENCY B IAS : P LACE C ELLS ON THE C IRCLE
Revised: D. L OW F REQUENCY B IAS : P LACE C ELLS ON THE C IRCLE

Original: We begin with the representation of an angle on a circle, introduce χ , and show the analytic form of the low frequency bias it produces.
Revised: We begin with the representation of an angle on a circle, introduce χ , and show the analytic form of the low frequency bias it produces.

Original: Since we’re on a circle χ must depend on distance on the circle, rather than on a line (i.e. not χ = 1 − e −
Revised: Since we’re on a circle χ must depend on distance on the circle, rather than on a line (i.e. not χ = 1 − e −

Original: [Equation]
Revised: [Equation]

Original: We deﬁne the obvious generalisation of this to periodic spaces, Figure 13A, which is normalised to integrate to one under a uniform p ( θ ) :
Revised: We deﬁne the obvious generalisation of this to periodic spaces, Figure 13A, which is normalised to integrate to one under a uniform p ( θ ) :

Original: [Equation]
Revised: [Equation]

Original: Where I 0 is the zeroth-order modiﬁed Bessel function.
Revised: Where I 0 is the zeroth-order modiﬁed Bessel function.

Original: Hence the loss is:
Revised: Hence the loss is:

Original: [Equation]
Revised: [Equation]

Original: In this expression k is a parameter that controls the generalisation width, playing the inverse role of l in the main paper.
Revised: In this expression k is a parameter that controls the generalisation width, playing the inverse role of l in the main paper.

Original: Taking the same steps as in Appendix C we arrive at:
Revised: Taking the same steps as in Appendix C we arrive at:

Original: [Equation]
Revised: [Equation]

Original: The β integral again kills the cross terms:
Revised: The β integral again kills the cross terms:

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: These integrals can be computed by relating them to modiﬁed Bessel functions of the ﬁrst kind, which can be deﬁned for integer n as:
Revised: These integrals can be computed by relating them to modiﬁed Bessel functions of the ﬁrst kind, which can be deﬁned for integer n as:

Original: [Equation]
Revised: [Equation]

Original: Hence the loss is:
Revised: Hence the loss is:

Original: [Equation]
Revised: [Equation]

Original: This is basically the same result as before, equation 42, all the frequencies decouple and decrease the loss in proportion to their magnitude.
Revised: This is basically the same result as before, equation 42, all the frequencies decouple and decrease the loss in proportion to their magnitude.

Original: However, we have added a weighting factor, that decreases with frequency, Figure 13B, i.e. a simple low frequency bias!
Revised: However, we have added a weighting factor, that decreases with frequency, Figure 13B, i.e. a simple low frequency bias!

Original: This analysis agrees with that of Sengupta et al.
Revised: This analysis agrees with that of Sengupta et al.

Original: (2018), who argued that the best representation of angles on a ring for a loss related to the one studied here should be place cells.
Revised: (2018), who argued that the best representation of angles on a ring for a loss related to the one studied here should be place cells.

Original: We disentangle the relationship between our two works in Appendix F. Regardless, our optimisation problem now asks us to build a module of lattice neurons with as many low frequencies as possible.
Revised: We disentangle the relationship between our two works in Appendix F. Regardless, our optimisation problem now asks us to build a module of lattice neurons with as many low frequencies as possible.

Original: The optimal solution to this is the lowest frequency lattice, i.e. place cells!
Revised: The optimal solution to this is the lowest frequency lattice, i.e. place cells!

Original: (we validate this numerically in
Revised: (we validate this numerically in

Original: Appendix M).
Revised: Appendix M).

Original: Therefore, it is only additional developments of the loss, namely the introduction of an inﬁnite space for which you prioritise the coding of visited regions, and the introduction of the neural lengthscale, σ , that lead us to conclude that modules of grid cells are better than place cells.
Revised: Therefore, it is only additional developments of the loss, namely the introduction of an inﬁnite space for which you prioritise the coding of visited regions, and the introduction of the neural lengthscale, σ , that lead us to conclude that modules of grid cells are better than place cells.

Original: D. H IGH F REQUENCY B IAS : O CCUPANCY D ISTRIBUTION
Revised: D. H IGH F REQUENCY B IAS : O CCUPANCY D ISTRIBUTION

Original: Now we’ll examine how a Gaussian p ( x ) affects the result, using a representation of x , a 1dimensional position:
Revised: Now we’ll examine how a Gaussian p ( x ) affects the result, using a representation of x , a 1dimensional position:

Original: [Equation]
Revised: [Equation]

Original: Developing the argument as before:
Revised: Developing the argument as before:

Original: [Equation]
Revised: [Equation]

Original: The cross terms again are zero, due to the symmetry of the β integral around 0.
Revised: The cross terms again are zero, due to the symmetry of the β integral around 0.

Original: [Equation]
Revised: [Equation]

Original: Where we’ve wrapped up the details into the following integrals of trigonometric functions over a gaussian measure:
Revised: Where we’ve wrapped up the details into the following integrals of trigonometric functions over a gaussian measure:

Original: [Equation]
Revised: [Equation]

Original: So we get our ﬁnal expression for the loss:
Revised: So we get our ﬁnal expression for the loss:

Original: [Equation]
Revised: [Equation]

Original: The downstream consequence of this loss is simple if a module encodes all areas of space evenly.
Revised: The downstream consequence of this loss is simple if a module encodes all areas of space evenly.

Original: Encoding evenly implies the phases of the neurons in the module are evenly distributed.
Revised: Encoding evenly implies the phases of the neurons in the module are evenly distributed.

Original: The phase of a particular neuron is encoded in the coefﬁcients of the sine and cosine of the base frequency, a n 1 and b n 1 .
Revised: The phase of a particular neuron is encoded in the coefﬁcients of the sine and cosine of the base frequency, a n 1 and b n 1 .

Original: Even distribution implies that a n 1 oscillates from A d , where A d = max n a n 1 , to − A d and back again as you sweep the neuron index.
Revised: Even distribution implies that a n 1 oscillates from A d , where A d = max n a n 1 , to − A d and back again as you sweep the neuron index.

Original: As you perform this sweep the subsidiary coefﬁcients, such as a n 2 , perform the same oscillation, but multiple times.
Revised: As you perform this sweep the subsidiary coefﬁcients, such as a n 2 , perform the same oscillation, but multiple times.

Original: a n 2 does it twice, since, by the arguments that led us to grids, the peaks of the two waves must align (Figure 3B), and that implies the peak of the wave at twice the frequency moves at the same speed as you sweep the neuron number.
Revised: a n 2 does it twice, since, by the arguments that led us to grids, the peaks of the two waves must align (Figure 3B), and that implies the peak of the wave at twice the frequency moves at the same speed as you sweep the neuron number.

Original: This implies its phase oscillates twice as fast, and hence that (cid:80) n a n 1 a n 2 = 0 .
Revised: This implies its phase oscillates twice as fast, and hence that (cid:80) n a n 1 a n 2 = 0 .

Original: Arguments like this imply the frequencies actually decouple for a uniform encoding of space, making the loss,
Revised: Arguments like this imply the frequencies actually decouple for a uniform encoding of space, making the loss,

Original: [Equation]
Revised: [Equation]

Original: This is exactly a high frequency bias!
Revised: This is exactly a high frequency bias!

Original: (Fig 13C Left)
Revised: (Fig 13C Left)

Original: However, uniform encoding of the space is actually not optimal for this loss, as can be seen either from the form of equation 57, and veriﬁed numerically using the loss in the next section, Figure 13EF.
Revised: However, uniform encoding of the space is actually not optimal for this loss, as can be seen either from the form of equation 57, and veriﬁed numerically using the loss in the next section, Figure 13EF.

