Modelling the Grid-like Encoding of Visual Space in Primates
Jochen Kerdels and Gabriele Peters
University of Hagen, Universit
¨
atsstrasse 1, D-58097 Hagen, Germany
Keywords:
Recursive Growing Neural Gas, Entorhinal Cortex, Primates, Grid-like Encoding, Grid Cell Model.
Abstract:
Several regions of the mammalian brain contain neurons that exhibit grid-like firing patterns. The most promi-
nent example of such neurons are grid cells in the entorhinal cortex (EC) whose activity correlates with the
animal’s location. Correspondingly, contemporary models of grid cells interpret this firing behavior as a spe-
cialized, functional part within a system for orientation and navigation. However, Killian et al. report on
neurons in the primate EC that show similar, grid-like firing patterns but encode gaze-positions in the field of
view instead of locations in the environment. We hypothesized that the phenomenon of grid-like firing pat-
terns may not be restricted to navigational tasks and may be related to a more general, underlying information
processing scheme. To explore this idea, we developed a grid cell model based on the recursive growing neural
gas (RGNG) algorithm that expresses this notion. Here we show that our grid cell model can – in contrast
to established grid cell models – also describe the observations of Killian et al. and we outline the general
conditions under which we would expect neurons to exhibit grid-like activity patterns in response to input
signals independent of a presumed, functional task of the neurons.
1 INTRODUCTION
Within the last decade neurons with grid-like firing
patterns were identified in several regions of the mam-
malian brain. Among these the so-called grid cells
are most prominent. Their firing pattern correlates
with the animal’s location resulting in a periodic, tri-
angular lattice of firing fields that cover the animal’s
environment. Grid cells were first discovered in the
entorhinal cortex of rats (Fyhn et al., 2004; Haft-
ing et al., 2005), and could later be shown to ex-
ist in mice (Domnisoru et al., 2013), bats (Yartsev
et al., 2011), and humans (Jacobs et al., 2013) as well.
Besides the entorhinal cortex (rats, mice, bats, hu-
mans), grid cells were also found in the pre- and para-
subiculum (rats) (Boccara et al., 2010), and the hip-
pocampus, parahippocampal gyrus, amygdala, cingu-
late cortex, and frontal cortex (humans) (Jacobs et al.,
2013), albeit in low numbers. In all reported cases
the observed firing patterns correlated with the ani-
mal’s location
1
. However, Killian et al. (Killian et al.,
2012) observed neurons with grid-like firing patterns
in the entorhinal cortex of primates that show a differ-
ent behavior. Instead of encoding positions in phys-
ical space the observed neurons encoded the visual
1
In some cases the subjects navigated a virtual environ-
ment (Domnisoru et al., 2013; Jacobs et al., 2013).
space of the animal with a grid-like pattern, i.e., the
activity of the neurons correlated with gaze-positions
in the animal’s field of view.
These observations suggests that grid-like neu-
ronal activity may be a phenomenon that is more
widespread and may represent a more general form
of information processing as assumed so far. Existing
computational models of grid cells (Moser and Moser,
2008; Welinder et al., 2008; Giocomo et al., 2011;
Barry and Burgess, 2014; Burak, 2014; Moser et al.,
2014) are based on priors that restrict their applica-
bility to the domain of path integration and naviga-
tion. Thus, to explore the hypothesis stated above we
developed a novel computational model based on the
recursive growing neural gas (RGNG) algorithm that
describes the behavior of neurons with grid-like activ-
ities without relying on priors that would restrict the
model to a single domain (Kerdels and Peters, 2013;
Kerdels and Peters, 2015b; Kerdels, 2016). Here we
show that our model is able to describe not only typi-
cal grid cells but also cells with similar grid-like firing
patterns that operate in different domains like the neu-
rons observed by Killian et al. (Killian et al., 2012).
The next section summarizes our RGNG-based
model and outlines the general conditions under
which the model’s neurons exhibit grid-like firing pat-
terns. Based on these conditions, a possible input sig-
42
Kerdels, J. and Peters, G.
Modelling the Grid-like Encoding of Visual Space in Primates.
DOI: 10.5220/0006045500420049
In Proceedings of the 8th International Joint Conference on Computational Intelligence (IJCCI 2016) - Volume 3: NCTA, pages 42-49
ISBN: 978-989-758-201-1
Copyright
c
 2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

