# A Survey of Entorhinal Grid Cell Properties

Jochen Kerdels  Gabriele Peters

###### Abstract

About a decade ago grid cells were discovered in the medial entorhinal cortex
of rat. Their peculiar firing patterns, which correlate with periodic
locations in the environment, led to early hypothesis that grid cells may
provide some form of metric for space. Subsequent research has since uncovered
a wealth of new insights into the characteristics of grid cells and their
neural neighborhood, the parahippocampal-hippocampal region, calling for a
revision and refinement of earlier grid cell models. This survey paper aims to
provide a comprehensive summary of grid cell research published in the past
decade. It focuses on the functional characteristics of grid cells such as the
influence of external cues or the alignment to environmental geometry, but
also provides a basic overview of the underlying neural substrate.

University of Hagen - Chair of Human-Computer Interaction

Universitätsstrasse 1, 58097 Hagen - Germany

##  1 Introduction

The parahippocampal-hippocampal region (PHR-HF) of the mammalian brain hosts a
variety of neurons whose activity correlates with a number of allocentric
variables. For example, the activity of so-called place cells correlates with
specific, mostly individual locations in the environment [50, 51], the
activity of head direction cells correlates with the absolute head direction
of an animal [66, 65], the activity of grid cells correlates with a regular
lattice of allocentric locations [16, 22], the activity of border cells
correlates with the proximity of an animal to specific borders in its
environment [60, 59], and, finally, the activity of speed cells correlates
with the current speed of an animal [32]. In addition, there are a number of
cells whose activity reflect a combination of those variables. Together, these
cells are commonly assumed to be part of a system that supports tasks of
spatial representation, orientation, and navigation. Within this system grid
cells stand out from the other neurons by their remarkable grid-like firing
patterns that resemble a set of carefully oriented rulers spanning the entire
environment. Hence, they are often seen as providing some kind of metric for
space to the animal [46, 45].

This survey aims to provide an overview of functional grid cell properties as
well as information on the underlying neuronal structures present in a major
area hosting grid cells, the medial entorhinal cortex (MEC). The survey is
based on previous work by one of the authors [28], but is significantly
extended and restructured to be comprehensive and self-contained. The survey
is structured as follows. Section 2 introduces key measures that are widely
used to identify and characterize grid cells on a functional level. Section 3
and section 4 report findings regarding the development of grid cells and
their topographical organization, respectively. Section 5 constitutes the main
part of this survey and covers important functional grid cell properties that
were discovered within the last decade. Subsequently, section 6 turns to the
underlying neural substrate reporting on findings regarding the general
structure of the MEC as well as specific findings regarding possible grid cell
microcircuits. Finally, section 7 concludes this survey.

##  2 Grid Cell Measures

Grid cells stand out from other neurons in the PHR-HF by their triangular,
grid-like firing patterns. To characterize the spatial properties of this grid
structure Hafting et al. [22] established four measures that are used
throughout the grid cell literature: spacing, orientation, field size, and
phase of a grid cell. In addition, Sargolini et al. [58] introduced a gridness
score which quantifies the degree of spatial periodicity of a cell’s firing
pattern.

The basis of all five measures is the firing rate map of the grid cell in
question. The firing rate map of a cell is typically constructed by
discretizing the environment into bins of equal size (e.g. $3\mathrm{cm}\times
3\mathrm{cm}$) and determining the spatially smoothed, average firing rate for
each bin. For instance, Sargolini et al. [58] estimate the average firing rate
$\lambda\left(x\right)$ of the bin centered on position $x$ as:

$\lambda\\!\left(x\right)=\sum_{i=1}^{n}g\\!\left(\frac{s_{i}-x}{h}\right)\bigg{/}\int_{0}^{T}g\\!\left(\frac{y\left(t\right)-x}{h}\right)dt$

with a Gaussian kernel $g$, a smoothing factor $h=3$, the number of spikes
$n$, the location $s_{i}$ of the $i$-th spike, the location $y\left(t\right)$
of the rat at time $t$, and the recording period $\left[0,T\right]$. To avoid
extrapolation errors bins further than the bin width apart from the tracked
path of the animal are considered as unvisited. In a more recent publication,
Stensola et al. [61] use a $5\times 5$ boxcar average instead of a Gaussian
kernel for smoothing. The use of this boxcar average results in firing fields
that appear more accentuated and crisp compared to the Gaussian smoothing.