Original: The weighting of the sine factor and the cosine factor differ, and in fact push toward sine, since it plateaus at lower frequencies.
Revised: The weighting of the sine factor and the cosine factor differ, and in fact push toward sine, since it plateaus at lower frequencies.

Original: Combine that with a low frequency push, and you get an incentive to use only sines, which is allowed if you can encode only some areas of space, as the euclidean loss permits.
Revised: Combine that with a low frequency push, and you get an incentive to use only sines, which is allowed if you can encode only some areas of space, as the euclidean loss permits.

Original: A ﬁnal interesting feature of this loss is the way the coupling between frequencies gracefully extends the decoupling between frequencies to the space of continuous frequencies.
Revised: A ﬁnal interesting feature of this loss is the way the coupling between frequencies gracefully extends the decoupling between frequencies to the space of continuous frequencies.

Original: In our most simpliﬁed loss, equation 42, each frequency contributed according to the square of its amplitude, (cid:107) a d (cid:107) 2 + (cid:107) b d (cid:107) 2 .
Revised: In our most simpliﬁed loss, equation 42, each frequency contributed according to the square of its amplitude, (cid:107) a d (cid:107) 2 + (cid:107) b d (cid:107) 2 .

Original: How should this be extended to continuous frequencies?
Revised: How should this be extended to continuous frequencies?

Original: Certainly, if two frequencies were very far from one another we should retrieve the previous result, and their contributions would be expected to decouple, (cid:107) a 1 (cid:107) 

Revised: Certainly, if two frequencies were very far from one another we should retrieve the previous result, and their contributions would be expected to decouple, (cid:107) a 1 (cid:107)

Original: + (cid:107) b 1 (cid:107) 2 + (cid:107) a 2 (cid:107) 2 + (cid:107) b 2 (cid:107) 2 .
Revised: + (cid:107) b 1 (cid:107) 2 + (cid:107) a 2 (cid:107) 2 + (cid:107) b 2 (cid:107) 2 .

Original: But if k 2 = k 1 + δ k , for a small δ k then the two frequencies are, for all intents and purposes, the same, so they should contribute an amplitude (cid:107) a 1 + a 2 (cid:107) 

Revised: But if k 2 = k 1 + δ k , for a small δ k then the two frequencies are, for all intents and purposes, the same, so they should contribute an amplitude (cid:107) a 1 + a 2 (cid:107)

Original: + (cid:107) b 1 + b 2 (cid:107) 2 (cid:54) =
Revised: + (cid:107) b 1 + b 2 (cid:107) 2 (cid:54) =

Original: (cid:107) a 1 (cid:107) 2 + (cid:107) b 1 (cid:107) 

Revised: (cid:107) a 1 (cid:107) 2 + (cid:107) b 1 (cid:107)

Original: + (cid:107) a 2 (cid:107) 2 + (cid:107) b 2 (cid:107) 2 .
Revised: + (cid:107) a 2 (cid:107) 2 + (cid:107) b 2 (cid:107) 2 .

Original: Equation 57, Figure 13C right, performs exactly this bit of intuition.
Revised: Equation 57, Figure 13C right, performs exactly this bit of intuition.

Original: It tells us that the key lengthscale in frequency space is 1 L .
Revised: It tells us that the key lengthscale in frequency space is 1 L .

Original: Two frequencies separated by more than this distance contribute in a decoupled way, as in equation 37.
Revised: Two frequencies separated by more than this distance contribute in a decoupled way, as in equation 37.

Original: Conversely, at very small distances, much smaller than 1 L the contribution is exactly (cid:107) a 1 + a 2 (cid:107) 2 + (cid:107) b
Revised: Conversely, at very small distances, much smaller than 1 L the contribution is exactly (cid:107) a 1 + a 2 (cid:107) 2 + (cid:107) b

Original: 1 + b 2 (cid:107) 2 , as it should be.
Revised: 1 + b 2 (cid:107) 2 , as it should be.

Original: But additionally, it tells us the functional form of the behaviour between these two limits: a Gaussian decay.
Revised: But additionally, it tells us the functional form of the behaviour between these two limits: a Gaussian decay.

Original: D. T HE C OMBINED E FFECT = H IGH AND L OW F REQUENCY B IAS
Revised: D. T HE C OMBINED E FFECT = H IGH AND L OW F REQUENCY B IAS

Original: In this section we show that the simple euclidean loss with both χ and p can be analysed, and it contains both the high and low frequency biases discussed previously in isolation.
Revised: In this section we show that the simple euclidean loss with both χ and p can be analysed, and it contains both the high and low frequency biases discussed previously in isolation.

Original: This is not very surprising, but we include it for completeness.
Revised: This is not very surprising, but we include it for completeness.

Original: [Equation]
Revised: [Equation]

Original: Following a very similar path to before we reach:
Revised: Following a very similar path to before we reach:

Original: [Equation]
Revised: [Equation]

Original: since L > l (63)
Revised: since L > l (63)

Original: This contains the low and high frequency penalisation, at lengthscales 1 L and 1 l , respectively.
Revised: This contains the low and high frequency penalisation, at lengthscales 1 L and 1 l , respectively.

Original: There are 4 terms, 2 are identical to equation 57, and hence perform the same role.
Revised: There are 4 terms, 2 are identical to equation 57, and hence perform the same role.

Original: The additional two terms are positive, so should be minimised, and they can be minimised by making the frequencies smaller than ∼ 1 l , Figure 13D.
Revised: The additional two terms are positive, so should be minimised, and they can be minimised by making the frequencies smaller than ∼ 1 l , Figure 13D.

Original: We verify this claim numerically by optimising a neural representation of 2D position, g ( x ) , to minimise L 1 subject to actionable and biological constraints.
Revised: We verify this claim numerically by optimising a neural representation of 2D position, g ( x ) , to minimise L 1 subject to actionable and biological constraints.

Original: It forms a module of hexagonal cells,
Revised: It forms a module of hexagonal cells,

Original: Figure 13E, but the representation is shoddy, because the cells only encode a small area of space well, as can be seen by zooming on the response, Figure 13.
Revised: Figure 13E, but the representation is shoddy, because the cells only encode a small area of space well, as can be seen by zooming on the response, Figure 13.

Original: E A NALYSIS OF F ULL L OSS : M ULTIPLE M ODULES OF G RIDS
Revised: E A NALYSIS OF F ULL L OSS : M ULTIPLE M ODULES OF G RIDS

Original: In this Section we will perturbatively analyse the full loss function, which includes the neural lengthscale, σ .
Revised: In this Section we will perturbatively analyse the full loss function, which includes the neural lengthscale, σ .

Original: The consequences of including this lengthscale will become most obvious for the simplest example, the representation of an angle on a ring, g ( θ ) , with a uniform weighting of the importance of separating different points.
Revised: The consequences of including this lengthscale will become most obvious for the simplest example, the representation of an angle on a ring, g ( θ ) , with a uniform weighting of the importance of separating different points.

Original: We’ll use it to derive the push towards non-harmonically related modules.
Revised: We’ll use it to derive the push towards non-harmonically related modules.

Original: Hence, we study the following loss:
Revised: Hence, we study the following loss:

Original: [Equation]
Revised: [Equation]

Original: And examine how the effects of this loss differ from the euclidean verison studied in Appendix C.
Revised: And examine how the effects of this loss differ from the euclidean verison studied in Appendix C.