nal that may underlie the phenomena reported by Kil-
lian et al. is described in section 3. Simulation results
demonstrating that such a signal can indeed cause the
observed grid-like firing patterns are presented in sec-
tion 4. Finally, conclusions reflecting on the results
are given in section 5.
2 GRID CELL MODEL
The majority of conventional grid cell models rely on
mechanisms that directly integrate information on the
velocity and direction of an animal into a periodic rep-
resentation of the animal’s location (Kerdels, 2016).
As a consequence, the particular models do not gen-
eralize well, i.e., they can not be used to describe or
investigate the behavior of neurons that receive other
kinds of input signals but may also exhibit grid-like
firing patterns. In contrast, the RGNG-based model
does not rely on specific types of information as input.
It describes the general behavior of a group of neurons
in response to inputs from arbitrary input spaces.
The main hypothesis of our RGNG-based model
is that the behavior observed in grid cells is just one
instance of a more general information processing
scheme. In this scheme each cell in a group of neu-
rons tries to learn the structure of its entire input space
while being in competition with its peers. Learn-
ing the input space structure is modelled on a per-
cell level with a growing neural gas – an unsuper-
vised learning algorithm that approximates the input
space structure with a network of units (Martinetz and
Schulten, 1994; Fritzke, 1995). Each unit in this net-
work is associated with a reference vector or proto-
type that represents a specific location in input space.
The network structure reflects the input space topol-
ogy, i.e., neighboring units in the network correspond
to neighboring regions of input space.
Interestingly, the competition between neurons
can be described by the same GNG dynamics as the
per-cell learning process resulting in a joint, recur-
sive model that describes both the learning processes
within each cell as well as the competition within a
group of neurons. At the model’s core lies the recur-
sive growing neural gas (RGNG) algorithm, which is
described formally in the next section. It represents
a generalization of the original GNG algorithm that
allows the unit’s prototypes to be entire GNGs them-
selves.
2.1 Recursive Growing Neural Gas
The recursive growing neural gas (RGNG) has essen-
tially the same structure as a regular GNG. Like a
GNG an RGNG g can be described by a tuple
2
:
g := (U,C, θ) ∈ G,
with a set U of units, a set C of edges, and a set θ of
parameters. Each unit u is described by a tuple:
u := (w, e) ∈ U, w ∈ W := R
n
∪ G, e ∈ R,
with the prototype w, and the accumulated error e.
Note that in contrast to the regular GNG the proto-
type w of an RGNG unit can either be a n-dimensional
vector or another RGNG. Each edge c is described by
a tuple:
c := (V, t) ∈ C, V ⊆ U ∧ |V | = 2, t ∈ N,
with the units v ∈ V connected by the edge and the
age t of the edge. The direct neighborhood E
u
of a
unit u ∈ U is defined as:
E
u
:=
{
k|∃ (V, t) ∈ C, V =
{
u, k
}
, t ∈ N
}
.
The set θ of parameters consists of:
θ :=
{
ε
b
, ε
n
, ε
r
, λ, τ, α, β, M
}
.
Compared to the regular GNG the set of parameters
has grown by θε
r
and θM. The former parameter is
a third learning rate used in the adaptation function
A (see below). The latter parameter is the maximum
number of units in an RGNG. This number refers
only to the number of “direct” units in a particular
RGNG and does not include potential units present in
RGNGs that are prototypes of these direct units.
Like its structure the behavior of the RGNG is ba-
sically identical to that of a regular GNG. However,
since the prototypes of the units can either be vectors
or RGNGs themselves, the behavior is now defined
by four functions. The distance function
D(x, y) : W ×W → R
determines the distance either between two vectors,
two RGNGs, or a vector and an RGNG. The interpo-
lation function
I(x, y) : (R
n
× R
n
) ∪ (G × G) → W
generates a new vector or new RGNG by interpolat-
ing between two vectors or two RGNGs, respectively.
The adaptation function
A(x, ξ, r) : W × R
n
× R → W
adapts either a vector or RGNG towards the input vec-
tor ξ by a given fraction r. Finally, the input function
F(g, ξ) : G × R
n
→ G × R
feeds an input vector ξ into the RGNG g and returns
the modified RGNG as well as the distance between ξ
and the best matching unit (BMU, see below) of g.
The input function F contains the core of the RGNG’s
behavior and utilizes the other three functions, but is
also used, in turn, by those functions introducing sev-
eral recursive paths to the program flow.
2
The notation gα is used to reference the element α
within the tuple.
Modelling the Grid-like Encoding of Visual Space in Primates
43

