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 introduces a general framework for modeling 2D navigation via grid cells. The main idea is to embed a 2D position x into a higher-dimensional positional encoding v(x) and model 2D translations via group transformations in this embedding space. A specific case are linear transformations via rotation matrices that act on the embedding v(x). Under this assumption, it can be shown that hexagonal patterns which are characteristic for neural grid cells naturally emerge as an optimal encoding.