HIPPOCAMPUS 24:912–919 (2014)
RAPID COMMUNICATION
The Transformation From Grid Cells to Place Cells is Robust
to Noise in the Grid Pattern
Amir H. Azizi,1,2 Natalie Schieferstein,2,3 and Sen Cheng1,2*
ABSTRACT: Spatial navigation in rodents has been attributed to placeselective cells in the hippocampus and entorhinal cortex. However, there
is currently no consensus on the neural mechanisms that generate the
place-selective activity in hippocampal place cells or entorhinal grid cells.
Given the massive input connections from the superficial layers of the
entorhinal cortex to place cells in the hippocampal cornu ammonis (CA)
regions, it was initially postulated that grid cells drive the spatial
responses of place cells. However, recent experiments have found that
place cell responses are stable even when grid cell responses are severely
distorted, thus suggesting that place cells cannot receive their spatial
information chiefly from grid cells. Here, we offer an alternative explanation. In a model with linear grid-to-place-cell transformation, the transformation can be very robust against noise in the grid patterns depending on
the nature of the noise. In the two more realistic noise scenarios, the
transformation was very robust, while it was not in the other two scenarios. Although current experimental data suggest that other types of placeselective cells modulate place cell responses, our results show that the
simple grid-to-place-cell transformation alone can account for the origin
C 2014 Wiley Periodicals, Inc.
of place selectivity in the place cells. V
KEY WORDS:
hippocampus; medial entorhinal cortex; spatial representation; neural networks; feedforward networks
INTRODUCTION
The hippocampal formation, with its place-selective cells, has long
been considered the location for the cognitive map in rodents (O’Keefe
and Nadel, 1979). The anatomical connections within the hippocampal
formation are well studied (Witter, 1993). Specifically, it has been
This article was published online on 06 June 2014. An error was subsequently identified. This notice is included in the online and print versions
to indicate that both have been corrected 13 June 2014.
1
Department of Psychology, Ruhr-University Bochum, Bochum,
Germany; 2 Mercator Research Group “Structure of Memory”, RuhrUniversity Bochum, Bochum, Germany; 3 Department of Mathematics,
Ruhr-University Bochum, Bochum, Germany
Grant sponsor: DFG; Grant number: SFB874-Project B2; Grant sponsor:
Stiftung Mercator.
*Correspondence to: Sen Cheng, Mercator Research Group “Structure
of Memory”, Department of Psychology, Ruhr-Universit€at Bochum,
Universit€atsstr. 150, Bochum, NRW 44801, Germany.
E-mail: [email protected]
Accepted for publication 16 May 2014.
DOI 10.1002/hipo.22306
Published online 6 June 2014 in Wiley Online Library
(wileyonlinelibrary.com).
C 2014 WILEY PERIODICALS, INC.
V
shown that there are massive projections from the
superficial layers of medial entorhinal cortex (MEC)
to the hippocampal CA regions (Zhang et al., 2013).
After the discovery of grid cells in MEC layer II
(Hafting et al., 2005), it was proposed that grid cells
provide the spatial input signal to place cells (Fuhs
and Touretzky, 2006; McNaughton et al., 2006; Rolls
et al., 2006; Solstad et al., 2006; Blair et al., 2007;
Franzius et al., 2007). Experimental evidence for the
presence of stable place cells, while non-MEC inputs
to CA1 were eliminated, indeed supports these models (Brun et al., 2002; Nakashiba et al., 2008; Van
Cauter et al., 2008; Cabral et al., 2014).
We showed previously that the various disparate
models find similar solutions for the grid-to-place-cell
transformation (Cheng and Frank, 2011). This solution is equivalent to a feedforward network, in which
the connection weights from grid to place cells are
monotonically decreasing with respect to the normalized spatial offset of the grid pattern.