F(g, ξ): The input function F is a generalized ver-
sion of the original GNG algorithm that facilitates the
use of prototypes other than vectors. In particular, it
allows to use RGNGs themselves as prototypes result-
ing in a recursive structure. An input ξ ∈ R
n
to the
RGNG g is processed by the input function F as fol-
lows:
• Find the two units s
1
and s
2
with the smallest dis-
tance to the input ξ according to the distance func-
tion D:
s
1
:= arg min
u∈gU
D(uw, ξ) ,
s
2
:= arg min
u∈gU\
{
s
1
}
D(uw, ξ) .
• Increment the age of all edges connected to s
1
:
∆ct = 1, c ∈ gC ∧ s
1
∈ cV .
• If no edge between s
1
and s
2
exists, create one:
gC ⇐ gC ∪
{
(
{
s
1
, s
2
}
, 0)
}
.
• Reset the age of the edge between s
1
and s
2
to
zero:
ct ⇐ 0, c ∈ gC ∧ s
1
, s
2
∈ cV .
• Add the squared distance between ξ and the pro-
totype of s
1
to the accumulated error of s
1
:
∆s
1
e = D(s
1
w, ξ)
2
.
• Adapt the prototype of s
1
and all prototypes of its
direct neighbors:
s
1
w ⇐ A(s
1
w, ξ, gθε
b
),
s
n
w ⇐ A(s
n
w, ξ, gθε
n
), ∀s
n
∈ E
s
1
.
• Remove all edges with an age above a given
threshold τ and remove all units that no longer
have any edges connected to them:
gC ⇐ gC \
{
c|c ∈ gC ∧ ct > gθτ
}
,
gU ⇐ gU \
{
u|u ∈ gU ∧ E
u
=
/
0
}
.
• If an integer-multiple of gθλ inputs was pre-
sented to the RGNG g and |gU | < gθM, add a
new unit u. The new unit is inserted “between” the
unit j with the largest accumulated error and the
unit k with the largest accumulated error among
the direct neighbors of j. Thus, the prototype uw
of the new unit is initialized as:
uw := I( jw, kw) , j = argmax
l∈gU
(le) ,
k = argmax
l∈E
j
(le) .
The existing edge between units j and k is re-
moved and edges between units j and u as well
as units u and k are added:
gC ⇐ gC \
{
c|c ∈ gC ∧ j, k ∈ cV
}
,
gC ⇐ gC ∪
{
(
{
j, u
}
, 0), (
{
u, k
}
, 0)
}
.
The accumulated errors of units j and k are de-
creased and the accumulated error ue of the new
unit is set to the decreased accumulated error of
unit j:
∆ je = −gθα· je, ∆ke = −gθα · ke,
ue := je .
• Finally, decrease the accumulated error of all
units:
∆ue = −gθβ · ue, ∀u ∈ gU .
The function F returns the tuple (g, d
min
) containing
the now updated RGNG g and the distance d
min
:=
D(s
1
w, ξ) between the prototype of unit s
1
and in-
put ξ. Note that in contrast to the regular GNG there
is no stopping criterion any more, i.e., the RGNG op-
erates explicitly in an online fashion by continuously
integrating new inputs. To prevent unbounded growth
of the RGNG the maximum number of units θM was
introduced to the set of parameters.
D(x, y): The distance function D determines the dis-
tance between two prototypes x and y. The calculation
of the actual distance depends on whether x and y are
both vectors, a combination of vector and RGNG, or
both RGNGs:
D(x, y) :=











