A Representational Model of Grid Cells' Path Integration Based on Matrix Lie Algebras

1 INTRODUCTION . Imagine walking in the darkness . Purely based on your sense of self-motion , you can gain a sense of self-position by integrating the self movement - a process often referred to as path integration ( Darwin , 1873 ; Etienne & Jeffery , 2004 ; Hafting et al. , 2005 ; Fiete et al. , 2008 ; McNaughton et al. , 2006 ) . While the exact neural underpinning of path integration remains unclear , it has been hypothesized that the grid cells ( Hafting et al. , 2005 ; Fyhn et al. , 2008 ; Yartsev et al. , 2011 ; Killian et al. , 2012 ; Jacobs et al. , 2013 ; Doeller et al. , 2010 ) in the mammalian medial entorhinal cortex ( mEC ) may be involved in this process ( Gil et al. , 2018 ; Ridler et al. , 2019 ; Horner et al. , 2016 ) . The grid cells are so named because individual neurons exhibit striking firing patterns that form hexagonal grid patterns when the agent ( such as a rat ) navigates in a 2D open field ( Fyhn et al. , 2004 ; Hafting et al. , 2005 ; Fuhs & Touretzky , 2006 ; Burak & Fiete , 2009 ; Sreenivasan & Fiete , 2011 ; Blair et al. , 2007 ; Couey et al. , 2013 ; de Almeida et al. , 2009 ; Pastoll et al. , 2013 ; Agmon & Burak , 2020 ) . The grid cells interact with the place cells in the hippocampus ( O ’ Keefe , 1979 ) . 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 ( or “ path integration calculations ” ) . We propose a representational model in which the self-position is represented by the population activity vector formed by grid cells , and the self-motion is represented by the rotation of this vector . Specifically , our model consists of two coupled systems : ( 1 ) When the agent moves along a certain direction , the vector is rotated by a generator matrix of a Lie algebra . ( 2 ) When the agent changes movement direction , the generator matrix itself is rotated by yet another generator matrix of a different Lie algebra . Our numerical experiments demonstrate that our model learns hexagon grid patterns which share many properties of the grid cells in the rodent brain . Furthermore , the learned model is capable of near exact path integration , and it is also capable of error correction . Our model is novel and simple , with explicit geometric and algebraic structures . The population activity vector formed by the grid cells rotates in the “ mental ” or neural space , monitoring the egocentric self-motion of the agent in the physical space . This model also connects naturally to the basis expansion model that decomposes the response maps of place cells as linear expansions of response maps of grid cells ( Dordek et al. , 2016 ; Sorscher et al. , 2019 ) . Overall , our model provides a new conceptual framework to study the grid cell systems in the brain by considering the structure of the intrinsic symmetry ( through Lie algebra ) of the task which the path integration system is solving . 2 REPRESENTATIONAL MODEL FOR PATH INTEGRATION . Consider an agent navigating within a squared domain ( theoretically the domain can be R2 ) . Let x = ( x1 , x2 ) be the self-position of the agent in a 2D environment . At self-position x , if the agent makes a displacement δr along the direction θ ∈ [ 0 , 2π ] , then the self-position is changed to x + δx , where δx = ( δx1 , δx2 ) = ( δr cos θ , δr sin θ ) . In our model , we use a polar coordinate system ( see figure 1a , b ) by directly using ( θ , δr ) , while only keeping ( δx1 , δx2 ) implicit . ( θ , δr ) is the biologically plausible egocentric representation of self-motion . We assume that the location x in the 2D environment is encoded by the response pattern of a population of d neurons ( e.g. , d = 200 ) , which correspond to a d-dimensional vector v ( x ) = ( vi ( x ) , i = 1 , ... , d ) > , with each element representing the firing rate of one neuron when the animal is at location x . From the embedding point of view , essentially we embed the 2D domain in R2 as a 2D manifold in a higher dimensional space Rd . Locally we embed the 2D local polar system centered at x ( see figure 1a , b ) into Rd so that it becomes a local system around v ( x ) ( see figure 1c ) . 