However, recent experimental results have cast doubt
on the simple view that the spatial selectivity in place
cells is driven mainly by grid cell inputs. Two studies
reported that, when theta oscillations in the hippocampus were disrupted by reversibly inactivating the medial
septum (MS; Mizumori et al., 1989), CA1 neurons
continued to have normal and stable place fields even
though the spatially periodic firing pattern of MEC
grid cells was severely degraded (Brandon et al., 2011;
Koenig et al., 2011). Two other studies found that in
young rat pups place and head-direction cells develop
adult-like firing patterns earlier than grid cells do (Langston et al., 2010; Wills et al., 2010). These studies conclude that spatially periodic activity patterns in grid
cells is not necessary to drive stable place cell responses,
thus indicating that inputs from other cells that encode
spatial information primarily drive place cell responses.
Recently, Bush et al. (2014) suggested that these inputs
are supplied by boundary cells (Solstad et al., 2008).
Here, we explore an alternative explanation and
hypothesize that grid cell inputs alone could account
for place cell firing, even when the periodicity of the
grid cells is lost. The key is that the transformation
from grid cells to place cells is very robust against biologically induced noise in the grid pattern. This is
ROBUSTNESS OF GRID-TO-PLACE-CELL TRANSFORMATION
913
FIGURE 1.
Definitions of the model. A) Three sinusoidal gratings, aligned at 60 , produce a hexagonal grid pattern. B) A typical example of grid cell firing pattern (upper panels) and their
auto-correlograms (lower panels) in the presence of large amounts
of corresponding noise. The titles in the upper panels indicate
how noise was added to the grid pattern, the number in the top-
right corner of the lower panels indicate the gridness score. C)
Illustration of the path integration noise. The longer the trajectory
is estimated via path integration, the larger the accumulated error
becomes. [Color figure can be viewed in the online issue, which is
available at wileyonlinelibrary.com.]
indeed what we find in neural network simulations of the gridto-place-cell transformation, where we add different types of
noise to the grid pattern. While we are well aware that inputs
from neurons other than grid cells strongly modulate the spatial firing of place cells, our results indicate that it is not inconceivable, given the current evidence, that grid cells provide the
primary spatial input to place cells.
The periodic hexagonal pattern of grid cell firing can be
defined by three parameters: the spacing between peaks, the
orientation, and the spatial offset of the fields with respect to a
reference point. In computational models, the grid pattern can
be formed by the summation of three cosine gratings, oriented
at 60 apart from each other (Fig. 1A). Indeed, a two dimensional Fourier analysis of experimental data has shown that
most grid cells can be characterized by three components with
similar wavelength in their corresponding Fourier series
(Krupic et al., 2012). We, therefore, modeled the activation of
grid cells as follows.
To assess the hexagonal periodicity of the grid pattern we
used the gridness score as defined in (Langston et al., 2010).
Electrophysiological recordings from a large ensemble of grid
cells have shown that grid spacing and orientation of grid cells
are organized in discrete modules across the dorsoventral axis
(Stensola et al., 2012). This topographic structure with independent modules of grid cells differs from the anatomical cortical column structure (Burgalossi et al., 2011; Ray et al., 2014).
Based on theoretical (Monaco and Abbott, 2011) and experimental (Stensola et al., 2012) studies, we used four distinct
modules in which all grid cells had the same spacing and orientation. The spatial offset of the grid pattern was chosen randomly for each of the cells.
To model the disrupted grid patterns, we studied four different mechanisms. First, we added noise to the amplitude of
each of the grid patterns, as a random number drawn from a
normal distribution around 0 with different variances
between 0 and 2 [na term in Eq. (1), Fig. 1B]. Second, we
added noise to the phase of the grid patterns, as a positive
random number drawn from a uniform distribution between
0 and 6 [np in Eq. (1)]. Our third strategy was motivated by
recent observations that grid cells with more stable Fourier
components tend to have more stable grid patterns. Disorientation of any of the components seems to be the reason for
deviation from the regular grid pattern (Krupic et al., 2012).
We modeled this process by adding noise to the direction of
unit vectors [nd in Eq. (1)]. The noise was distributed uniformly between 2f and f, where 0 <f< 60 . Since these
noise mechanisms appear somewhat ad-hoc, we finally studied noise due to the drift in path integration (Fig. 1C), as
grid cells have been suggested to play an important role in
0
1
1
1
!