D
RR
(x, y) if x, y ∈ R
n
,
D
GR
(x, y) if x ∈ G ∧ y ∈ R
n
,
D
RG
(x, y) if x ∈ R
n
∧ y ∈ G,
D
GG
(x, y) if x, y ∈ G.
In case the arguments of D are both vectors, the
Minkowski distance is used:
D
RR
(x, y) := (
∑
n
i=1
|
x
i
− y
i
|
p
)
1
p
, x = (x
1
, . . . , x
n
),
y = (y
1
, . . . , y
n
),
p ∈ N.
Using the Minkowski distance instead of the Eu-
clidean distance allows to adjust the distance measure
with respect to certain types of inputs via the param-
eter p. For example, setting p to higher values results
in an emphasis of large changes in individual dimen-
sions of the input vector versus changes that are dis-
tributed over many dimensions (Kerdels and Peters,
2015a). However, in the case of modeling the behav-
ior of grid cells the parameter is set to a fixed value
of 2 which makes the Minkowski distance equivalent
to the Euclidean distance. The latter is required in
this context as only the Euclidean distance allows the
GNG to form an induced Delaunay triangulation of
its input space.
NCTA 2016 - 8th International Conference on Neural Computation Theory and Applications
44

In case the arguments of D are a combination of
vector and RGNG, the vector is fed into the RGNG
using function F and the returned minimum distance
is taken as distance value:
D
GR
(x, y) := F(x, y)d
min
,
D
RG
(x, y) := D
GR
(y, x).
In case the arguments of D are both RGNGs, the dis-
tance is defined to be the pairwise minimum distance
between the prototypes of the RGNGs’ units, i.e., sin-
gle linkage distance between the sets of units is used:
D
GG
(x, y) := min
u∈xU, k∈yU
D(uw, kw).
The latter case is used by the interpolation function
if the recursive depth of an RGNG is at least 2. As
the RGNG-based grid cell model has only a recursive
depth of 1 (see next section), the case is considered
for reasons of completeness rather than necessity. Al-
ternative measures to consider could be, e.g., average
or complete linkage.
I(x, y): The interpolation function I returns a new
prototype as a result from interpolating between the
prototypes x and y. The type of interpolation depends
on whether the arguments are both vectors or both
RGNGs:
I(x, y) :=
(
I
RR
(x, y) if x, y ∈ R
n
,
I
GG
(x, y) if x, y ∈ G.
In case the arguments of I are both vectors, the result-
ing prototype is the arithmetic mean of the arguments:
I
RR
(x, y) :=
x + y
2
.
In case the arguments of I are both RGNGs, the result-
ing prototype is a new RGNG a. Assuming w.l.o.g.
that |xU| ≥ |yU | the components of the interpolated
RGNG a are defined as follows:
a := I(x, y),
aU :=







(w, 0)








w = I(uw, kw),
∀u ∈ xU,
k = argmin
l∈yU
D(uw, lw)







,
aC :=











(
{
l, m
}
, 0)










∃c ∈ xC
∧ u, k ∈ cV
∧ lw = I(uw, ·)
∧ mw = I(kw, ·)











,
aθ := xθ .
The resulting RGNG a has the same number of units
as RGNG x. Each unit of a has a prototype that was
Figure 1: Illustration of the RGNG-based neuron model.
The top layer is represented by three units (red, green, blue)
connected by dashed edges. The prototypes of the top layer
units are themselves RGNGs. The units of these RGNGs
are illustrated in the second layer by corresponding colors.
interpolated between the prototype of the correspond-
ing unit in x and the nearest prototype found in the
units of y. The edges and parameters of a correspond
to the edges and parameters of x.
A(x, ξ, r): The adaptation function A adapts a proto-
type x towards a vector ξ by a given fraction r. The
type of adaptation depends on whether the given pro-
totype is a vector or an RGNG:
A(x, ξ, r) :=
(
A
R
(x, ξ, r) if x ∈ R
n
,
A
G
(x, ξ, r) if x ∈ G.
In case prototype x is a vector, the adaptation is per-
formed as linear interpolation:
A
R
(x, ξ, r) := (1 − r)x + r ξ.
In case prototype x is an RGNG, the adaptation is per-
formed by feeding ξ into the RGNG. Importantly, the
parameters ε
b
and ε
n
of the RGNG are temporarily
changed to take the fraction r into account:
θ
∗
:= ( r, r · xθε
r
, xθε
r
, xθλ, xθτ,
xθα, xθβ, xθM),
x
∗
:= (xU, xC, θ
∗
),
A
G
(x, ξ, r) := F(x
∗
, ξ)x .
Note that in this case the new parameter θε
r
is used
to derive a temporary ε
n
from the fraction r.
This concludes the formal definition of the RGNG al-
gorithm.
2.2 RGNG-based Neuron Model
The RGNG-based neuron model uses a single RGNG
to describe a group of neurons that compete against
each other. The RGNG has a recursive depth of one
resulting in a two-layered structure (Fig. 1). The units
Modelling the Grid-like Encoding of Visual Space in Primates
45

