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 develop a model for learning the observed responses of grid cells (GC) in the entorhinal cortex from the animal movement vectors. Their key assumption is that the GC activity vector rotates with the movement magnitude according to the Lie group formalism and the corresponding Lie group generator is also rotated by the change in movement orientation. Their claim is supported by a numerical optimization of the objective function reflecting these assumptions as well as the projection onto the place cell representation in the hippocampus.