0
!
!
4p
r
2r
@cos @pffiffiffi !
Gi ð!
r Þ5
6ki A1np A1na A;
u ðul 1ui 1nd Þ @
ai
3
l51
3
X
0
0
(1)
where ai is the spacing between neighboring grid firing fields,
!
!
r 0 is the location of the reference point, ki is the normalized
spatial offset. !
u ðuÞ5ðcos ðuÞ; sin ðuÞÞ are the directional unit
vectors of the grid, in which u1 530 , u2 590 , and u3 5150
define the main three components. na , np , and nd represent the
noise added to amplitude, phase, and the direction of the unit
vectors, respectively. These noise terms will be discussed below.
Hippocampus
914
AZIZI ET AL.
path integration (McNaughton et al., 2006; Moser and
Moser, 2008).
!
x ðtÞ5x!
01
ðt 0
! !
v ðuÞ1 n pi du;
(2)
where !
x0 is the estimated initial position and !
x ðtÞ is the trajectory estimated from the instantaneous velocity of the animal,
!
!
v ðtÞ. Path integration is corrupted by the noise vector n pi, in
which each component assumes a uniformly distributed random number between 2h and h, where 0<h< 90. We used a
model of spiking neurons in which the probability of each grid
cell to spike is derived from the underlying grid pattern of that
cell at the estimated location of the simulated animal. We constructed the firing rate map at the end of the simulated exploration data much as electrophysiologists would do in an
experiment. The environment was divided into 2 3 2 cm2
bins. The firing rate in each bin was calculated by dividing the
number of spikes fired by the cell, while the simulated animal
was located in each bin by the time spent in that bin. The
duration of random exploration in the environment determined the variability of the resulting firing rate map. We used
an exploration time of 600 s.
We first studied the effect that different types of noise have
on the gridness score. Since noise was added to qualitatively
different parameters in these four scenarios, the noise parameters differ greatly in range, which prevents direct comparisons
between noise mechanisms. We, therefore, chose to quantify
the amount of noise added by using scaled noise levels. For
each noise mechanism, we chose 15 noise levels such that the
entire range of effects on the gridness score becomes visible
(Fig. 2). In all four noise scenarios, the mean gridness score
decreases rapidly with the noise level (Fig. 2, right column),
but there are differences in the distributions of gridness scores
(Fig. 2, left column). When noise is added to the amplitude or
the phase, the gridness score largely remains positive, irrespective of the noise level (Figs. 2A,B). Except for intermediate
noise levels, the distributions of gridness scores are fairly narrow. By contrast, adding noise to the direction or in path integration leads to much broader distributions of the gridness
scores, which also extend to negative numbers (Figs. 2C,D).
Since cells with negative gridness scores have been observed
experimentally (Brandon et al., 2011; Koenig et al., 2011;
Krupic et al., 2012), only the latter two noise scenarios are
consistent with experimental data. Therefore, noise in the
direction of unit vectors or in path integration is more plausible than the other two noise scenarios. Cells with negative gridness scores show nonhexagonal regular patterns (e.g., 20.26
for direction and 20.28 for path integration in Fig. 1B), somewhat similar to stripe cells reported in (Krupic et al., 2012). In
the path integration simulation, gridness scores were computed
from firing rate maps that were constructed from the simulated
trajectories and grid cell spiking. As a result, the maps are less
accurate than in the other three scenarios and the highest gridness score is lower compared to the other noise scenarios.
Interestingly, the two more plausible noise scenarios, noise to
Hippocampus
direction and path integration, also exhibit qualitatively similar
distributions of gridness scores.