Table 1: Parameters of the RGNG-based model.
θ
1
θ
2
ε
b
= 0.04 ε
b
= 0.01
ε
n
= 0.04 ε
n
= 0.0001
ε
r
= 0.01 ε
r
= 0.01
λ = 1000 λ = 1000
τ = 300 τ = 300
α = 0.5 α = 0.5
β = 0.0005 β = 0.0005
M = 100 M =
{
20, 80
}
in the top layer (TL) correspond to the individual neu-
rons of the group. The prototypes of these TL units
are RGNGs themselves and can be interpreted as the
dendritic trees of the corresponding neurons. They
constitute the bottom layer (BL) of the model. The
prototypes of the BL units are regular vectors repre-
senting specific locations in input space.
2.2.1 Parameterization
Top and bottom layer of the model have their own
set of parameters θ
1
and θ
2
, respectively. Parameter
set θ
1
controls the main TL RGNG while parameter
set θ
2
controls all BL RGNGs, i.e., the prototypes of
the TL units. Table 1 summarizes typical parameter
values for an RGNG-based neuron model. For a de-
tailed characterization of these parameters we refer
to (Kerdels, 2016). Parameter θ
1
M sets the maxi-
mum number of neurons in the model and Parame-
ter θ
2
M controls the maximum number of specific
input space locations that a single neuron can repre-
sent within its dendritic tree. The learning rates θ
2
ε
b
and θ
2
ε
n
determine to which degree each neuron per-
forms a form of sub-threshold, competition indepen-
dent learning. In contrast, learning rate θ
1
ε
b
con-
trols how strongly the single best
3
neuron in the group
adapts towards a particular input. Finally, learning
rate θ
1
ε
n
can be interpreted as being inversely pro-
portional to the lateral inhibition that the most active
neuron exerts on its peers.
2.2.2 Learning
The RGNG algorithm does not have an explicit train-
ing phase. It learns continuously and updates its in-
put space approximation with every input. In case
of the RGNG-based neuron model the learning pro-
cess can be understood as a mixture of two processes.
For each input the TL RGNG has to determine which
TL unit lies closest to the input based on the distance
function D. The distance function will in turn feed
3
The neuron that is most active for a given input.
Figure 2: Typical visualization of a grid cell’s firing pattern
as introduced by Hafting et al. (Hafting et al., 2005). Left:
trajectory (black lines) of a rat in a circular environment
with marked locations (red dots) where the observed grid
cell fired. Middle: color-coded firing rate map of the ob-
served grid cell ranging from dark blue (no activity) to red
(maximum activity). Right: color-coded spatial autocor-
relation of the firing rate map ranging from blue (negative
correlation, -1) to red (positive correlation, +1) highlighting
the hexagonal structure of the firing pattern. Figure from
Moser et al. (Moser and Moser, 2008).
the input to each BL RGNG allowing the BL RGNGs
to learn the input space structure independently from
each other. Once the closest TL unit is identified
this single unit and its direct neighbors are allowed to
preferentially adapt towards the respective input. This
second learning step aligns the individual input space
representations of the BL RGNGs in such a way that
they evenly interleave and collectively cover the input
space as well as possible (Kerdels, 2016).
2.2.3 Activity Approximation
Observations of biological neurons typically use the
momentary firing rate of a neuron as indicator of the
neuron’s activity. This activity can then be correlated
with other observed variables such as the animal’s lo-
cation. The result of such a correlation can be visual-
ized as, e.g., rate map (Fig. 2), which is then used
as the basis for further analysis. Accordingly, the
RGNG-based neuron model has to estimate the ac-
tivity of each TL unit in response to a given input ξ as
well. To this end, the “activity” a
u
of a TL unit u is
defined as:
a
u
:= e
−
(1−r)
2
2σ
2
,
with σ = 0.2 and ratio r:
r :=
D(s
2
w, ξ) − D(s
1
w, ξ)
D(s
1
w, s
2
w)
, s
1
, s
2
∈ uwU,
with BL units s
1
and s
2
being the BMU and second
BMU in uwU with respect to input ξ. Based on this
measure of activity it is possible to correlate the re-
sponses of individual, simulated neurons (TL units)
to the given inputs and compare the resulting artificial
rate maps with observations reported in the literature.
NCTA 2016 - 8th International Conference on Neural Computation Theory and Applications
46