Four of the five grid measures, i.e., spacing, orientation, field size, and
gridness require the calculation of a spatial autocorrelogram of the grid
cell’s firing rate map. Sargolini et al. [58] construct this autocorrelogram
using the Pearson product-moment correlation coefficient
$r\\!\left(\tau_{x},\tau_{y}\right)$ to calculate the autocorrelation between
rate map bins $\lambda$ separated by $\left(\tau_{x},\tau_{y}\right)$:

$\frac{n\sum\lambda\\!\left(x,y\right)\lambda\\!\left(x-\tau_{x},y-\tau_{y}\right)-\sum\lambda\\!\left(x,y\right)\sum\lambda\\!\left(x-\tau_{x},y-\tau_{y}\right)}{\sqrt{n\sum\lambda\\!\left(x,y\right)^{2}-\left(\sum\lambda\\!\left(x,y\right)\right)^{2}}\sqrt{n\sum\lambda\\!\left(x-\tau_{x},y-\tau_{y}\right)^{2}-\left(\sum\lambda\\!\left(x-\tau_{x},y-\tau_{y}\right)\right)^{2}}}$

where the summation is over all $n$ bins for which both
$\lambda\\!\left(x,y\right)$ and $\lambda\\!\left(x-\tau_{x},\right.$
$\left.y-\tau_{y}\right)$ have valid entries in the firing rate map.
Autocorrelations for shifts where $n<20$ are not included in the resulting
autocorrelogram.

Figure 1: Grid cell spacing and orientation.(a) The spacing of a grid cell is
defined as the median distance between the center peak and the six surrounding
peaks in the autocorrelogram. (b) The orientation of a grid cell is defined as
the angle $\alpha$ between a fixed reference line (dashed) going through the
central peak and the closest of the three main diagonals of the grid cell
hexagon in counterclockwise direction. Figures based on autocorrelogram from
Sargolini et al. [58]. Figure 2: Gridness scores of two grid cells. The shown
autocorrelograms include only the area containing the six peaks surrounding
the center that was used for the calculation of the gridness scores. The graph
to the right of each autocorrelogram shows the correlation of the particular
autocorrelogram with a rotated version of itself in $6^{\circ}$ steps.
Gridness scores are given to the left of each autocorrelogram. Figure adapted
from Sargolini et al. [58].

Based on the autocorrelogram of the grid cell’s firing rate map spacing,
orientation, field size, and gridness are defined as follows. The spacing of a
grid cell (fig. 1a) is defined as the median distance between the central peak
of the autocorrelogram and its six surrounding peaks. The orientation of a
grid cell (fig. 1b) is defined as the angle between a fixed reference line (0
degrees) going through the central peak of the autocorrelogram and the closest
of the three main diagonals of the surrounding hexagon in counterclockwise
direction. The field size of a grid cell refers to the size of the individual
firing fields. It is estimated as the area occupied by the central peak in the
autocorrelogram with respect to a fixed threshold, e.g., $r=0.2$ as used by
Hafting et al. [22]. To calculate the gridness score of a grid cell only the
six peaks surrounding the central peak in the autocorrelogram are taken into
account. All other regions of the autocorrelogram including the central peak
are masked out. Then, the correlation values between the masked
autocorrelogram and rotated versions of itself at $30^{\circ}$, $60^{\circ}$,
$90^{\circ}$, $120^{\circ}$, and $150^{\circ}$ are computed. The gridness
score is then calculated as the difference between the lowest correlation
value at $60^{\circ}$ and $120^{\circ}$ and the highest correlation value at
$30^{\circ}$, $90^{\circ}$, and $150^{\circ}$. Figure 2 illustrates the
underlying motivation for this measure. The two graphs show the results for
successive correlations between the masked autocorrelogram of a grid cell with
rotated versions of itself in $6^{\circ}$ steps. The graph shown on the right
of figure 2 is an example for a grid cell with a highly periodic, triangular
firing pattern resulting in high correlation values at multiples of
$60^{\circ}$ and low correlation values in between. In contrast, the graph on
the left is an example for a grid cell with a less regular firing pattern
resulting in a much weaker difference between the expected correlation maxima
at $60^{\circ}$ and $120^{\circ}$ and the expected correlation minima at
$30^{\circ}$, $90^{\circ}$, and $150^{\circ}$. Thus, the difference between
the lowest of the expected correlation maxima and the highest of the expected
correlation minima provides a suitable measure of a grid cells triangular
periodicity. Sargolini et al. [58] classify all cells with a gridness score
greater zero as grid cells. Others, e.g., Wills et al. [69] use thresholds
determined by the 95th percentile of a shuffled gridness score distribution.
Typically, the shuffled distribution is obtained by randomly shifting the
spike times of each cell by more than $20$ seconds and less than trial
duration minus $20$ seconds before calculating the cell’s gridness score. This
way the correlation with the animal’s position is broken while the temporal
firing characteristics are preserved. In case of Wills et al. [69] the
resulting threshold was $0.27$. Hence, unlike Sargolini et al. [58] Wills et
al. [69] would not classify the cell shown in figure 2 on the left as grid
cell.