We next modeled a downstream place cell response as a linear weighted summation of the firing rates of the grid cells,
hð!
r Þ5
N
X
w i G i ð!
r Þ:
(3)
i51
The strength of the connecting weight between the i-th grid
cell and the place cell, wi , is a function of the normalized offset of the corresponding grid pattern. We used the linear function wðkÞ52k21, which was acquired as a result of a
Hebbian learning mechanism (Cheng and Frank, 2011). This
!
transformation would lead to a place field centered on r 0 . We
deemed the output cell to be a model for a place cell, if a
threshold could be determined such that the activation of the
output cell hð!
r Þ that is above threshold, that is, the firing rate
!
map, exhibits a single contiguous area that includes r 0 and is
larger than 50 cm2 and smaller than 2,250 cm2. We call this
area the place field. Noise heavily influences the transformation
from grid cells to place cells. Even the addition of small
amounts of noise to the amplitude or the phase of the grid
pattern leads to an abrupt loss of the output place field (Fig.
3A; columns 1 and 2, respectively). By contrast, the transformation is very robust to large amounts of noise added to the
direction of unit vectors and the accumulated error due to path
integration drift (Fig. 3A; columns 3 and 4, respectively). This
analysis only counts those hippocampal neurons that have a
single, identifiable place field and, therefore, does not reveal
what spatial firing pattern, if any, the other cells exhibit. We,
therefore, analyzed the spatial selectivity of all cells using the
spatial information measure as used, for example, by Langston
et al., (2010). As one would expect, in all four noise scenarios
the spatial information in the hippocampal neuron drops when
noise was added to the grid cell inputs (Fig. 3B). However,
even for highly degraded grid cell inputs the output cell encodes a finite amount of spatial information in all four noise scenarios, even though in many cases the activation of the output
cell does not exhibit a clear place field.
To directly compare the periodicity of grid cells and the spatial selectivity of place cells, we plotted the fraction of simulations, in which a single place field could be identified, as a
function of the gridness score (Fig. 3C). We found that the
dependence of place fields on the grid cell inputs critically
depends on the type of noise that is added to the grid pattern.
When noise is added to the amplitude or the phase, place fields
can only be observed, when the mean gridness score is high. In
other words, in these two noise scenarios, robust place cell
activity cannot be driven by noisy grid cell firing. However, in
the more plausible noise scenarios 3 and 4, place fields can be
observed in a large fraction of simulations and even when the
grid inputs have low gridness scores, as long as there are at
least a few hundred grid cell inputs.
Finally, we explored the stability of place fields when grid
cells were degraded. To quantify stability, we calculated the
ROBUSTNESS OF GRID-TO-PLACE-CELL TRANSFORMATION
FIGURE 2.
The effect of adding different types of noise to
the grid pattern. Left column: Distribution of the gridness score
(refer to the text) for a network of 100 cells, when the noise is
added to!the amplitude na (A), phase np (B), direction of the unit
vectors n d (C), and to path integration ||np || (D). Grid cells are
915
organized into four different modules, each with the same orientation and spacing. Right column: The corresponding decrease in
average gridness scores for different noise strategies. [Color figure
can be viewed in the online issue, which is available at
wileyonlinelibrary.com.]
Hippocampus
916
AZIZI ET AL.
FIGURE 3.
Robustness of the grid to place-cell transformation. (A) Fraction of simulations that yield an identifiable place
field as a function of noise added to the amplitude, phase, direction of the grid pattern, and to path integration. Different lines
represent networks with different number of grid cells as indicated
in the legend. For large grid cell networks, the transformation is
robust to the addition of noise to the direction of unit vectors or
to path integration. (B) Average spatial information across all
cells, including those that do not have a clear place field, as a
function of noise level. (C) The fraction of simulations that yield
an identifiable place field as a function of average gridness score.
The dependence of place fields on the grid cell inputs critically
depends on the type of noise that is added to the grid pattern.
When noise is added to amplitude and phase, place fields are only
observed when gridness scores are large. By contrast, when noise is
added to the directions of the unit vectors, place fields are quite
insensitive to the noise in large networks.
average Euclidean distance between the locations of the resulting place fields with and without noise added to the grid patterns (Fig. 4A). In all four noise scenarios, the displacement of
the detected place field is small when noise is added to grid
cell activity, which is consistent with the experimental data
(Koenig et al., 2011). We next asked whether place fields in
the model might change in other ways when noise is added to
the grid cells. As a simple measure, we chose to study the maximal size of a place field that could be obtained in the firing
rate map by thresholding the hippocampal activation [Eq. (3)].