Figure 3: Eye and orbit anatomy with motor nerves by
Patrick J. Lynch, medical illustrator; C. Carl Jaffe, MD, car-
diologist (CC BY 2.5).
2.3 Input Space Properties
The RGNG-based neuron model can approximate ar-
bitrary input spaces as long as the set of inputs con-
stitute a sufficiently dense sampling of the underly-
ing space. However, not every input space will result
in a grid-like firing pattern. To observe such a pat-
tern, the input space has to exhibit certain characteris-
tics. Specifically, the input samples have to originate
from a two-dimensional, uniformly distributed mani-
fold. Since the BL RGNGs will approximate such a
manifold with an induced Delaunay triangulation the
BL prototypes will be spread evenly across the mani-
fold in a periodic, triangular pattern.
This predictability of the RGNG-based model al-
lows to form testable hypothesis about potential input
signals with respect to the activity observed in biolog-
ical neurons. An example for such a hypothesis and
the resulting simulation outcomes are presented in the
next sections.
3 GAZE-POSITION INPUT
Killian et al. (Killian et al., 2012) report on neurons in
the primate medial entorhinal cortex (MEC) that show
grid-like firing patterns similar to those of typical grid
cells present in the MEC of rats (Fyhn et al., 2004;
Hafting et al., 2005). Their finding is especially re-
markable since it is the first observation of a grid-like
firing pattern that is not correlated with the animal’s
location. Instead, the observed activity is correlated
with gaze-positions in the animal’s field of view.
Given a suitable input space the RGNG-based
neuron model can replicate this firing behavior. The
gaze-position in primates is essentially determined by
the four main muscles attached to the eye (Fig. 3).
Thus, a possible input signal to the RGNG-based
model could originate
4
from the population of mo-
tor neurons that control these muscles. In such a
signal the number of neurons that are active for a
particular muscle determines how strongly this mus-
cle contracts. A corresponding input signal ξ :=
(v
x
0
, v
x
1
, v
y
0
, v
y
1
) for a given normalized gaze posi-
tion (x, y) can be implemented as four concatenated
d-dimensional vectors v
x
0
, v
x
1
, v
y
0
and v
y
1
:
v
x
0
i
:= max

min

1 − δ

i+1
d
− x

, 1

, 0

,
v
x
1
i
:= max

min

1 − δ

i+1
d
− (1 − x)

, 1

, 0

,
v
y
0
i
:= max

min

1 − δ

i+1
d
− y

, 1

, 0

,
v
y
1
i
:= max

min

1 − δ

i+1
d
− (1 − y)

