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 [1]). 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. [3]). 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 [4]). 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.
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
,
0
≤
θ
<
π
/
3
,
(1)
the base length
0
<
b
<
∞
,
0
<
b
<
∞
,
(2)
and the offset δ=(ρ,φ),δ=(ρ,φ), which has a magnitude
0
<
ρ
<
b
0
<
ρ
<
b
(3)
and a direction
0
≤
φ
<
2
π
.
0
≤
φ
<
2
π
.
(4)
We will make frequent use of the grid height
h
=
b
tan
(
π
/
3
)
/
2
h
=
b
tan
(
π
/
3
)
/
2
(5)
and grid center
c
=
(
ρ
cos
φ
,
ρ
sin
φ
)
.
c
=
(
ρ
cos
φ
,
ρ
sin
φ
)
.
(6)
The set of grid points, G(θ,b,δ),G(θ,b,δ), forms hexagonal patterns. This set is the union of two sets, G1(θ,b,δ)G1(θ,b,δ) and G2(θ,b,δ),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
)
,
G
(
θ
,
b
,
δ
)
=
G
1
(
θ
,
b
,
δ
)
⋃
G
2
(
θ
,
b
,
δ
)
,
(7)
where
G
1
(
θ
,
b
,
δ
)
=
c
+
k
b
(
cos
θ
,
sin
θ
)
+
2
j
h
(
-
sin
θ
,
cos
θ
)
:
j
,
k
∈
Z
,
G
1
(
θ
,
b
,
δ
)
=
c
+
k
b
(
cos
θ
,
sin
θ
)
+
2
j
h
(
-
sin
θ
,
cos
θ
)
:
j
,
k
∈
Z
,
(8)
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
.
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
.
(9)
Figure 1 shows an example of G,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.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
2
=
c
2
-
y
1
-
c
1
m
if
θ
≠
0
,
(10)
M
=
(
y
1
,
y
2
)
∈
ℜ
2
:
y
1
=
0
if
θ
=
0
.
M
=
(
y
1
,
y
2
)
∈
ℜ
2
:
y
1
=
0
if
θ
=
0
.
(11)
The grid meridian is marked with a bold line in Figure 4.
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 Table 1 and Figure 2.
Table 1: Parameter Definitions. The rat's motion is determined by its position, velocity, and acceleration vectors. We assume the rat is confined in a rectangular enclosure defined by its width s1 and height s2.s2. We also assume the rat is rectangular and specify its its radius r=(r1,r2),r=(r1,r2), where r1 is half the rat's length, and r2 is half the rat's width. We take the rat's position, x,x, to be the location of the rat's center.
|
s
=
(
s
1
,
s
2
)
s
=
(
s
1
,
s
2
)
|
(width, height) of enclosure |
|
r
=
(
r
1
,
r
2
)
r
=
(
r
1
,
r
2
)
|
rat's dimensions |
|
x
=
(
x
1
,
x
2
)
x
=
(
x
1
,
x
2
)
|
rat's position |
|
v
=
(
v
1
,
v
2
)
v
=
(
v
1
,
v
2
)
|
rat's velocity |
|
a
=
(
a
1
,
a
2
)
a
=
(
a
1
,
a
2
)
|
rat's acceleration |
|
Δ
t
Δ
t
|
length of each time interval |
We let ti=iΔt∀i>0.ti=iΔt∀i>0. We assume the rat's acceleration is constant over each time interval and set ai=a(t)ai=a(t) for ti≤t<ti+1.ti≤t<ti+1. We begin by initializing the rat's position and velocity, x(t0)x(t0) and v(t0).v(t0). Then, ∀i>0∀i>0, given x(ti)x(ti) and v(ti)v(ti), we follow the algorithm outlined below to find x(ti+1)x(ti+1) and v(ti+1).v(ti+1). This algorithm is adapted from the algorithm presented by Samsonovich and Ascoli [2].
Algorithm: Simulation of the Rat's Motion
- Select a˜ia˜i from a Gaussian distribution with mean μ=0μ=0 and standard deviation σ.σ.
- Set x˜(ti+1)=x(ti)+v(ti)Δt+a˜i(Δt)2/2.x˜(ti+1)=x(ti)+v(ti)Δt+a˜i(Δt)2/2.
- If rj≤xj˜(ti+1)≤sj-rjrj≤xj˜(ti+1)≤sj-rj for j=1,2,j=1,2, the proposed new position, x˜(ti+1),x˜(ti+1), is within the enclosure. Set x(ti+1)=x˜(ti+1)x(ti+1)=x˜(ti+1) and ai=a˜i.ai=a˜i. Proceed to step (4). If not, the proposed position is outside the enclosure. Reject x˜(ti+1)x˜(ti+1) and a˜ia˜i and repeat steps (1) through (3).
- Set v(ti+1)=v(ti)+aΔt.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.vmax. In step (4), if |v(ti+1)|≥|vmax|,|v(ti+1)|≥|vmax|, reduce v(ti+1)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˜ia˜i has not been found after several tries, set v(ti)=(0,0).v(ti)=(0,0). This signifies that the rat has been stopped by a wall of the enclosure.