Original: This loss is difﬁcult to analyse, but insight can be gained by making an approximation.
Revised: This loss is difﬁcult to analyse, but insight can be gained by making an approximation.

Original: We assume that the distance in neural space between the representations of two inputs depends only on the difference between those two inputs,
Revised: We assume that the distance in neural space between the representations of two inputs depends only on the difference between those two inputs,

Original: i
Revised: i

Original: [Equation]
Revised: [Equation]

Original: e.:
Revised: e.:

Original: [Equation]
Revised: [Equation]

Original: Looking at the constrained form of the representation
Revised: Looking at the constrained form of the representation

Original: [Equation]
Revised: [Equation]

Original: we can see this is satisﬁed if frequencies are orthogonal:
Revised: we can see this is satisﬁed if frequencies are orthogonal:

Original: [Equation]
Revised: [Equation]

Original: Then:
Revised: Then:

Original: [Equation]
Revised: [Equation]

Original: Now we will combine this convenient form with the following expansion:
Revised: Now we will combine this convenient form with the following expansion:

Original: [Equation]
Revised: [Equation]

Original: Developing equation 64 using eqns.
Revised: Developing equation 64 using eqns.

Original: 69 and 68:
Revised: 69 and 68:

Original: [Equation]
Revised: [Equation]

Original: We will now look at each of these two terms separately and derive from each an intuitive conclusion.
Revised: We will now look at each of these two terms separately and derive from each an intuitive conclusion.

Original: Term A will tell us that frequency amplitudes will tend to be capped at around the legnthscale σ .
Revised: Term A will tell us that frequency amplitudes will tend to be capped at around the legnthscale σ .

Original: Term B will tell us how well different frequencies combine, and as such, will exert the push towards non-harmonically related frequency combinations
Revised: Term B will tell us how well different frequencies combine, and as such, will exert the push towards non-harmonically related frequency combinations

Original: [Equation]
Revised: [Equation]

Original: E. T ERM A E NCOURAGES A D IVERSITY OF H IGH A MPLITUDE F REQUENCIES IN THE
Revised: E. T ERM A E NCOURAGES A D IVERSITY OF H IGH A MPLITUDE F REQUENCIES IN THE

Original: ODE
Revised: ODE

Original: Fig 14A shows a plot of I 

Revised: Fig 14A shows a plot of I

Original: ( A 2 d 4 σ 2 ) .
Revised: ( A 2 d 4 σ 2 ) .

Original: As can be seen there are a couple of regimes, if A d , the amplitude of frequency n d , is smaller than σ then I 

Revised: As can be seen there are a couple of regimes, if A d , the amplitude of frequency n d , is smaller than σ then I

Original: ( A 2 d 4 σ 2 )
Revised: ( A 2 d 4 σ 2 )

Original: ≈ 1 .
Revised: ≈ 1 .

Original: On the other hand, as A d grows, I 0 also grows.
Revised: On the other hand, as A d grows, I 0 also grows.

Original: Asymptotically:
Revised: Asymptotically:

Original: [Equation]
Revised: [Equation]

Original: Correspondingly Term A also behaves differently in the two regimes, when A d is smaller than σ the loss decreases exponentially with the amplitude of frequency d :
Revised: Correspondingly Term A also behaves differently in the two regimes, when A d is smaller than σ the loss decreases exponentially with the amplitude of frequency d :

Original: [Equation]
Revised: [Equation]

Original: Alternatively:
Revised: Alternatively:

Original: [Equation]
Revised: [Equation]

Original: So, up to a threshold region deﬁned by the lengthscale σ , there is an exponential improvement in the loss as you increase the amplitude.
Revised: So, up to a threshold region deﬁned by the lengthscale σ , there is an exponential improvement in the loss as you increase the amplitude.

Original: Beyond that the improvements drop to being polynomial.
Revised: Beyond that the improvements drop to being polynomial.

Original: This makes a lot of sense, once you have used a frequency to separate a subset of points sufﬁciently well from one another you only get minimal gains for more increase in that frequency’s amplitude,
Revised: This makes a lot of sense, once you have used a frequency to separate a subset of points sufﬁciently well from one another you only get minimal gains for more increase in that frequency’s amplitude,

Original: Figure 14B. Since there are many frequencies, and the norm constraint ensures that few can have an amplitude of order σ , it encourages the optimiser to put more power into the other frequencies.
Revised: Figure 14B. Since there are many frequencies, and the norm constraint ensures that few can have an amplitude of order σ , it encourages the optimiser to put more power into the other frequencies.

Original: In other words, amplitudes are only usefully increased to a lengthscale σ , encouraging a diversity in high amplitude frequencies.
Revised: In other words, amplitudes are only usefully increased to a lengthscale σ , encouraging a diversity in high amplitude frequencies.

Original: This is moderately interesting, but will be a useful piece of information for the next derivation.
Revised: This is moderately interesting, but will be a useful piece of information for the next derivation.

Original: E. T ERM B E NCOURAGES THE C ODE ’ S F REQUENCIES TO BE N ON -H ARMONICALLY R ELATED
Revised: E. T ERM B E NCOURAGES THE C ODE ’ S F REQUENCIES TO BE N ON -H ARMONICALLY R ELATED

Original: Term B is an integral of the product of D inﬁnite sums, which seems very counter-productive to examine.
Revised: Term B is an integral of the product of D inﬁnite sums, which seems very counter-productive to examine.

Original: We are fortunate, however.
Revised: We are fortunate, however.

Original: First, the coefﬁcients in the inﬁnite sum decay very rapidly with m when the ratio of A d σ is O (1) , Figure 14C.
Revised: First, the coefﬁcients in the inﬁnite sum decay very rapidly with m when the ratio of A d σ is O (1) , Figure 14C.

Original: The previous section’s arguments make it clear that, in any optimal code, A d will be smaller or of the same order as σ.
Revised: The previous section’s arguments make it clear that, in any optimal code, A d will be smaller or of the same order as σ.

Original: As such, the iniﬁte sum is rapidly truncated.
Revised: As such, the iniﬁte sum is rapidly truncated.

Original: Additionally, all of the coefﬁcient ratios, (cid:18)
Revised: Additionally, all of the coefﬁcient ratios, (cid:18)

Original: [Equation]
Revised: [Equation]

Original: , are smaller than 1, as such we can understand the role of Term B by analysing just the ﬁrst few terms in the following series expansion,
Revised: , are smaller than 1, as such we can understand the role of Term B by analysing just the ﬁrst few terms in the following series expansion,

Original: [Equation]
Revised: [Equation]

Original: where, due to the rapid decay of
Revised: where, due to the rapid decay of

Original: [Equation]
Revised: [Equation]

Original: [Table] [Table]
Revised: [Table] [Table]

Original: The 0 th order term is a constant, and the 1 st order term is zero when you integrate over α .
Revised: The 0 th order term is a constant, and the 1 st order term is zero when you integrate over α .

Original: As such, the leading order effect of this loss on the optimal choice of frequencies arrives via the 2 nd order term.
Revised: As such, the leading order effect of this loss on the optimal choice of frequencies arrives via the 2 nd order term.

Original: The integral is easy,
Revised: The integral is easy,

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: We want to minimise the loss, and this contribution is positive, so any time the 2 nd order term is non-zero it is a penalty.
Revised: We want to minimise the loss, and this contribution is positive, so any time the 2 nd order term is non-zero it is a penalty.

Original: As such, we want as few frequencies to satisfy the condition in the delta function, mn d
Revised: As such, we want as few frequencies to satisfy the condition in the delta function, mn d

Original: = m (cid:48) n d (cid:48) .
Revised: = m (cid:48) n d (cid:48) .