, 1

, 0

,
∀i ∈
{
0. . . d − 1
}
,
with δ = 4 defining the “steepness” of the population
signal and the size d of each motor neuron population.
4 SIMULATION RESULTS
To test the input space described in the previous sec-
tion we conducted a set of simulation runs with vary-
ing sizes d of the presumed motor neuron populations.
All simulations used the fixed set of parameters given
in table 1 (with θ
2
M = 20) and processed random
gaze-positions. The resulting artificial rate maps that
correlate each TL unit’s activity with corresponding
gaze-positions were used to calculate gridness score
distributions for each run. The gridness score was
introduced by Sargolini et al. as a measure of how
grid-like an observed firing pattern is (Sargolini et al.,
2006). Gridness scores range from −2 to 2 with
scores greater than zero indicating a grid-like firing
pattern, albeit more conservative thresholds between
0.3 and 0.4 are chosen in recent publications.
Figure 4 summarizes the results of the simulation
runs. The shown rate maps as well as the gridness
score distributions show that the neurons described by
the RGNG-based model form grid-like firing patterns
in response to the population signal defined above.
Even small population sizes yield a significant pro-
portion of simulated neurons with gridness scores
above a 0.4 threshold.
The firing patterns shown in figure 4 cover the en-
tire two-dimensional manifold of the population sig-
nal, i.e., the entire field of view. At its borders a strong
4
As a so-called efference copy
Modelling the Grid-like Encoding of Visual Space in Primates
47

Figure 4: Artificial rate maps and gridness distributions of
simulation runs that processed input from a varying num-
ber d of presumed motor neurons per muscle (columns) that
control the gaze-position. All simulation runs used a fixed
set of parameters as given in table 1 (θ
2
M = 20) and pro-
cessed random gaze locations. Each artificial rate map was
chosen randomly from the particular set of rate maps and
displays all of the respective TL unit’s firing fields. The
gridness distributions show the gridness values of all TL
units. Gridness threshold of 0.4 indicated by red marks.
alignment of the firing fields can be observed which
may influence the resulting gridness scores. Since it
is unlikely that experimental observations of natural
neurons will cover the entire extent of an underlying
input space, we investigated how the partial obser-
vation of firing fields may influence gridness score
distributions. Figure 5 shows the results of a sec-
ond series of simulation runs. Again, all simulation
runs used the set of parameters given in table 1 but
with θ
2
M = 80 prototypes per neuron instead of 20.
In addition to artificial rate maps that contain all fir-
ing fields of the respective neurons, we also gener-
ated rate maps containing only one-quarter or one-
sixteenth of the particular firing fields. The result-
ing gridness score distributions show that rate maps
based on local subsets of firing fields tend to have
higher gridness scores than the rate maps containing
all firing fields. This indicates that local distortions
of the overall grid pattern remain local with respect
to their influence on the gridness of other regions. As
a consequence, grid-like firing patterns may be ob-
served in natural neurons that receive signals from in-
put spaces that are only partially two-dimensional and
evenly distributed. As long as the experimental con-
ditions restrict the input signals to these regions the
resulting firing rate maps will exhibit grid-like pat-
terns. If the input signals shift to other regions of in-
put space, the grid-like firing patterns may then get
distorted or fully disappear.
Figure 5: Artificial rate maps and gridness distributions of
simulation runs presented like in figure 4, but with parame-
ter θ
2
M = 80 and containing either all, one-quarter, or one-
sixteenth (rows) of the respective TL unit’s firing fields.
5 CONCLUSIONS
The contemporary view on grid cells interprets their
behavior as a specialized part within a system for ori-
entation and navigation that contributes to the inte-
gration of speed and direction. However, recent ex-
perimental findings indicate that grid-like firing pat-
terns are more prevalent and functionally more di-
verse than previously assumed. Based on these ob-
servations we hypothesized that a more general infor-
mation processing scheme may underlie the firing be-
havior of grid cells. We developed a RGNG-based
neuron model to test this hypothesis. In this paper
we demonstrate that the RGNG-based model can –
in contrast to established grid cell models – describe
the grid-like activity of neurons other than typical grid
NCTA 2016 - 8th International Conference on Neural Computation Theory and Applications
48