In contrast to the measures described so far, the phase of a grid cell is not
an absolute measure. It describes the relative displacement between the firing
grids of two co-located grid cells, i.e, grid cells with similar spacing,
orientation, and field size. The relative displacement is determined by
calculating the cross-correlation between the firing rate maps of the
particular grid cells. Due to the cells’ similarity in spacing, orientation,
and field size the resulting cross-correlogram looks similar to the
autocorrelogram of a single grid cell, with the main difference that the
central peak is offset from the cross-correlogram’s center. This offset is the
relative phase between the two grid cells.

The previously described measures are well established and widely used across
the grid cell literature. In particular, the gridness score introduced by
Sargolini et al. [58] is the primary measure to identify neurons as grid
cells. As a result, cells with a gridness score below the respective threshold
used in a study are typically not included in the analysis. In a recent study
Krupic et al. [34] introduced a different approach to characterize the firing
patterns of cells in the MEA and adjacent parasubiculum (PaS). They calculated
the Fourier power spectrum of each cell’s firing rate map to identify the main
plane waves that give rise to the firing pattern. To reduce the effects of
noise they further filtered the power spectrum by subtracting the 50th
percentile value of the power spectrum generated from a spatially shuffled
version of the data and setting negative values to zero. Furthermore, main
peaks in the vicinity of a higher peak were treated as local maxima and
ignored in the subsequent analysis. To qualify as a cell with a spatially
periodic firing pattern the maximal component of the Fourier power spectrum
had to exceed 95% of all components in the power spectrum of shuffled data.
Using this measure, it is possible to not only identify grid cells, i.e.,
cells with hexagonal firing patterns, but also other cells that exhibit firing
patterns with a different spatial periodicity.

Figure 3: Examples of three spatially periodic (A-C) and one non-periodic cell
(D). Data for two successive trials per cell are shown: unsmoothed firing rate
maps (first row), autocorrelograms of the firing rate maps (middle row), and
filtered Fourier power spectrograms (last row). Color scale from blue (low
values) to red (high values). Stability of the firing patterns across trials
is given as Pearson product-moment correlation coefficient between the firing
rate maps. Peak firing rate, gridness score, and maximum Fourier power are
indicated above the particular maps. Figure adapted from Krupic et al. [34].

Krupic et al. recorded 351 cells in seven rats from layers II and III of the
MEA and adjacent PaS. 65% of all recorded cells in MEA and 79% of all recorded
cells in PaS were classified as spatially periodic cells. Of these 48% in MEA
and 18% in PaS were classified as grid cells based on their gridness score. As
expected grid cells exhibited three main components in the Fourier power
spectrum which were separated by multiples of 60°. The other, non-grid cells
had one to four main components with varying relative orientations and
wavelengths. Figure 3 shows examples of firing rate maps, autocorrelograms,
and Fourier power spectra for cells with periodic and non-periodic firing
patterns.

