On Path Integration of Grid Cells: Group Representation and Isotropic Scaling

1 Introduction . Imagine walking in the darkness . Purely based on the sense of self-motion , one can gain a sense of self-position by integrating the self motion - a process often referred to as path integration [ 10 , 14 , 21 , 15 , 27 ] . While the exact neural underpinning of path integration remains unclear , it has been hypothesized that the grid cells [ 21 , 17 , 40 , 24 , 23 , 12 ] in the mammalian medial entorhinal cortex ( mEC ) may be involved in this process [ 20 , 30 , 22 ] . The grid cells are so named because individual neurons exhibit striking firing patterns that form hexagonal grids when the agent ( such as a rat ) navigates in a 2D open field [ 18 , 21 , 16 , 6 , 34 , 5 , 7 , 11 , 29 , 1 ] . The grid cells also interact with the place cells in the hippocampus [ 28 ] . Unlike a grid cell that fires at the vertices of a lattice , a place cell often fires at a single ( or a few ) locations . The purpose of this paper is to understand how the grid cells may perform path integration calculations . We study a general optimization-based representational model in which the 2D self-position is ∗The author is now a Research Scientist at Google Brain team . 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . represented by a higher dimensional vector and the 2D self-motion is represented by a transformation of the vector . The vector representation can be considered position encoding or position embedding , where the elements of the vector may be interpreted as activities of a population of grid cells . The transformation can be realized by a recurrent network that acts on the vector . Our focus is to study the properties of the transformation . Specifically , we identify two conditions for the transformation : a group representation condition and an isotropic scaling condition , under which we demonstrate that the local neighborhood around each self-position in the 2D physical space is embedded conformally as a 2D neighborhood around the vector representation of the self-position in the neural space . We then investigate the simplest special case of the transformation , i.e. , linear transformation , that forms a matrix Lie group of rotation , under which case we show that the isotropic scaling condition is connected to a special class of hexagonal grid patterns . Our numerical experiments demonstrate that our model learns clear hexagon grid patterns of multiple scales which share observed properties of the grid cells in the rodent brain , by optimizing a simple loss function . The learned model is also capable of accurate long distance path integration . Contributions . Our work contributes to understanding the grid cells from the perspective of representation learning . We conduct theoretical analysis of ( 1 ) general transformation for path integration by identifying two key conditions and a local conformal embedding property , ( 2 ) linear transformation by revealing the algebraic and geometric structure and connecting the isotropic scaling condition and a special class of hexagon grid patterns , and ( 3 ) integration of linear transformation model and linear basis expansion model via unitary group representation theory . Experimentally we learn clear hexagon grid patterns that are consistent with biological observations , and the learned model is capable of accurate path integration . 