cells, i.e., neurons that encode gaze-positions in the
animal’s field of view rather than the animal’s loca-
tion in its environment.
In addition, we outlined the general conditions un-
der which we would expect grid-like firing patterns
to occur in neurons that utilize the general informa-
tion processing scheme expressed by the RGNG al-
gorithm. As these conditions depend solely on char-
acteristics of the input signal, i.e., on the data that
are processed by the respective neurons, the RGNG-
based model allows to form testable predictions on the
input-output relations of biological, neuronal circuits.
Shifting interpretations of neurobiological circuits
from models based on application-specific priors to
models based primarily on general, computational
principles may prove to be beneficial for the wider
understanding of high-level cortical circuits. They
may allow to relate experimental observations made
in very different contexts on an abstract, computa-
tional level and thus promote a deeper understanding
of common neuronal principles and structures.
REFERENCES
Barry, C. and Burgess, N. (2014). Neural mechanisms of
self-location. Current Biology, 24(8):R330 – R339.
Boccara, C. N., Sargolini, F., Thoresen, V. H., Solstad, T.,
Witter, M. P., Moser, E. I., and Moser, M.-B. (2010).
Grid cells in pre- and parasubiculum. Nat Neurosci,
13(8):987–994.
Burak, Y. (2014). Spatial coding and attractor dynamics of
grid cells in the entorhinal cortex. Current Opinion
in Neurobiology, 25(0):169 – 175. Theoretical and
computational neuroscience.
Domnisoru, C., Kinkhabwala, A. A., and Tank, D. W.
(2013). Membrane potential dynamics of grid cells.
Nature, 495(7440):199–204.
Fritzke, B. (1995). A growing neural gas network learns
topologies. In Advances in Neural Information Pro-
cessing Systems 7, pages 625–632. MIT Press.
Fyhn, M., Molden, S., Witter, M. P., Moser, E. I., and
Moser, M.-B. (2004). Spatial representation in the en-
torhinal cortex. Science, 305(5688):1258–1264.
Giocomo, L., Moser, M.-B., and Moser, E. (2011). Com-
putational models of grid cells. Neuron, 71(4):589 –
603.
Hafting, T., Fyhn, M., Molden, S., Moser, M.-B., and
Moser, E. I. (2005). Microstructure of a spatial map
in the entorhinal cortex. Nature, 436(7052):801–806.
Jacobs, J., Weidemann, C. T., Miller, J. F., Solway, A.,
Burke, J. F., Wei, X.-X., Suthana, N., Sperling, M. R.,
Sharan, A. D., Fried, I., and Kahana, M. J. (2013). Di-
rect recordings of grid-like neuronal activity in human
spatial navigation. Nat Neurosci, 16(9):1188–1190.
Kerdels, J. (2016). A Computational Model of Grid Cells
based on a Recursive Growing Neural Gas. PhD the-
sis, Hagen.
Kerdels, J. and Peters, G. (2013). A computational model
of grid cells based on dendritic self-organized learn-
ing. In Proceedings of the International Conference
on Neural Computation Theory and Applications.
Kerdels, J. and Peters, G. (2015a). Analysis of high-
dimensional data using local input space histograms.
Neurocomputing, 169:272 – 280.
Kerdels, J. and Peters, G. (2015b). A new view on grid cells
beyond the cognitive map hypothesis. In 8th Confer-
ence on Artificial General Intelligence (AGI 2015).
Killian, N. J., Jutras, M. J., and Buffalo, E. A. (2012). A
map of visual space in the primate entorhinal cortex.
Nature, 491(7426):761–764.
Martinetz, T. M. and Schulten, K. (1994). Topology repre-
senting networks. Neural Networks, 7:507–522.
Moser, E. I. and Moser, M.-B. (2008). A metric for space.
Hippocampus, 18(12):1142–1156.
Moser, E. I., Moser, M.-B., and Roudi, Y. (2014). Net-
work mechanisms of grid cells. Philosophical Trans-
actions of the Royal Society B: Biological Sciences,
369(1635).
Sargolini, F., Fyhn, M., Hafting, T., McNaughton, B. L.,
Witter, M. P., Moser, M.-B., and Moser, E. I.
(2006). Conjunctive representation of position, di-
rection, and velocity in entorhinal cortex. Science,
312(5774):758–762.
Welinder, P. E., Burak, Y., and Fiete, I. R. (2008). Grid
cells: The position code, neural network models of
activity, and the problem of learning. Hippocampus,
18(12):1283–1300.
Yartsev, M. M., Witter, M. P., and Ulanovsky, N. (2011).
Grid cells without theta oscillations in the entorhinal
cortex of bats. Nature, 479(7371):103–107.
Modelling the Grid-like Encoding of Visual Space in Primates
49