To determine the maximal place field size, we found the lowest
threshold for which a cell had a single contiguous place field.
With a lower bound of 50 cm2 fixed as a result of our definition of a place field, the average maximal place field sizes range
up to 420 cm2 depending on the type and amount of noise
injected into the grid cells (Fig. 4B). The overall range is compatible with experimental data (Langston et al., 2010; Wills
et al., 2010). For the two more plausible noise scenarios, the
possible place field sizes are plausible for a large range of average gridness scores.
Here, we have studied the robustness of the linear transformation of grid patterns into place fields against the addition of
four different kinds of noise to the grid patterns. We found
that the robustness critically depends on how the grid patterns
are degraded. In two cases, the transformation is not robust,
that is, no place fields arise even if only small amounts of noise
are added to the amplitude or phase. In the other two cases,
when noise is added to the direction of grid components or to
path integration, the transformation from grid cells to place
cells is very robust, that is, even for larger amounts of noise
Hippocampus
ROBUSTNESS OF GRID-TO-PLACE-CELL TRANSFORMATION
917
FIGURE 4.
Stability of the resultant place field. (A) (from left
to right) Average displacement of the place field center from the
no-noise condition caused by the addition of noise to amplitude,
phase, direction of the grid pattern, and to path integration. In all
cases, place fields are very stable since the largest displacement is
only around 8.3 cm and most displacements are even smaller. (B)
Average maximal size of the place fields as a function of the average gridness score in the input population. For the gridness score
ranges, where a place cell is detected, the place fields sizes are
comparable to the experimental data.
added to the grid pattern the place cells still have stable place
fields. Our results show that the observation of stable place
fields while grid pattern in the MEC are severely degraded
could be accounted for by the simple grid-to-place-cell transformation. In other words, this observation does not rule out
that spatial selectivity of place cells is predominantly driven by
grid cell inputs.
While it remains open how the degradation of grid patterns
during early development (Langston et al., 2010; Wills et al.,
2010) or after inactivation of the MS (Brandon et al., 2011;
Koenig et al., 2011) should be modeled precisely, experimental
observations place strong constraints. As mentioned above, we
regard the noise addition to the direction and path integration
as more plausible scenarios than the noise addition to amplitude or phase, since the two former scenarios yield negative
gridness scores, while the latter two do not. In addition,
Krupic et al. showed that, compared to grid cells, spatially
modulated nongrid cells have less stable Fourier components
(Krupic et al., 2012). Our interpretation of this observation is
that grid cells have correctly aligned unit vectors, while the
addition of noise to the direction of unit vectors generates the
spatially modulated nongrid cells.
A question that remains is whether the grid-to-place-cell
transformation alone is sufficient to account for the spatial
selectivity of place cells. What other inputs might account for
the spatial selectivity of place cells? Numerous studies suggest
that place cells receive a variety of spatial as well as nonspatial
inputs. The conventional view in the field until recently was
that spatial and nonspatial inputs are divided along anatomical
lines between medial and lateral entorhinal cortex, respectively
(Hargreaves et al., 2005; Ahmed and Mehta, 2009; Deshmukh
and Knierim, 2011). However, recently Zhang et al. found
that, although fewer in number, some MEC cells that project
to place cells have irregular or nonspatial firing correlates
(Zhang et al., 2013). Nonspatial inputs are widely thought to
modulate the spatial firing of place cells such as in rate remapping (Lu et al., 2013), which does not alter the location of spatial firing (Leutgeb et al., 2005).