The main Fourier components of all spatially periodic cells were clustered
around a limited number of orientations and wavelengths confirming the
findings of Stensola et al. [61] on the topographical organization of grid
cells in MEA (see section 4). In addition, the use of the Fourier spectrum
revealed that the orientations of the main Fourier components of spatially
periodic non-grid cells were similar to the orientations of grid cells as well
and only differed in a wider distribution of relative orientations within the
non-grid cells.

The firing patterns of all cells classified as spatially periodic were more
stable than chance within as well as across trial days. Of these patterns the
firing patterns of grid cells were the most stable. However, the number and
relative orientation of the main Fourier components of some cells changed
gradually over time such that, e.g., cells classified as grid cells became
non-grid cells or vice versa. This change could be observed for trials within
the same environment (11% of spatially periodic cells changed) as well as
across different111In this case “different” refers to a change from a square
to a circular environment. environments (32% of spatially periodic cells
changed). Diehl et al. [13] investigated the stability of the spatial and non-
spatial firing patterns of cells in the MEA in more detail and could confirm
the observations of Krupic et al. In particular, they observed that between 9%
and 24% of cells in MEA depending on cell classification and environment
changed their spatial firing patterns and/or firing rate across recording
sessions with inter-session time intervals ranging from 5 minutes to 6 hours.
In addition, they were able to characterize the change in firing patterns of
individual cells as varying around a stable set point rather than random
drift.

These results suggest, that grid cells may be just one instance in a continuum
of spatially periodic cells [12, 13]. This hypothesis could, e.g., provide an
explanation for the multi-peaked, irregular firing patterns of early, putative
grid cells in developing rats observed by Wills et al. [69] and Langston et
al. [37] (see next section).

##  3 Development

The development of spatial representations as expressed by head direction
cells, place cells, and grid cells in the PHR-HF of young rats was recently
investigated by Wills et al. [69] and Langston et al. [37]. Both teams used
comparable experimental procedures and obtained consistent results. However,
the teams differ slightly in their respective interpretations of the results.

Wills et al. [69] recorded putative place cells from CA1, as well as putative
grid and head direction cells from MEA in rats between the ages of P16
(postnatal day 16) and P30. During recording the rats foraged for food in a
$62cm\times 62cm$ box. Head direction cells exhibited strong directional
tuning and were found in adult-like proportion right from age P16, i.e.,
during the rat’s first exploration of its environment222The rat’s eyelids
unseal at around P14 to P15.. Similarly, place cells could also be observed in
a significant proportion from day P16 with adult-like stability and quality in
their firing pattern. Throughout the observed development period the number of
place cells increased steadily towards adult-like levels. In case of grid
cells, putative grid cells with multi-peaked firing fields could also be
observed from day P16. However, the recorded firing patterns were irregular at
first. Significant proportions of cells with adult-like, hexagonal firing
patterns emerged around P20 and increased fast to near-adult proportions by
P22. Based on these results Wills et al. question the hypothesis that MEA grid
cells provide the only spatial input to place cells. They point out that the
observed differential developmental time course suggest that the
interconnectivity between grid cells and place cells develops only after the
pups begin to explore their environment.

Figure 4: Firing rate maps and their autocorrelations of grid cells in rats of
ages between P16 and P33. Color scale from blue (low values) to red (high
values). Peak firing rates and gridness scores indicated above each rate map.
Figure from Langston et al. [37].