Original: Recall from equation 77 that these terms are weighted by coefﬁcients that decay rapidly as m grows.
Revised: Recall from equation 77 that these terms are weighted by coefﬁcients that decay rapidly as m grows.

Original: As such, the condition that the loss is penalising, mn d = m (cid:48) n d (cid:48) , is really saying: penalise all pairs of frequencies in the code, n d , n d (cid:48) , that can be related by some simple harmonic ratio, n d = m (cid:48)
Revised: As such, the condition that the loss is penalising, mn d = m (cid:48) n d (cid:48) , is really saying: penalise all pairs of frequencies in the code, n d , n d (cid:48) , that can be related by some simple harmonic ratio, n d = m (cid:48)

Original: m n d (cid:48) , where m and m (cid:48) are small integers, roughly less than 5, Figure 14C.
Revised: m n d (cid:48) , where m and m (cid:48) are small integers, roughly less than 5, Figure 14C.

Original: This may seem obtuse, but is very interpretable!
Revised: This may seem obtuse, but is very interpretable!

Original: Simply harmonically related frequencies are exactly those that encode many inputs with the same response!
Revised: Simply harmonically related frequencies are exactly those that encode many inputs with the same response!

Original: Hence, they should be penalised, Fig 4C.
Revised: Hence, they should be penalised, Fig 4C.

Original: E. A H ARMONIC T USSLE - N ON -H ARMONICS L EADS TO M ULTIPLE M ODULES
Revised: E. A H ARMONIC T USSLE - N ON -H ARMONICS L EADS TO M ULTIPLE M ODULES

Original: As described in Section 3.3, we now arrive at a dilemma.
Revised: As described in Section 3.3, we now arrive at a dilemma.

Original: The arguments of Sections 3.1 and C suggest that the optimal representation is comprised of lattice modules, i.e. the neurons must build a code from a lattice of frequencies.
Revised: The arguments of Sections 3.1 and C suggest that the optimal representation is comprised of lattice modules, i.e. the neurons must build a code from a lattice of frequencies.

Original: A lattice is exactly the arrangement that ensures all the frequencies are related to one another by simple harmonic ratios.
Revised: A lattice is exactly the arrangement that ensures all the frequencies are related to one another by simple harmonic ratios.

Original: Yet, the arguments of the previous section suggest that frequencies with simple harmonic relations are a bad choice - i.e. a frequency lattice would be the worst option!
Revised: Yet, the arguments of the previous section suggest that frequencies with simple harmonic relations are a bad choice - i.e. a frequency lattice would be the worst option!

Original: How can this be resolved?
Revised: How can this be resolved?

Original: The resolution will depend on the value of two parameters: the ﬁrst, σ , is the neural lengthscale; the second, N , is the number of neurons, which, through the boundedness constraint, sets the scale of the amplitudes, A d .
Revised: The resolution will depend on the value of two parameters: the ﬁrst, σ , is the neural lengthscale; the second, N , is the number of neurons, which, through the boundedness constraint, sets the scale of the amplitudes, A d .

Original: We can already see this key ratio, A d σ , appearing in the preceding arguments.
Revised: We can already see this key ratio, A d σ , appearing in the preceding arguments.

Original: Its role is to choose the compromise point in this harmonic tussle, and in doing so it chooses the number of modules in the code.
Revised: Its role is to choose the compromise point in this harmonic tussle, and in doing so it chooses the number of modules in the code.

Original: A simple Taylor expansion illustrates that as σ tends towards ∞ the full loss reverts to the euclidean approximation that was the focus of Sections C and D, in which the effect of the harmonic penalisation drops to zero.
Revised: A simple Taylor expansion illustrates that as σ tends towards ∞ the full loss reverts to the euclidean approximation that was the focus of Sections C and D, in which the effect of the harmonic penalisation drops to zero.

Original: [Equation]
Revised: [Equation]

Original: Hence, as σ → ∞ we recover our original optimal solution: all the neurons belong to one module, whose shape is determined by χ and p .
Revised: Hence, as σ → ∞ we recover our original optimal solution: all the neurons belong to one module, whose shape is determined by χ and p .

Original: This explains why, with few neurons (equivalent to σ → ∞ ), the optimal solution is a single module of perfectly hexagonal grids, ﬁg 4B.
Revised: This explains why, with few neurons (equivalent to σ → ∞ ), the optimal solution is a single module of perfectly hexagonal grids, ﬁg 4B.

Original: But as σ decreases the effect of the harmonic penalisation becomes larger and larger.
Revised: But as σ decreases the effect of the harmonic penalisation becomes larger and larger.

Original: Eventually, one module is no longer optimal.
Revised: Eventually, one module is no longer optimal.

Original: Yet, all the previous arguments that modularised codes are easily made non-negative, and hence valuable, still apply.
Revised: Yet, all the previous arguments that modularised codes are easily made non-negative, and hence valuable, still apply.

Original: The obvious solution is to have two modules.
Revised: The obvious solution is to have two modules.

Original: Within each module all the neurons are easily made non-negative, but between modules you can have very non-harmonic relations, trading off both requirements.
Revised: Within each module all the neurons are easily made non-negative, but between modules you can have very non-harmonic relations, trading off both requirements.

Original: As such increasing the ratio A d σ , either by increasing the number of neurons or decreasing σ , creates an optimisation problem for which the high quality representations are modules of grids with more modules as the ratio gets bigger.
Revised: As such increasing the ratio A d σ , either by increasing the number of neurons or decreasing σ , creates an optimisation problem for which the high quality representations are modules of grids with more modules as the ratio gets bigger.

Original: We observed exactly this effect numerically, and it explains the presence of multiple modules of grids in the full version of our problem for a range of values of σ .
Revised: We observed exactly this effect numerically, and it explains the presence of multiple modules of grids in the full version of our problem for a range of values of σ .

Original: Finally, we can extract from this analysis a guiding light: that the frequencies in different modules should be maximally non-harmonically related.
Revised: Finally, we can extract from this analysis a guiding light: that the frequencies in different modules should be maximally non-harmonically related.

Original: This additional principle, beyond the simple low and high frequency biases of Section D, frames our normative arguments for the optimal arrangement of modules, that we explore Section 4.2 and
Revised: This additional principle, beyond the simple low and high frequency biases of Section D, frames our normative arguments for the optimal arrangement of modules, that we explore Section 4.2 and

Original: Appendix K.
Revised: Appendix K.

Original: E. N ON -H ARMONICALLY R ELATED C ONTINUOUS F REQUENCIES - A S MOOTHING E FFECT
Revised: E. N ON -H ARMONICALLY R ELATED C ONTINUOUS F REQUENCIES - A S MOOTHING E FFECT

Original: We will address one ﬁnal concern.
Revised: We will address one ﬁnal concern.

Original: When representing inﬁnite variables the frequencies in the code become real numbers.
Revised: When representing inﬁnite variables the frequencies in the code become real numbers.

Original: For example, when representing a 1-dimensional continuous input, x , the code, g ( x ) , has the form,
Revised: For example, when representing a 1-dimensional continuous input, x , the code, g ( x ) , has the form,

Original: [Equation]
Revised: [Equation]

Original: And it seems like the equivalent penalisation as in equation 78, δ ( mω d − m (cid:48)
Revised: And it seems like the equivalent penalisation as in equation 78, δ ( mω d − m (cid:48)

Original: ω d (cid:48) ) , is very easy to avoid.
Revised: ω d (cid:48) ) , is very easy to avoid.