2.1 THE PROPOSED REPRESENTATIONAL MODEL : COUPLING TWO ROTATION SYSTEMS . Assuming δr to be infinitesimal , we propose the following model v ( x + δx ) = ( I + B ( θ ) δr ) v ( x ) + o ( δr ) , ( 1 ) which parameterizes a recurrent neural network ( Hochreiter & Schmidhuber , 1997 ) , where I is the identity matrix , and B ( θ ) is a d-dimensional matrix depending on the direction θ , which will need to be learned . Rotation . We assume B ( θ ) = −B ( θ ) > , i.e. , skew-symmetric , so that I + B ( θ ) δr is a rotation or orthogonal matrix , due to that ( I + B ( θ ) δr ) ( I + B ( θ ) δr ) > = I +O ( δr2 ) . Because the upper triangle part of B ( θ ) is the negative of the transpose of the lower triangle part ( the diagonal elements are zeros ) , in fact we only need to learn its lower triangle part . The geometric interpretation is that , if the agent moves along the direction θ , the vector v ( x ) is rotated by the matrix B ( θ ) , while the ` 2 norm ‖v ( x ) ‖2 remains stable ( figure 1c ) . We may interpret ‖v ( x ) ‖2 = ∑d i=1 vi ( x ) 2 as the total energy of grid cells , which is stable across different locations . From embedding point of view , the local polar system in figure 1a is embedded into a d-dimensional sphere in neural response space . When the agent makes an infinitesimal change of direction from θ to θ + δθ , B ( θ ) is changed to B ( θ + δθ ) . We assume B ( θ + δθ ) = ( I + Cδθ ) B ( θ ) + o ( δθ ) , ( 2 ) where C is a d-dimensional matrix , which is also to be learned . We again assumes C = −C > , so that I + Cδθ is a rotation matrix . The geometric interpretation is that if the agent changes direction , B ( θ ) is rotated by C. Equations ( 1 ) and ( 2 ) together define our proposed model for path integration , which couples two rotation systems . 2.2 A MIRROR OF EGOCENTRIC MOTION : PRESERVING LOCAL GEOMETRIC RELATIONS . As a representational model , equations ( 1 ) and ( 2 ) form a mirror in the d-dimensional “ mental ” ( or neural ) space for the egocentric motion in the 2D physical space . Importantly , the embedding preserves the local geometric relations of the local polar system . Let δθv ( x ) = v ( x + δx ) − v ( x ) be the displacement of v ( x ) when the agent moves from x by δr along direction θ . It follows from equation ( 1 ) that δθv ( x ) = ( B ( θ ) δr + o ( δr ) ) v ( x ) . Ignoring high order terms , we obtain δθ+δθv ( x ) = ( I + Cδθ ) δθv ( x ) . That is , with δr fixed , the local change of v ( x ) along different θ are rotated versions of each other , mirroring the local polar system at x . See figure 1 for an illustration . As for the angle between v ( x ) and v ( x + δx ) , i.e. , how much the vector v rotates in the neural space as the agent moves in 2D physical space by δr , we have Proposition 1 In the above notation , let δα be the angle between v ( x ) and v ( x + δx ) , we have δα = βδr +O ( δr2 ) , where δr = ‖δx‖ , and β = ‖B ( θ ) v ( x ) ‖2/‖v ( x ) ‖2 is independent of θ . See Supplementary A.1 for a proof . That means , the angle δα in the d-dimensional neural space is proportional to the Euclidean distance δr in the 2D space , and more importantly the angle δα is independent of direction θ , i.e. , β is isotropic . 2.3 HEXAGON GRID PATTERNS . For the learned model , β can be much bigger than 1 , so that the vector v ( x ) will rotate back to itself in a short distance , causing the periodic patterns of v ( x ) . Moreover , β does not depend on the direction of self-motion , and this isotropic property appears to underly the emergent hexagonal periodic patterns , as suggested by the following result . The hexagon grid patterns can be created by linearly mixing three Fourier plane waves whose directions are 2π/3 apart . In the following , we state a theoretical result adapted from Gao et al . ( 2018 ) that connects such linearly mixed Fourier waves to the geometric property in Proposition 1 in the previous subsection . Proposition 2 Let e ( x ) = ( exp ( i〈aj , x〉 ) , j = 1 , 2 , 3 ) > , where ( aj , j = 1 , 2 , 3 ) are three 2D vectors of equal norm , and the angle between every pair of them is 2π/3 . Let v ( x ) = Ue ( x ) , where U is an arbitrary unitary matrix , i.e. , U∗U = I . Let δα be the angle between v ( x ) and v ( x + δx ) , we have δα = βδr + O ( δr2 ) , where δr = ‖δx‖ , and β ∝ ‖aj‖ is independent of the direction of δx . See Supplementary A.1 for a proof , which relies on the fact that ( aj , j = 1 , 2 , 3 ) forms a tight frame in 2D . Proposition 2 says that the geometric property that emerges from our model as elucidated by Proposition 1 is satisfied by the orthogonal mixing of three Fourier plane waves that creates hexagonal grid patterns . We are currently pursuing a more general analysis of our model , i.e. , equations ( 1 ) and ( 2 ) .

Summary: The authors propose a simple recurrent network as a model of spatial navigation in the MEC/Hippocampal network. This model assumes that grid cells only regularly receive egocentric movement information, an important aspect for understanding the origin of these functional cell types in-vitro. Overall, this article should be of interest for any ICLR members interested in biological  spatial navigation.