2 General transformation 2.1 Position embedding Consider an agent ( e.g. , a rat ) navigating within a 2D open field . Let x = ( x1 , x2 ) be the selfposition of the agent . We assume that the selfposition x in the 2D physical space is represented by the response activities of a population of d neurons ( e.g. , d = 200 ) , which form a vector v ( x ) = ( vi ( x ) , i = 1 , ... , d ) > in the ddimensional “ neural space ” , with each element vi ( x ) representing the firing rate of one neuron when the animal is at location x. v ( x ) can be called position encoding or position embedding . Collectively , ( v ( x ) , ∀x ) forms a codebook of x ∈ R2 , and ( v ( x ) , ∀x ) is a 2D manifold in the d-dimensional neural space , i.e. , globally we embed R2 as a 2D manifold in the neural space . Locally , we identify two condi- tions under which the 2D local neighborhood around each x is embedded conformally as a 2D neighborhood around v ( x ) with a scaling factor . See Fig . 1 . As shown in Section 3.3 , the conformal embedding is connected to the hexagon grid patterns . 2.2 Transformation and path integration . At self-position x , if the agent makes a self-motion ∆x = ( ∆x1 , ∆x2 ) , then it moves to x+∆x . Correspondingly , the vector representation v ( x ) is transformed to v ( x+∆x ) . The general form of the transformation can be formulated as : v ( x+∆x ) = F ( v ( x ) , ∆x ) . ( 1 ) The transformation F ( · , ∆x ) can be considered a representation of ∆x , which forms a 2D additive group . We call Eq . ( 1 ) the transformation model . It can be implemented by a recurrent network to derive a path integration model : if the agent starts from x0 , and makes a sequence of moves ( ∆xt , t = 1 , ... , T ) , then the vector is updated by vt = F ( vt−1 , ∆xt ) , where v0 = v ( x0 ) , and t = 1 , ... , T . 2.3 Group representation condition . The solution to the transformation model ( Eq . ( 1 ) ) should satisfy the following condition . Condition 1 . ( Group representation condition ) ( v ( x ) , ∀x ) and ( F ( · , ∆x ) , ∀∆x ) form a representation of the 2D additive Euclidean group R2 in the sense that F ( v ( x ) ,0 ) = v ( x ) , ∀x ; ( 2 ) F ( v ( x ) , ∆x1 +∆x2 ) = F ( F ( v ( x ) , ∆x1 ) , ∆x2 ) , ∀x , ∆x1 , ∆x2 . ( 3 ) ( F ( · , ∆x ) , ∀∆x ) is a Lie group of transformations acting on the codebook manifold ( v ( x ) , ∀x ) . The reason for ( 2 ) is that if ∆x = 0 , then F ( ·,0 ) should be the identity transformation . Thus the codebook manifold ( v ( x ) , ∀x ) consists of fixed points of the transformation F ( ·,0 ) . If F ( ·,0 ) is furthermore a contraction around ( v ( x ) , ∀x ) , then ( v ( x ) , ∀x ) are the attractor points . The reason for ( 3 ) is that the agent can move in one step by ∆x1 +∆x2 , or first move by ∆x1 , and then move by ∆x2 . Both paths would end up at the same x+∆x1 +∆x2 , which is represented by the same v ( x+∆x1 +∆x2 ) . The group representation condition is a necessary self-consistent condition for the transformation model ( Eq . ( 1 ) ) . 2.4 Egocentric self-motion . Self-motion ∆x can also be parametrized egocentrically as ( ∆r , θ ) , where ∆r is the displacement along the direction θ ∈ [ 0,2π ] , so that ∆x= ( ∆x1 = ∆r cosθ , ∆x2 = ∆r sinθ ) . The egocentric self-motion may be more biologically plausible where θ is encoded by head direction , and ∆r can be interpreted as the speed along direction θ . The transformation model then becomes v ( x+∆x ) = F ( v ( x ) , ∆r , θ ) , ( 4 ) where we continue to use F ( · ) for the transformation ( with slight abuse of notation ) . ( ∆r , θ ) form a polar coordinate system around x . 2.5 Infinitesimal self-motion and directional derivative . In this subsection , we derive the transformation model for infinitesimal self-motion . While we use ∆x or ∆r to denote finite ( non-infinitesimal ) self-motion , we use δx or δ r to denote infinitesimal self-motion . At self-position x , for an infinitesimal displacement δ r along direction θ , δx= ( δx1 = δ r cosθ , δx2 = δ r sinθ ) . See Fig . 