Langston et al. [37] recorded putative place cells from CA1, putative grid
cells from MEA, as well as putative head direction cells from pre- and
parasubiculum in rats between the ages P16 and P35. In addition they recorded
cells from an adult control group. During recording the rats foraged for food
in a $50cm\times 50cm$ box. In essence, their observations are consistent with
the results obtained by Wills et al. [69]. 61.9% of all cells recorded in the
pre- and parasubiculum in rats of age P15 and P16 could be classified as head
direction cells showing strong directional tuning. This proportion of HD cells
was similar to the proportion found in the adult control group (64.0%). 41.4%
of the recorded cells in CA1 could be classified as place cells in rats of age
P16 to P18. This proportion continued to increase steadily throughout
development to adult-like levels at around 60%. During development the
stability of the firing fields generally increased within as well as between
trials. This finding matches earlier reports on the development of place cells
by Martin et al. [41]. In the MEA putative grid cells recorded in rats of age
P16 to P18 showed multi-peaked, irregular firing fields. Yet, 12.8% of the
cells recorded at age P16 to P18 could already be classified as grid cells.
The proportion of cells passing the grid cell criterion increased slightly
during development to about 17.5% in rats of age P31 to P34 being
substantially lower than the proportion of grid cells measured in the adult
control group ($\sim$ 30%). However, the periodic properties of cells
classified as grid cells increased noticeably during development reaching
gridness scores of near-adult levels at age P34. Figure 4 illustrates this
development showing the firing rate maps and their autocorrelograms of several
grid cells in rats of increasing age. In contrast to Wills et al. [69]
Langston et al. suggest that the three observed cell types (HD cells, place
cells, and grid cells) may interact from the outset of exploration. They
hypothesize that the observed, rudimentary grid cells provide a sufficiently
patterned input to CA1 cells to enable the generation of place-specific firing
fields in the hippocampus.

##  4 Topographical Organization

Grid cells in the MEA are topographically organized. Neighboring grid cells
exhibit similar grid spacing, field size and grid orientation, but have
dissimilar phases. Starting from the postrhinal border grid spacing and field
size increase along the dorsoventral axis of the MEA. A similar, systematic
change of grid orientation along this axis could not be observed [22].

| dorsal | intermediate | ventral
---|---|---|---
mean number of fields | 8. | 4 | $\pm$ | 0. | 3 | 5. | 9 | $\pm$ | 0. | 4 | 4. | 6 | $\pm$ | 0. | 4
mean field width | 56. | 0 | $\pm$ | 1. | 0 cm | 92. | 0 | $\pm$ | 6. | 0 cm | 119. | 0 | $\pm$ | 7. | 0 cm
largest field width | 90. | 0 | $\pm$ | 4. | 0 cm | 129. | 0 | $\pm$ | 8. | 0 cm | 190. | 0 | $\pm$ | 13. | 0 cm
minimum spacing | 91. | 0 | $\pm$ | 12. | 0 cm | 202. | 0 | $\pm$ | 24. | 0 cm | 269. | 0 | $\pm$ | 47. | 0 cm
median spacing | 171. | 0 | $\pm$ | 13. | 0 cm | 301. | 0 | $\pm$ | 23. | 0 cm | 370. | 0 | $\pm$ | 46. | 0 cm
mean firing rate | 3. | 6 | $\pm$ | 0. | 2 Hz | 4. | 5 | $\pm$ | 0. | 4 Hz | 2. | 0 | $\pm$ | 0. | 2 Hz
peak firing rate | 21. | 3 | $\pm$ | 0. | 9 Hz | 17. | 4 | $\pm$ | 1. | 1 Hz | 11. | 4 | $\pm$ | 0. | 9 Hz
Table 1:  Summary of grid cell properties obtained by Brun et al. [5] for 143
grid cells in 15 rats. Cells were partitioned into dorsal, intermediate, and
ventral groups according to their position along the dorsoventral axis of the
MEA.

