8.3 A model of a grid cell

Introduction

A network spanning several regions in the brain provides the mechanisms for spatial representation. The hippocampus, a center for both learning and memory, is an important component of this network (O'Keefe and Nadel  [link] ). The majority of cells in the hippocampus will primarily spike at one location in an environment. These cells, known as place cells, can collectively represent the position within an environment (Solstad et al.  [link] ). One large source of input to place cells are a group of cells in the medial entorhinal cortex (MEC) known as grid cells, which primarily spike in hexagonal patterns in the environment (Witter and Moser  [link] ). Thus, in order to understand the brain's spatial representation of its environment, it is essential to understand the behavior of grid cells in the MEC, place cells in the hippocampus, and the interaction between them. The work presented here focuses on one portion of this problem: modelling grid cells. We consider the setting of a rat exploring a rectangular enclosure, but this model could easily be extended to a variety of settings.

Properties of the grid

To simulate the behavior of a grid cell, it is first necessary to set its grid. We parameterize a grid with three scalars, the tilt

the base length

and the offset $\delta =(\rho ,\phi ),$ which has a magnitude

and a direction

We will make frequent use of the grid height

and grid center

The set of grid points, $G(\theta ,b,\delta ),$ forms hexagonal patterns. This set is the union of two sets, $G_1(\theta ,b,\delta )$ and $G_2(\theta ,b,\delta ),$ that are staggered with respect to each other. The set is defined by

where

[link] shows an example of $G,$ where the elements of G ₁ are marked with black circles, and the elements of G ₂ are marked with red diamonds. The three grid parameters, height, and center are also shown.

The latitudes of the grid have slope $m=tan\theta .$ The grid meridian, M , is the line that intersects the grid center and runs perpendicular to the grid latitudes, given by

The grid meridian is marked with a bold line in [link] .