1 ( a ) for an illustration . Given that δ r is infinitesimal , for any fixed θ , a first order Taylor expansion of F ( v ( x ) , δ r , θ ) with respect to δ r gives us v ( x+δx ) = F ( v ( x ) , δ r , θ ) = F ( v ( x ) ,0 , θ ) +F ′ ( v ( x ) ,0 , θ ) δ r+o ( δ r ) = v ( x ) + fθ ( v ( x ) ) δ r+o ( δ r ) , ( 5 ) where F ( v ( x ) ,0 , θ ) = v ( x ) according to Condition 1 , and fθ ( v ( x ) ) : = F ′ ( v ( x ) ,0 , θ ) is the first derivative of F ( v ( x ) , ∆r , θ ) with respect to ∆r at ∆r = 0. fθ ( v ( x ) ) is the directional derivative of F ( · ) at self-position x and direction θ . For a fixed θ , ( F ( · , ∆r , θ ) , ∀∆r ) forms a one-parameter Lie group of transformations , and fθ ( · ) is the generator of its Lie algebra . 2.6 Isotropic scaling condition . With the directional derivative , we define the second condition as follows , which leads to locally conformal embedding and is connected to hexagon grid pattern . Condition 2 . ( Isotropic scaling condition ) For any fixed x , ‖ fθ ( v ( x ) ) ‖ is constant over θ . Let f0 ( v ( x ) ) denote fθ ( v ( x ) ) for θ = 0 , and fπ/2 ( v ( x ) ) denote fθ ( v ( x ) ) for θ = π/2 . Then we have the following theorem : Theorem 1 . Assume group representation condition 1 and isotropic scaling condition 2 . At any fixed x , for the local motion δx= ( δ r cosθ , δ r sinθ ) around x , let δv = v ( x+δx ) −v ( x ) be the change of vector and s = ‖ fθ ( v ( x ) ) ‖ , then we have ‖δv‖= s‖δx‖ . Moreover , δv = fθ ( v ( x ) ) δ r+o ( δ r ) = f0 ( v ( x ) ) δ r cosθ + fπ/2 ( v ( x ) ) δ r sinθ +o ( δ r ) , ( 6 ) where f0 ( v ( x ) ) and fπ/2 ( v ( x ) ) are two orthogonal basis vectors of equal norm s. See Supplementary for a proof and Fig . 1 ( b ) for an illustration . Theorem 1 indicates that the local 2D polar system around self-position x in the 2D physical space is embedded conformally as a 2D polar system around vector v ( x ) in the d-dimensional neural space , with a scaling factor s ( our analysis is local for any fixed x , and s may depend on x ) . Conformal embedding is a generalization of isometric embedding , where the metric can be changed by a scaling factor s. If s is globally constant for all x , then the intrinsic geometry of the codebook manifold ( v ( x ) , ∀x ) remains Euclidean , i.e. , flat . Why isotropic scaling and conformal embedding ? The neurons are intrinsically noisy . During path integration , the errors may accumulate in v. Moreover , when inferring self-position from visual image , it is possible that v is inferred first with error , and then x is decoded from the inferred v. Due to isotropic scaling and conformal embedding , locally we have ‖δv‖= s‖δx‖ , which guarantees that the ` 2 error in v translates proportionally to the ` 2 error in x , so that there will not be adversarial perturbations in v ( x ) that cause excessively big errors in x . Specifically , we have the following theorem . Theorem 2 . Assume the general transformation model ( Eq . ( 4 ) ) and the isotropic scaling condition . For any fixed x , let s = ‖ fθ ( v ( x ) ) ‖ , which is independent of θ . Suppose the neurons are noisy : v = v ( x ) + ε , where ε ∼N ( 0 , τ2Id ) and d is the dimensionality of v. Suppose the agent infers its 2D position x̂ from v by x̂ = argminx′ ‖v−v ( x′ ) ‖2 , i.e. , v ( x̂ ) is the projection of v onto the 2D manifold formed by ( v ( x′ ) , ∀x′ ) . Then we have E‖x̂−x‖2 = 2τ2/s2 . ( 7 ) See Supplementary for a proof . Connection to continuous attractor neural network ( CANN ) defined on 2D torus . The group representation condition and the isotropic scaling condition appear to be satisfied by the CANN models [ 2 , 6 , 7 , 29 , 1 ] that are typically hand-designed on a 2D torus . See Supplementary for details .

Summary: The paper derives a model of grid cells from first principles, starting with the basic assumption that position is represented as a high-D vector and that it is updated as a function of direction and speed.  The authors stipulate an isotropic scaling condition in order to make the representation robust to neural noise.  The elements of the resulting high-D embedding resemble grid cells, and importantly, the ability for error correction during path integration.