By contrast, cells that encode spatial information such as
border cells (Solstad et al., 2008) might be able to drive the
place-selective responses of place cells (Zhang et al., 2013;
Bush et al., 2014). This possibility is intriguing and consistent
with studies that have investigated the role of MEC inputs to
the hippocampus. Initially, when grid cells were discovered,
these studies had been interpreted as supporting the view that
grid cells drive the spatial firing of place cells. However, they
did not differentiate between inputs from grid cells and those
from other spatially selective cells in MEC, which were discovered only later. For instance, Brun et al. showed that after
lesioning CA3 inputs to CA1, place cells in CA1 continue to
have stable place fields across sessions, even if there are subtle
changes (Brun et al., 2002). Nakashiba et al. obtained similar
results for place cells in transgenic mice (Nakashiba et al.,
2008). These results show that direct MEC inputs provide spatial information to CA1 place cells. By contrast, changes in
CA1 were more marked when layer III of MEC, which projects to CA1, was lesioned (Brun et al., 2008; Van Cauter
et al., 2008). After reversible inactivation of the MEC, the
object-location map of many CA1 place cells changed (Navawongse and Eichenbaum, 2013), pointing to the crucial role of
Hippocampus
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AZIZI ET AL.
the MEC signal in the stability of CA1 place cells. Nevertheless, inputs from both CA3 and MEC-III to CA1 are required
for normal learning in hippocampally dependent tasks (Brun
et al., 2002; Remondes and Schuman, 2004; Nakashiba et al.,
2008; Suh et al., 2011).
One potential explanation is that CA1 acts as a comparator
of both inputs and only highly synchronized EC and CA3
inputs can make a CA1 cell fire (Ahmed and Mehta, 2009).
Alternatively, we have hypothesized elsewhere that instantaneous CA1 activity patterns driven by both MEC and CA3
inputs are similar, because CA1 performs pattern completion
on both inputs (Cheng, 2013). The special role of CA3 is that
its recurrent network enables it to intrinsically generate sequential neural activity (Buhry et al., 2011; Azizi et al., 2013),
which is used to store episodic memories (Cheng, 2013). This
idea is supported by a recent experiment, in which transgenic
mice with impaired plasticity in the CA1 input synapses
showed deficits in navigational tasks requiring sequential memory. However, they performed normally in tasks requiring place
memory (Cabral et al., 2014). Based on LFP oscillations, the
authors of that study attributed sequential memory to CA3
inputs and place memory to the entorhinal inputs.
Some evidence, however, favors the hypothesis that grid
cells, rather than border cells, drive the spatial activity of
place cells. First, grid cells project to the hippocampus in
larger numbers than border cells do (Zhang et al., 2013).
CA1 apparently receives inputs not only from grid cells in
layer III of MEC, but also from grid cells in layer II. It was
recently reported that pyramidal cells arranged in clustered
patches in MEC II might be grid cells (Ray et al., 2014) and
that these cells directly project to the interneurons in the
CA1 region (Kitamura et al., 2014). Second, grid cells show
phase precession like place cells. Hafting et al. proposed that
the phase advance originates locally in the principal cells in
layer II of the MEC and is passed on to place cells in the
CA3 and CA1 regions of hippocampus (Hafting et al.,
2008). While it is possible that grid cells and place cells generate phase precession independently or that nonspatial, theta
modulated cells in the septo-hippocampal region drive phase
precession in place cells (Burgess, 2008), preliminary results
strongly suggest that phase precession in CA1 requires inputs
from MEC (M. I. Schlesiger, C. C. Cannova, E. A. Mankin,
B. B. Boublil, J. B. Hales, J. K. Leutgeb, C. Leibold; unpublished observation reported at SfN 2013) and not from CA3
(S.J. Middleton, T. J. McHugh; unpublished observation
reported at SfN 2013).
The results we discussed here show that it is quite possible
that future experiments will find more evidence that the grid
cell -to place-cell transformation alone is sufficient to account
for the spatial selectivity of place cells. A possible experiment
can be to selectively silence grid cell inputs to place cells while
leaving those from other cells intact. Moreover, our results
indicate that having a larger ensemble of grid cells makes the
grid-to-place transformation more robust (Figs. 3 and 4). This
would predict that in the aforementioned experiment, silencing
more grid cells will lead to fewer and less stable place cells.
Hippocampus
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