Original: If mω d = m (cid:48) ω d (cid:48) and the code is being penalised, just increase ω d by an inﬁnitely small amount, δω , such that mω d + mδω (cid:54) = m (cid:48)
Revised: If mω d = m (cid:48) ω d (cid:48) and the code is being penalised, just increase ω d by an inﬁnitely small amount, δω , such that mω d + mδω (cid:54) = m (cid:48)

Original: ω
Revised: ω

Original: d (cid:48) , and suddenly this term does not contribute to the loss.
Revised: d (cid:48) , and suddenly this term does not contribute to the loss.

Original: This, however, is not how the loss manifests.
Revised: This, however, is not how the loss manifests.

Original: Instead frequencies are penalised for being close to one another, where close is deﬁned by the size of the region the animal encodes, set by lengthscale
Revised: Instead frequencies are penalised for being close to one another, where close is deﬁned by the size of the region the animal encodes, set by lengthscale

Original: L .
Revised: L .

Original: This can be seen by analysing the following loss,
Revised: This can be seen by analysing the following loss,

Original: [Equation]
Revised: [Equation]

Original: Making the same assumption as in the previous Section, equation 67, and developing the loss identically,
Revised: Making the same assumption as in the previous Section, equation 67, and developing the loss identically,

Original: [Equation]
Revised: [Equation]

Original: We again analyse this through the series expansion in equation 77.
Revised: We again analyse this through the series expansion in equation 77.

Original: The 0 th order term is again constant.
Revised: The 0 th order term is again constant.

Original: The 1 st order term is proporational to,
Revised: The 1 st order term is proporational to,

Original: [Equation]
Revised: [Equation]

Original: i
Revised: i

Original: [Equation]
Revised: [Equation]

Original: e., as in Section D.2, p ( x ) implements a high frequency bias.
Revised: e., as in Section D.2, p ( x ) implements a high frequency bias.

Original: In fact, the mathematical form of this penalty, a gaussian, is identical to that in diagonal terms of the previously derived sum, equation
Revised: In fact, the mathematical form of this penalty, a gaussian, is identical to that in diagonal terms of the previously derived sum, equation

Original: [Equation]
Revised: [Equation]

Original: The 2 nd order term is where we ﬁnd the answer to this Section’s original concern,
Revised: The 2 nd order term is where we ﬁnd the answer to this Section’s original concern,

Original: [Equation]
Revised: [Equation]

Original: Which is similar to the cross terms in equation 57, but with the addition of harmonic interactions.
Revised: Which is similar to the cross terms in equation 57, but with the addition of harmonic interactions.

Original: This is the continuous equivalent to the penalty in equation 78.
Revised: This is the continuous equivalent to the penalty in equation 78.

Original: Frequencies and their low order harmonics are penalised for landing closer than ∼ 1 L from one another, Fig 14D.
Revised: Frequencies and their low order harmonics are penalised for landing closer than ∼ 1 L from one another, Fig 14D.

Original: In conclusion, the logic gracefully transfers to continuous frequencies.
Revised: In conclusion, the logic gracefully transfers to continuous frequencies.

Original: F L INKS TO S ENGUPTA ET AL .
Revised: F L INKS TO S ENGUPTA ET AL .

Original: Sengupta et al.
Revised: Sengupta et al.

Original: (2018) derive an optimal representation of structured variables, where their deﬁnition of structured exactly aligns with ours: the variable’s transformations form a group (though the term they use is symmetric manifolds).
Revised: (2018) derive an optimal representation of structured variables, where their deﬁnition of structured exactly aligns with ours: the variable’s transformations form a group (though the term they use is symmetric manifolds).

Original: Further, there are analytic similarities between the loss they study and ours.
Revised: Further, there are analytic similarities between the loss they study and ours.

Original: However, their optimal representations are place cells, and ours are multiple modules of grid cells.
Revised: However, their optimal representations are place cells, and ours are multiple modules of grid cells.

Original: In this section we will disentangle these two threads, explaining this difference, which arises from our actionability principle, the neural lengthscale σ , and our use of inﬁnite inputs.
Revised: In this section we will disentangle these two threads, explaining this difference, which arises from our actionability principle, the neural lengthscale σ , and our use of inﬁnite inputs.

Original: Sengupta et al.
Revised: Sengupta et al.

Original: (2018) study the optimisation of a loss derived from a dot product similarity matching objective, an objective that has prompted numerous interesting discoveries (Pehlevan & Chklovskii, 2019).
Revised: (2018) study the optimisation of a loss derived from a dot product similarity matching objective, an objective that has prompted numerous interesting discoveries (Pehlevan & Chklovskii, 2019).

Original: Input vectors, x , stacked into the matrix X , are represented by vectors of neural activity, y , stacked into Y , that have to be non-negative and bounded.
Revised: Input vectors, x , stacked into the matrix X , are represented by vectors of neural activity, y , stacked into Y , that have to be non-negative and bounded.

Original: The loss measures the similarity between pairs of inputs, x T x (cid:48) , and pairs of outputs, y T y (cid:48) , and tries to optimise the neural representation, y , such that all input similarities above a threshold, α , are faithfully reproduced in the similarity structure of the neural representation.
Revised: The loss measures the similarity between pairs of inputs, x T x (cid:48) , and pairs of outputs, y T y (cid:48) , and tries to optimise the neural representation, y , such that all input similarities above a threshold, α , are faithfully reproduced in the similarity structure of the neural representation.

Original: [Equation]
Revised: [Equation]

Original: (87) where E is a matrix of ones, and diag ( Y T Y ) ≤ β 1 enforces the neural activity’s norm constraint.
Revised: (87) where E is a matrix of ones, and diag ( Y T Y ) ≤ β 1 enforces the neural activity’s norm constraint.

Original: Our loss is strikingly similar.
Revised: Our loss is strikingly similar.

Original: Our inputs are variables, such as the angle, that play the role of x .
Revised: Our inputs are variables, such as the angle, that play the role of x .

Original: Sengupta et al.
Revised: Sengupta et al.

Original: ’s input dot product similarity, x T x (cid:48) − α , plays the role of χ ( θ, θ (cid:48) ) , measuring similarity in input space.
Revised: ’s input dot product similarity, x T x (cid:48) − α , plays the role of χ ( θ, θ (cid:48) ) , measuring similarity in input space.

Original: The neural activities, g , must, like y , be non-negative and bounded in L  norm.
Revised: The neural activities, g , must, like y , be non-negative and bounded in L  norm.

Original: And minimising the similarity matching loss is like minimising the simple euclidean form of the loss studied in Appendix D, with the integrals playing the role of the trace,
Revised: And minimising the similarity matching loss is like minimising the simple euclidean form of the loss studied in Appendix D, with the integrals playing the role of the trace,

Original: [Equation]
Revised: [Equation]

Original: The analogy then, is that minimising L 1 is asking the most similar elements of the input to be represented most similarly, and dissimilar dissimilarly, as in the non-negative similarity matching objective; an orthogonal motivation for our loss!
Revised: The analogy then, is that minimising L 1 is asking the most similar elements of the input to be represented most similarly, and dissimilar dissimilarly, as in the non-negative similarity matching objective; an orthogonal motivation for our loss!

Original: Sengupta et al.
Revised: Sengupta et al.

Original: (2018) use beautiful convex optimisation techniques (that cannot, unfortunately, be applied directly to our problem, Appendix G) to show that the optimal representation of the ﬁnite spaces they consider (circles and spheres) are populations of place cells.
Revised: (2018) use beautiful convex optimisation techniques (that cannot, unfortunately, be applied directly to our problem, Appendix G) to show that the optimal representation of the ﬁnite spaces they consider (circles and spheres) are populations of place cells.

