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8.3 A model of a grid cell

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The brain has the ability to build an internal map of its environment, and an active area of research concerns understanding how the brain performs this important function. One type of cell believed to play a vital role is known as a grid cell. Many experiments are being performed in which the researcher records the firing patterns of grid cells as a rat explores an environment, and it is often necessary to simulate these experiments. This module explains how to set the grid, simulate the rat's motion, calculate the rat's distance to the grid, and determine the firing pattern 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

0 ≤ θ < π / 3 ,

the base length

0 < b < ∞ ,

and the offset δ = ( ρ , φ ) , which has a magnitude

0 < ρ < b

and a direction

0 ≤ φ < 2 π .

We will make frequent use of the grid height

h = b tan ( π / 3 ) / 2

and grid center

c = ( ρ cos φ , ρ sin φ ) .

The set of grid points, G ( θ , b , δ ) , forms hexagonal patterns. This set is the union of two sets, G 1 ( θ , b , δ ) and G 2 ( θ , b , δ ) , that are staggered with respect to each other. The set is defined by

G ( θ , b , δ ) = G 1 ( θ , b , δ ) ⋃ G 2 ( θ , b , δ ) ,

where

G 1 ( θ , b , δ ) = c + k b ( cos θ , sin θ ) + 2 j h ( - sin θ , cos θ ) : j , k ∈ Z ,
G 2 ( θ , b , δ ) = c + ( k + ( 1 / 2 ) ) b ( cos θ , sin θ ) + ( 2 j - 1 ) h ( - sin θ , cos θ ) : j , k ∈ Z .

[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 θ . The grid meridian, M , is the line that intersects the grid center and runs perpendicular to the grid latitudes, given by

M = ( y 1 , y 2 ) ∈ ℜ 2 : y 2 = c 2 - y 1 - c 1 m if θ ≠ 0 ,
M = ( y 1 , y 2 ) ∈ ℜ 2 : y 1 = 0 if θ = 0 .

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

Graphical demonstration of parameters. This plot shows an example of a grid, G ( θ , b , δ ) , where θ = 0 . 925 , b = 1 . 146 , ρ = 0 . 496 , and φ = 0 . 925 . The black circles and red diamonds represent the elements of G 1 and G 2 , respectively. We have drawn a large circle around the grid center, c .
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Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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