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 ). 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. ). 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 ). 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, G1(θ,b,δ) and G2(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 . shows an example of G, where the elements of G₁ are marked with black circles, and the elements of G₂ are marked with red diamonds. The three grid parameters, height, and center are also shown. The latitudes of the grid have slope m=tanθ. 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 .

Motion of the Rat We model experiments in which a rat explores a rectangular enclosure. Before determining the behavior of the grid cells, we must simulate the rat's motion within its enclosure. We begin by uniformly discretizing over time and setting the parameters defined in and . s = ( s 1 , s 2 ) (width, height) of enclosure r = ( r 1 , r 2 ) rat's dimensions x = ( x 1 , x 2 ) rat's position v = ( v 1 , v 2 ) rat's velocity a = ( a 1 , a 2 ) rat's acceleration Δ t length of each time interval

We let ti=iΔt∀i>0. We assume the rat's acceleration is constant over each time interval and set ai=a(t) for ti≤t<ti+1. We begin by initializing the rat's position and velocity, x(t0) and v(t0). Then, ∀i>0, given x(ti) and v(ti), we follow the algorithm outlined below to find x(ti+1) and v(ti+1). This algorithm is adapted from the algorithm presented by Samsonovich and Ascoli . Algorithm: Simulation of the Rat's Motion Select a˜i from a Gaussian distribution with mean μ=0 and standard deviation σ. Set x˜(ti+1)=x(ti)+v(ti)Δt+a˜i(Δt)2/2. If rj≤xj˜(ti+1)≤sj-rj for j=1,2, the proposed new position, x˜(ti+1), is within the enclosure. Set x(ti+1)=x˜(ti+1) and ai=a˜i. Proceed to step (4). If not, the proposed position is outside the enclosure. Reject x˜(ti+1) and a˜i and repeat steps (1) through (3). Set v(ti+1)=v(ti)+aΔt. By following the above algorithm, the rat's motion can be simulated for any finite length of time. There are two modifications to the above algorithm, however, that make the resulting path more realistic. First, specify a maximum velocity, vmax. In step (4), if |v(ti+1)|≥|vmax|, reduce v(ti+1) to 90 percent of its value, effectively forcing the rat to decelerate over the next time interval. Second, if an acceptable value for a˜i has not been found after several tries, set v(ti)=(0,0). This signifies that the rat has been stopped by a wall of the enclosure.

Distance to the Grid We define the distance of the rat to the grid using the metric d ( x , G ( θ , b , δ ) ) = min { | x - y | : y ∈ G ( θ , b , δ ) } , where x denotes the position of the rat. We follow the algorithm outlined below to efficiently calculate d(x,G). All formulas given are derived by examining the geometry of the grid and are based on the assumption that θ≠0. and each show an example of G(θ,b,δ) that may assist in the understanding of the algorithm.

Algorithm: Calculation of d(x,G) Find p=(p1,p2), the projection of x onto the meridian, given by p1=m2x1-mx2+mc2+c1m2+1, and p2=c2-p1-c1m, where c=(c1,c2). We have drawn a large square around this projection in . Compute the two distances ℓ1≡|p-c| and ℓ2≡|p-x| and the associated integers n1= floor (ℓ1/h) and n2= floor ℓ2(b/2). These values permit us to box in the position, x. The four lines used to create this box are shown by dotted lines in and are given by b1k=(y1,y2)∈ℜ2:y2=q2k+(y1-q1k)m, for k=0,1,b2k=(y1,y2)∈ℜ2:y2=w2k-(y1-w1k)/m, for k=0,1, where (q1k,q2k)=c+sign(p2-c2)(n1+k)h(-sinθ,cosθ) and (w1k,w2k)=p+sign(x1-p1)(n2+k)(b/2)(cosθ,sinθ). The line b₁^k parallels the latitudes, and the line b₂^k parallels the meridian, for k=0,1. Find g1∈G1 and g2∈G2, the two elements of the grid G(θ,b,δ) that lie on the boundary of the box. These two vertices are the closest vertices to the rat with respect to the metric d(x,G). The four corners of the box are given by the four intersections of the lines b₁^k and b₂^k for k=0,1. For j=0,1, if n_j is even, b_j⁰ crosses only elements of G₁, and b_j¹ crosses only elements of G2. On the other hand, if n_j is odd, b_j⁰ crosses only elements of G2, and b_j¹ crosses only elements of G1. Thus, the two corners of the box corresponding to the grid vertices g₁ and g₂ form a diagonal pair given by solving b10(g)=b20(g) and b11(g')=b21(g') if n1+n2 is even ,b10(g)=b21(g) and b11(g')=b20(g') if n1+n2 is odd . Calculate d2(x,G(θ,b,δ))=min{|x-g1|2,|x-g2|2}.

Firing Pattern All that remains is to determine the firing pattern of the grid cell. Each grid cell has an efficacy value, ε, that incorporates a refractory period. When the cell spikes, ε is set to 0. Then, ε recovers exponentially to its base value of 1 at a rate determined by the time constant τ, ε ( t ) = 1 - exp ( - ( t - t s ) / τ ) , where t_s is the time of the most recent spike of the cell. The probability, P(x,G,ε,γ), that a grid cell fires is then taken from a Gaussian distribution with a width dependent on the rat's distance to the grid, the grid's base, the cell's efficacy value, and the scaling parameter γ, P ( x , G , ε , γ ) = exp - ε d 2 ( x , G ) γ b 2 . The parameter γ is a constant used to determine the spread of the firing fields of the grid cell. After calculating the probability P, we select a random number η from a uniform distribution. If P≥η, the grid cell fires; if P<η, the grid cell does not fire. Figure 4 shows the resulting firing fields for three different values of γ. As expected, the firing fields of the grid cell form hexagonal patterns as the rat explores its enclosure.