Original: What is it about our losses that leads us to different conclusions?
Revised: What is it about our losses that leads us to different conclusions?

Original: Our problems differ in a few ways.
Revised: Our problems differ in a few ways.

Original: First, since they use dot-product similarities, their loss is closest to our simpliﬁed losses, i.e. those without the presence of the neural lengthscale, σ .
Revised: First, since they use dot-product similarities, their loss is closest to our simpliﬁed losses, i.e. those without the presence of the neural lengthscale, σ .

Original: So, due to this difference, we would expect them to produce a single module of responses, as observed.
Revised: So, due to this difference, we would expect them to produce a single module of responses, as observed.

Original: The second difference is the addition of actionability to our code which produces a harsh frequency content constraint, and is pivotal to our derivation (as can be seen from ablation studies, Appendix I).
Revised: The second difference is the addition of actionability to our code which produces a harsh frequency content constraint, and is pivotal to our derivation (as can be seen from ablation studies, Appendix I).

Original: They, however, assume inﬁnitely many neurons, at which point the frequency constraint becomes, especially when there is only one module, relatively vacuous.
Revised: They, however, assume inﬁnitely many neurons, at which point the frequency constraint becomes, especially when there is only one module, relatively vacuous.

Original: So we could still see their analysis as the inﬁnite neuron version of ours.
Revised: So we could still see their analysis as the inﬁnite neuron version of ours.

Original: Third, we use a different measure of input similarity, though we do not believe this would have particularly strong effects on our conclusions.
Revised: Third, we use a different measure of input similarity, though we do not believe this would have particularly strong effects on our conclusions.

Original: Finally, we consider inﬁnite spaces, like position on a line, by introducing a probability distribution over inputs, p ( x ) .
Revised: Finally, we consider inﬁnite spaces, like position on a line, by introducing a probability distribution over inputs, p ( x ) .

Original: As such, for representations of ﬁnite variables, like angle, the inﬁnite neuron predictions of our simpliﬁed loss and Sengupta et al.
Revised: As such, for representations of ﬁnite variables, like angle, the inﬁnite neuron predictions of our simpliﬁed loss and Sengupta et al.

Original: should align.
Revised: should align.

Original: Thankfully, they do!
Revised: Thankfully, they do!

Original: As shown in Appendix M we predict place-cell like tuning for the representation of circular variables.
Revised: As shown in Appendix M we predict place-cell like tuning for the representation of circular variables.

Original: This follows from our previous discussions, Appendix D.1: for ﬁnite spaces including a low frequency bias via the input similarity χ was argued to make the optimal representation a module of the lowest frequency lattices, place cells!
Revised: This follows from our previous discussions, Appendix D.1: for ﬁnite spaces including a low frequency bias via the input similarity χ was argued to make the optimal representation a module of the lowest frequency lattices, place cells!

Original: Hence, relative to Sengupta et al., by extending to the representation of inﬁnite spaces with actionability we reach grids, rather than place cells.
Revised: Hence, relative to Sengupta et al., by extending to the representation of inﬁnite spaces with actionability we reach grids, rather than place cells.

Original: And by additionally including σ , the neural lengthscale, we produce multiple modules.
Revised: And by additionally including σ , the neural lengthscale, we produce multiple modules.

Original: G O PTIMAL N ON - NEGATIVE A CTIONABLE C ODE - A N ON -C ONVEX P ROBLEM
Revised: G O PTIMAL N ON - NEGATIVE A CTIONABLE C ODE - A N ON -C ONVEX P ROBLEM

Original: In this section we highlight that actionability is what makes our problem non-convex.
Revised: In this section we highlight that actionability is what makes our problem non-convex.

Original: Speciﬁcally, choosing the set of frequencies is difﬁcult: for a ﬁxed choice of frequencies the simpliﬁed loss (equation 7) and constraints are convex in the neural coefﬁcients.
Revised: Speciﬁcally, choosing the set of frequencies is difﬁcult: for a ﬁxed choice of frequencies the simpliﬁed loss (equation 7) and constraints are convex in the neural coefﬁcients.

Original: We include it here to point towards tools that may be useful in proving grid’s optimality, rather than heuristically justifying them with numerically support, as we have done.
Revised: We include it here to point towards tools that may be useful in proving grid’s optimality, rather than heuristically justifying them with numerically support, as we have done.

Original: Consider a representation g ( θ ) .
Revised: Consider a representation g ( θ ) .

Original: For a ﬁxed set of frequencies { n d } Dd =1 , we’ll aim to optimise the coefﬁcients a 0 , { a d , b d } Dd =1 .
Revised: For a ﬁxed set of frequencies { n d } Dd =1 , we’ll aim to optimise the coefﬁcients a 0 , { a d , b d } Dd =1 .

Original: The losses take simple forms, the functional objective to be minimised is min − (cid:80) d (cid:107) a d (cid:107) 

Revised: The losses take simple forms, the functional objective to be minimised is min − (cid:80) d (cid:107) a d (cid:107)

Original: + (cid:107) b d (cid:107) 2 .
Revised: + (cid:107) b d (cid:107) 2 .

Original: The boundedness constraint is similarly quadratic in the coefﬁcients, equation 8.
Revised: The boundedness constraint is similarly quadratic in the coefﬁcients, equation 8.

Original: The only potential trouble is the non-negativity constraint.
Revised: The only potential trouble is the non-negativity constraint.

Original: Sengupta et al.
Revised: Sengupta et al.

Original: (2018) solve this problem by having no actionabtility and inﬁntely many neurons, we can take a different approach.
Revised: (2018) solve this problem by having no actionabtility and inﬁntely many neurons, we can take a different approach.

Original: Our code is a Positive Trigonometric Polynomial, a function of the following form:
Revised: Our code is a Positive Trigonometric Polynomial, a function of the following form:

Original: [Equation]
Revised: [Equation]

Original: where D max = max d n d , a type of polynomial studied in engineering and optimisation settings
Revised: where D max = max d n d , a type of polynomial studied in engineering and optimisation settings

Original: Dumitrescu (2007).
Revised: Dumitrescu (2007).

Original: Due to actionability our code is a sparse positive trigonometric polynomial, only 2 D + 1 of the r d are non-zero.
Revised: Due to actionability our code is a sparse positive trigonometric polynomial, only 2 D + 1 of the r d are non-zero.

Original: While positivity is a hard constraint to enforce, Bochner’s theorem gives us hope.
Revised: While positivity is a hard constraint to enforce, Bochner’s theorem gives us hope.

Original: It tells us that for g n ( θ ) to be positive for all θ , its fourier transform, g n ( ω ) must be positive-deﬁnite.
Revised: It tells us that for g n ( θ ) to be positive for all θ , its fourier transform, g n ( ω ) must be positive-deﬁnite.

Original: This means that the following matrix, made from the fourier coefﬁcients, must be positive-deﬁnite:
Revised: This means that the following matrix, made from the fourier coefﬁcients, must be positive-deﬁnite:

Original: [Equation]
Revised: [Equation]

Original: The set of positive-deﬁnite matrices forms a convex set.
Revised: The set of positive-deﬁnite matrices forms a convex set.

Original: Even the set of matrices Q n where ﬁxed diagonals are zero is a convex set.
Revised: Even the set of matrices Q n where ﬁxed diagonals are zero is a convex set.

