A REPRESENTATIONAL MODEL OF GRID CELLS' PATH INTEGRATION BASED ON MATRIX LIE ALGEBRAS Anonymous

Abstract

The grid cells in the mammalian medial entorhinal cortex exhibit striking hexagon firing patterns when the agent navigates in the open field. It is hypothesized that the grid cells are involved in path integration so that the agent is aware of its selfposition by accumulating its self-motion. Assuming the grid cells form a vector representation of self-position, we elucidate a minimally simple recurrent model for grid cells' path integration based on two coupled matrix Lie algebras that underlie two coupled rotation systems that mirror the agent's self-motion: (1) When the agent moves along a certain direction, the vector is rotated by a generator matrix. (2) When the agent changes direction, the generator matrix is rotated by another generator matrix. Our experiments show that our model learns hexagonal grid response patterns that resemble the firing patterns observed from the grid cells in the 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.

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 R 2 ). Let x = (x 1 , x 2 ) 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 = (δx 1 , δx 2 ) = (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 (δx 1 , δx 2 ) 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) = (v i (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 R 2 as a 2D manifold in a higher dimensional space R d . Locally we embed the 2D local polar system centered at x (see figure 1a, b ) into R d 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), 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(δr 2 ). 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 v i (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), ( ) 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.



Figure 1: Illustration of the proposed representational model. (a) 2D local polar system centered at x for egocentric self-motion, to be embedded in R d . (b) 2D local displacement δr and local change of direction δθ. (c) Mirroring relations in (b). x is mirrored by v(x). Local displacement δr from x along direction θ is mirrored by B(θ)δr applied to v(x). Local change of direction δθ is mirrored by Cδθ applied to B(θ).