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
)
}
,
d
(
x
,
G
(
θ
,
b
,
δ
)
)
=
min
{
|
x
-
y
|
:
y
∈
G
(
θ
,
b
,
δ
)
}
,
(12)
where x denotes the position of the rat. We follow the algorithm outlined below to efficiently calculate d(x,G).d(x,G). All formulas given are derived by examining the geometry of the grid and are based on the assumption that θ≠0.θ≠0. Figure 3 and (Reference) each show an example of G(θ,b,δ)G(θ,b,δ) that may assist in the understanding of the algorithm.
Algorithm: Calculation of d(x,G)d(x,G)
- Find p=(p1,p2)p=(p1,p2), the projection of x onto the meridian, given by
p1=m2x1-mx2+mc2+c1m2+1, and p2=c2-p1-c1m,p1=m2x1-mx2+mc2+c1m2+1, and p2=c2-p1-c1m,(13)
where c=(c1,c2).c=(c1,c2). We have drawn a large square around this projection in Figure 4.
- Compute the two distances
ℓ1≡|p-c| and ℓ2≡|p-x|ℓ1≡|p-c| and ℓ2≡|p-x|(14)
and the associated integers
n1= floor (ℓ1/h) and n2= floor ℓ2(b/2).n1= floor (ℓ1/h) and n2= floor ℓ2(b/2).(15)
These values permit us to box in the position, x.x. The four lines used to create this box are shown by dotted lines in Figure 4 and are given by
b1k=(y1,y2)∈ℜ2:y2=q2k+(y1-q1k)m, for k=0,1,b1k=(y1,y2)∈ℜ2:y2=q2k+(y1-q1k)m, for k=0,1,(16)
b2k=(y1,y2)∈ℜ2:y2=w2k-(y1-w1k)/m, for k=0,1,b2k=(y1,y2)∈ℜ2:y2=w2k-(y1-w1k)/m, for k=0,1,(17)
where
(q1k,q2k)=c+sign(p2-c2)(n1+k)h(-sinθ,cosθ)(q1k,q2k)=c+sign(p2-c2)(n1+k)h(-sinθ,cosθ)(18)
and
(w1k,w2k)=p+sign(x1-p1)(n2+k)(b/2)(cosθ,sinθ).(w1k,w2k)=p+sign(x1-p1)(n2+k)(b/2)(cosθ,sinθ).(19)
The line b1k parallels the latitudes, and the line b2k parallels the meridian, for k=0,1.k=0,1. - Find g1∈G1g1∈G1 and g2∈G2,g2∈G2, the two elements of the grid G(θ,b,δ)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).d(x,G). The four corners of the box are given by the four intersections of the lines b1k and b2k for k=0,1.k=0,1. For j=0,1,j=0,1, if nj is even, bj0 crosses only elements of G1, and bj1 crosses only elements of G2.G2. On the other hand, if nj is odd, bj0 crosses only elements of G2,G2, and bj1 crosses only elements of G1.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)=b20(g) and b11(g')=b21(g') if n1+n2 is even ,(20)
b10(g)=b21(g) and b11(g')=b20(g') if n1+n2 is odd .b10(g)=b21(g) and b11(g')=b20(g') if n1+n2 is odd .(21)
- Calculate
d2(x,G(θ,b,δ))=min{|x-g1|2,|x-g2|2}.d2(x,G(θ,b,δ))=min{|x-g1|2,|x-g2|2}.(22)
All that remains is to determine the firing pattern of the grid cell. Each grid cell has an efficacy value, ε,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,τ,
ε
(
t
)
=
1
-
exp
(
-
(
t
-
t
s
)
/
τ
)
,
ε
(
t
)
=
1
-
exp
(
-
(
t
-
t
s
)
/
τ
)
,
(23)
where ts is the time of the most recent spike of the cell.
The probability, P(x,G,ε,γ)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,γ,
P
(
x
,
G
,
ε
,
γ
)
=
exp
-
ε
d
2
(
x
,
G
)
γ
b
2
.
P
(
x
,
G
,
ε
,
γ
)
=
exp
-
ε
d
2
(
x
,
G
)
γ
b
2
.
(24)
The parameter γ is a constant used to determine the spread of the firing fields of the grid cell. After calculating the probability P,P, we select a random number η from a uniform distribution. If P≥η,P≥η, the grid cell fires; if P<η,P<η, the grid cell does not fire. Figure 4 shows the resulting firing fields for three different values of γ.γ. As expected, the firing fields of the grid cell form hexagonal patterns as the rat explores its enclosure.
-
O'Keefe, J. and Nadel, L. (1978). The Hippocampus as a Cognitive Map. Clarendon: Oxford.
-
Samsonovich, A. and Ascoli, G. (2005). A simple neural network model of the hippocampus suggesting its pathfinding role in episodic memory retrieval. Learning and Memory, 12, 193-208.
-
Solstad, T. and Moser, E. and Einevoll, G. (2006). From grid cells to place cells: a mathematical model. Hippocampus, 16, 1021-1031.
-
Witter, M.P. and Moser, E.I. (2006). Spatial representation and the architecture of the entorhinal cortex. Trends in Neuroscience, 29, 671-678.