Original: Further, this matrix displays all the right tradeoffs for positivity: r n 0 must be non-zero for the matrix to be positive-deﬁnite, i.e. there must be a constant offset to achieve non-negativity.
Revised: Further, this matrix displays all the right tradeoffs for positivity: r n 0 must be non-zero for the matrix to be positive-deﬁnite, i.e. there must be a constant offset to achieve non-negativity.

Original: Similarly, including harmonics makes achieving non-negativity easier than non-harmonics, just as in Figure 2A.
Revised: Similarly, including harmonics makes achieving non-negativity easier than non-harmonics, just as in Figure 2A.

Original: But even having made this link we cannot make progress, since we had to choose which frequencies had non-zero amplitude.
Revised: But even having made this link we cannot make progress, since we had to choose which frequencies had non-zero amplitude.

Original: Hence, the choice of frequency sparsity implied by actionability stops our problem from being convex.
Revised: Hence, the choice of frequency sparsity implied by actionability stops our problem from being convex.

Original: A plausible solution might be to solve the convex optimisation problem for a given set of frequencies, then to take gradients through that solution to update the frequencies, but we have not implemented this.
Revised: A plausible solution might be to solve the convex optimisation problem for a given set of frequencies, then to take gradients through that solution to update the frequencies, but we have not implemented this.

Original: H L INKS TO SOME OTHER P ATH I NTEGRATION M ODELS
Revised: H L INKS TO SOME OTHER P ATH I NTEGRATION M ODELS

Original: Our theory is a theory of path-integrating representations.
Revised: Our theory is a theory of path-integrating representations.

Original: Actionability means you understand the rules of space (a representation should be unchanged after taking a north then east then south then west), but it does not mean you understand where you are, or how ﬁne-grained this understanding is (a constant representation satisﬁes the actionable rules for example).
Revised: Actionability means you understand the rules of space (a representation should be unchanged after taking a north then east then south then west), but it does not mean you understand where you are, or how ﬁne-grained this understanding is (a constant representation satisﬁes the actionable rules for example).

Original: Our functional constraint rectiﬁes this, as it requires all relevant locations to be represented differently, up to a certain spatial scale.
Revised: Our functional constraint rectiﬁes this, as it requires all relevant locations to be represented differently, up to a certain spatial scale.

Original: Understanding rules, and representing locations differently is the same as path-integrationnow a representation can be updated according to the rules, and it is different in different locations meaning the underlying spatial variables can be decoded.
Revised: Understanding rules, and representing locations differently is the same as path-integrationnow a representation can be updated according to the rules, and it is different in different locations meaning the underlying spatial variables can be decoded.

Original: H. L INEAR P ATH I NTEGRATION M ODELS
Revised: H. L INEAR P ATH I NTEGRATION M ODELS

Original: Linear path integration models of the entorhinal cortex, like ours, have, to the best of our knowledge, been suggested three times before (Issa & Zhang, 2012; Whittington et al., 2020; Gao et al., 2021).
Revised: Linear path integration models of the entorhinal cortex, like ours, have, to the best of our knowledge, been suggested three times before (Issa & Zhang, 2012; Whittington et al., 2020; Gao et al., 2021).

Original: Whittington et al.
Revised: Whittington et al.

Original: and
Revised: and

Original: Gao et al.
Revised: Gao et al.

Original: are both simulation based, optimising losses related to ours, and show the resulting optimisations look like grid cells, though both papers only obtain multiple modules of grids by manually enforcing block diagonal transformation matrices.
Revised: are both simulation based, optimising losses related to ours, and show the resulting optimisations look like grid cells, though both papers only obtain multiple modules of grids by manually enforcing block diagonal transformation matrices.

Original: In this section we will discuss the differences between these two losses and ours, to explain their results in our framework as much as possible.
Revised: In this section we will discuss the differences between these two losses and ours, to explain their results in our framework as much as possible.

Original: Gao et al.
Revised: Gao et al.

Original: optimise a representation of position and a set of linear neural transition matrices to minimise a series of losses.
Revised: optimise a representation of position and a set of linear neural transition matrices to minimise a series of losses.

Original: One loss enforces path integration, like our actionability constraint.
Revised: One loss enforces path integration, like our actionability constraint.

Original: Another bounds the ﬁring rate of vectors, similar to our boundedness constraint.
Revised: Another bounds the ﬁring rate of vectors, similar to our boundedness constraint.

Original: A ﬁnal similarity is their version of the functional objective, which enforces the ability to linearly readout place-cell-like activity from the grid code.
Revised: A ﬁnal similarity is their version of the functional objective, which enforces the ability to linearly readout place-cell-like activity from the grid code.

Original: However, additionally they enforce an ‘isotropic scaling term’.
Revised: However, additionally they enforce an ‘isotropic scaling term’.

Original: This term measures how much the neural activity vector moves when you take a small step in direction ψ in the input space, and encourages all the steps to be the same size independent of ψ .
Revised: This term measures how much the neural activity vector moves when you take a small step in direction ψ in the input space, and encourages all the steps to be the same size independent of ψ .

Original: This can be understood through the strong force towards hexagonality it produces: of all the two dimensional grid the one that is most symmetric is hexagons, with it’s C 6 symmetry.
Revised: This can be understood through the strong force towards hexagonality it produces: of all the two dimensional grid the one that is most symmetric is hexagons, with it’s C 6 symmetry.

Original: Hence it will minimise this isotropic scaling.
Revised: Hence it will minimise this isotropic scaling.

Original: We have shown that it is not necessary to include this term in the loss to produce hexagonal grids.
Revised: We have shown that it is not necessary to include this term in the loss to produce hexagonal grids.

Original: However, there will be differences in behaviour between a code minimising this constraint and a code that doesn’t.
Revised: However, there will be differences in behaviour between a code minimising this constraint and a code that doesn’t.

Original: The stronger push towards hexagonality that isotopy exerts will make deviations from perfect hexagonality, such as shearing Stensola et al. (2015) less optimal.
Revised: The stronger push towards hexagonality that isotopy exerts will make deviations from perfect hexagonality, such as shearing Stensola et al. (2015) less optimal.

Original: A careful analysis of the behaviour of the two losses, both with functional, actionable, and biological constraints, but adding or removing isotropy, in different rooms could produce different predictions.
Revised: A careful analysis of the behaviour of the two losses, both with functional, actionable, and biological constraints, but adding or removing isotropy, in different rooms could produce different predictions.

Original: Further, each module would be more robustly hexagonal with this term added, which might be necessary to make many many modules hexagonal.
Revised: Further, each module would be more robustly hexagonal with this term added, which might be necessary to make many many modules hexagonal.

Original: Finally, the relation between our non-negative ﬁring rates, and the concept of non-negativity in Gao et al. is slightly murky.
Revised: Finally, the relation between our non-negative ﬁring rates, and the concept of non-negativity in Gao et al. is slightly murky.

Original: Gao et al.
Revised: Gao et al.

Original: use non-negative grid-to-place-cell weights, but also show this non-negativity is not required to produce grids.
Revised: use non-negative grid-to-place-cell weights, but also show this non-negativity is not required to produce grids.

Original: How this meshes with our ﬁndings that nonnegativity is vital for grid formation remains unclear.
Revised: How this meshes with our ﬁndings that nonnegativity is vital for grid formation remains unclear.

Original: This discrepancy could be caused by the isotropy constraint, but we remain unsure.
Revised: This discrepancy could be caused by the isotropy constraint, but we remain unsure.

Original: Theoretically, Gao et al. develop a rich group theory view for studying grid cells.
Revised: Theoretically, Gao et al. develop a rich group theory view for studying grid cells.

