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 1