Original: We add to this work using Representation theory to make clear what constraints linear actionability implies (equation 5).
Revised: We add to this work using Representation theory to make clear what constraints linear actionability implies (equation 5).

Original: Given this starting point we are able to explain intuitively and with analytic insight why this loss leads to grid cells, extend to multiple modules, and make neural predictions.
Revised: Given this starting point we are able to explain intuitively and with analytic insight why this loss leads to grid cells, extend to multiple modules, and make neural predictions.

Original: Whittington et al.
Revised: Whittington et al.

Original: train a recurrent neural network to path integrate, but also to separate the representation of different points in the input space by predicting different observations at different positions.
Revised: train a recurrent neural network to path integrate, but also to separate the representation of different points in the input space by predicting different observations at different positions.

Original: Path integration is linear with action dependent matrices, exactly like our actionability, and the neural activities are bounded.
Revised: Path integration is linear with action dependent matrices, exactly like our actionability, and the neural activities are bounded.

Original: In recent versions of the model, Whittington et al.
Revised: In recent versions of the model, Whittington et al.

Original: (2021), non-negative ﬁring rates are used, which gave hexagonal grid cells.
Revised: (2021), non-negative ﬁring rates are used, which gave hexagonal grid cells.

Original: This can be understood as a direct application of our framework, though the pressures on the representation are naturally more blurry since it is optimised with behavioural trajectories using a recurrent neural network.
Revised: This can be understood as a direct application of our framework, though the pressures on the representation are naturally more blurry since it is optimised with behavioural trajectories using a recurrent neural network.

Original: In particular, the goldilocks frequency bounds (section 3.2) can be thought of as coming from the ﬁnite map the agent explores (high frequency bias) and the discrete step-size in the environment, which creates a low frequency bias.
Revised: In particular, the goldilocks frequency bounds (section 3.2) can be thought of as coming from the ﬁnite map the agent explores (high frequency bias) and the discrete step-size in the environment, which creates a low frequency bias.

Original: H. S ORSCHER , M EL , ET AL .’ S N ON -L INEAR P ATH I NTEGRATION M ODEL
Revised: H. S ORSCHER , M EL , ET AL .’ S N ON -L INEAR P ATH I NTEGRATION M ODEL

Original: Sorscher et al.
Revised: Sorscher et al.

Original: (2019) provide a nice theoretical analysis of how non-negativity and  reconstruction ofmexican-hat place cells can lead to hexagonal grids, providing valuable insight into previous observations in the ﬁeld such as that of Dordek et al.
Revised: (2019) provide a nice theoretical analysis of how non-negativity and linear reconstruction of place cells can lead to hexagonal grids, providing valuable insight into previous observations in the ﬁeld such as that of Dordek et al.

Original: At one level the arguments that lead to hexagons in our work and  that  of Sorscher et al.
Revised: The key difference between our theoretical analysis and theirs, is that they rely on the covariance structure of place cell input to obtain a representation made from plane waves, whereas we show that actionability (path integration) alone demands a representation built from plane waves.

Original: (2019) are similar, both rely on a band-pass ﬁlter. However, the mechanism by which this ﬁlter arises, and the way in which it leads to hexagons are both quite different between the two works. First, our bandpass ﬁlter arises because animals only want to encode a bounded region of space, and they want to encode it only up to  some useful resolution.
Revised: The reliance on a particular place cell covariance structure makes a strict constraint on the form place cells take - whether this is compatible with observed place cells is a topic of much debate (Schaeffer et al., 2022; Sorscher et al., 2022). The subsequent arguments that lead to hexagonal grid cells bear some resemblance, but are different:

Original: Sorscher et al.
Revised: Sorscher et al.

Original: (2019)’s arise due toreconstructing particular place cells, whose fourier transform is peaked in a small band of frequencies (probably unlike those in the brain!
Revised: (2019) discuss how ﬁnite room effects (i.e. the animal only exploring a limited region of space) lead to a discretisation of Fourier space onto a lattice.

Original: Schaeffer et al.
Revised: Their place cell covariance structure argument then says allowed frequencies should lie on a ring since the spectrum is peaked in a small band of frequencies. Lastly, including a 3rd order regularisation on the code (such as non-negativity) induces a triplet interaction between frequencies on the ring, meaning they should add to zero - this is satisﬁed by a hexagonal code. Our argument also leads to lattice frequencies, but this is due to actionability and non-negativity.

Original: Second, in our work non-negativity leads us to griddy tuning curves, and the  band-pass ﬁlter  chooses hexagons from amongst all 2D grids. In contrast, Sorscher et al.
Revised: Then, from the space of all lattices, a band-pass ﬁlter argument chooses hexagons: the high-pass arises because the animal only explores a limited region of space, and the low-pass because the animal only wants to encode space up to some useful resolution.

Original: (2019) argue that non-negativity leads to triplet interactions that pick out a hexagonal pattern.
Revised: This creates a preferred an annulus in frequency space, and leads to a preference for a hexagonal lattice, since it most densely packs frequencies into the annulus that is optimal for encoding space. So the arguments have similar components - lattice frequencies, non-negativity, band-pass ﬁltersbut they put these pieces together in different ways.

Original: These technical differencs, among others, lead our two theories to make different predictions.
Revised: These technical differences, among others, lead our two theories to make different predictions.

Original: For example, the alignment of grid axes in our model is explained normatively, unlike in Sorscher et al.

Original: Further, we provide a normative derivation of multiple modules, which we believe only arise in Sorscher et al.
Revised: For example, we provide a normative derivation of multiple modules, which we believe only arise in Sorscher et al.

Original: due to a quantization effect.
Revised: due to a quantization effect.

Original: Finally, and crucially in our opinion, our predictions result from a normative framework for understanding generic representations.
Revised: Finally, and crucially in our opinion, our predictions result from a normative framework for understanding generic representations.

Original: I A BLATION S TUDIES
Revised: I A BLATION S TUDIES

Original: We have argued that three principles - biological, functional, and actionable - are necessary to produce grid cells.
Revised: We have argued that three principles - biological, functional, and actionable - are necessary to produce grid cells.

Original: In this section we show that removing any of these three and numerically optimising produces fundamentally different patterns, and we explain these results with the theory we’ve developed in this work.
Revised: In this section we show that removing any of these three and numerically optimising produces fundamentally different patterns, and we explain these results with the theory we’ve developed in this work.

Original: We study the representation of a small torus, since for periodic spaces the three terms appear cleanly in the loss, so can be easily excised, Appendix B. Figure 15 shows that removing any term stops grid cells emerging.
Revised: We study the representation of a small torus, since for periodic spaces the three terms appear cleanly in the loss, so can be easily excised, Appendix B. Figure 15 shows that removing any term stops grid cells emerging.

Original: No Non-negativity Without non-negativity the code wants to maximise the size of each of the limited number of frequencies in the code, ensuring that they are non-harmonically related.
Revised: No Non-negativity Without non-negativity the code wants to maximise the size of each of the limited number of frequencies in the code, ensuring that they are non-harmonically related.

Original: As such, it builds a wavey but fairly random looking code.
Revised: As such, it builds a wavey but fairly random looking code.

Original: No Functional Without a functional term the loss simply wants the code to contain few frequencies and be non-negative, so produces random looking responses with a sparse fourier spectrum and a large constant offset to ensure non-negativity.
Revised: No Functional Without a functional term the loss simply wants the code to contain few frequencies and be non-negative, so produces random looking responses with a sparse fourier spectrum and a large constant offset to ensure non-negativity.

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.