In order to determine the range of grid spacings and field sizes present in
the MEA Brun et al. [5] recorded 143 grid cells in 15 rats that shuttled back
and forth on a 18m long, linear track. The cells were sampled from all
entorhinal cell layers with an emphasis on the superficial layers (layer II:
26, layer III: 35, layer II or III: 40, layers V and VI: 42) and their
locations were distributed between $1\%$ and $75\%$ along the dorsoventral
axis. To compare grid spacing and field size with respect to the cells’
position the cells were grouped into a dorsal ($0\%$ \- $25\%$, 55 cells),
intermediate ($25\%$ \- $50\%$, 59 cells), and ventral group ($50\%$ \-
$75\%$, 29 cells). Table 1 summarizes the obtained results. Grid spacing as
well as field size increase from dorsal to ventral positions and,
correspondingly, the number of firing fields along the 18m track decreases. In
addition, an increase in field size appears to be accompanied by a reduction
in peak firing rate. Grid spacing is characterized not only by median values
but also by minimum values. The minimum values were provided as the algorithm
used to detect the firing fields missed a substantial number of visually
discernable fields due to low firing rates. Thus, the median values may
overestimate the true grid spacing.

Using the same experimental setup as Brun et al. [5] Kjelstrup et al. [29]
could show, that the increase in grid spacing and field size along the
dorsoventral axis of the MEA is reflected in the field sizes of place cells in
CA3 which receive input from the MEA. The width of the place fields ranged
from 1.41m in the dorsal region up to 13.59m in the ventral region of CA3.

In a recent study Stensola et al. [61] examined the topographic organization
of MEA grid cells in more detail. In particular, they investigated whether the
increase in grid spacing and field size along the dorsoventral axis is
continuous or modular, the latter option being indicated by earlier
experiments [71, 2] and theoretical considerations [42, 14]. In total 968 grid
cells from 15 animals were recorded while the animals foraged in 100cm to
220cm wide, square boxes. The high number of recorded cells per animal (up to
186 grid cells) and the use of two sampling strategies that covered large
parts of the MEA were key to enable the determination whether the topographic
organization of grid cells is continuous or modular. In every single animal a
modular organization could be observed. Grid cells within a module share
similar grid spacing, field size, and grid orientation. Modules with
increasing grid spacing and field size along the dorsoventral axis overlap in
their extent, i.e., the positions of grid cells belonging to different modules
are not separated and may interleave. Across all animals the distribution of
mean grid spacing values covers a wide range with no apparent peaks. However,
within animals the scale relation between grid spacings of successive modules
is governed by a fixed factor of approximately 1.42 ($\sqrt{2}$) leading to a
doubling of the area covered by each grid hexagon between modules. The
circumstance that the same scale ratio was found in all animals despite
different, absolute values for the grid spacings implies that a genetic
mechanism is responsible for the scale relation while the different absolute
values may be influenced or determined by external factors.

Many of the hexagonal grid patterns observed by Stensola et al. were elongated
in one direction and it could be shown that grid cells sharing the same
distortion also shared grid spacing, field size, and orientation, i.e.,
belonged to the same module. By exploiting an experimental paradigm that
provokes a rescaling of the hexagonal pattern in grid cells (see section 5.4),
it could also be shown that grid cell modules are functionally independent.
Each grid cell module exhibited the induced rescaling phenomena independent
from each other, i.e., some modules exhibited rescaling while others did not.
This indicates that inputs based on the same environment are processed
independently by each module.

Stensola et al. found a maximum of five grid scale modules per animal and they
estimate that the total number of grid cell modules in the MEA is in the upper
single digit range.

##  5 Functional Properties

###  5.1 Phase Precession

The phenomenon of phase precession refers to a specific relation between the
firing behavior of an individual neuron and the overall, extracellular
activity, the electroencephalogram (EEG), of a brain region. In case of the
rat hippocampus two major classes of overall, extracellular activity can be
observed. Behaviors like walking, running, swimming, rearing, or jumping are
accompanied by a characteristic, sinusoidal 7Hz to 12Hz activity called theta
activity or just theta for short. Other behaviors like eating, drinking, or
grooming, i.e., behaviors that do not change the location of the animal,
correlate with large irregular activity covering a broad spectrum of
frequencies. During theta activity place cells fire in bursts at specific
points of the theta phase, e.g., at the trough. When an animal crosses the
firing field of a place cell the point of the theta phase at which the cell
fires shifts with the animal‘s relative position within the place field. This
shift in the timing of the place cell’s activity relative to the overall theta
activity is called phase precession [52].

Hafting et al. [21] investigated whether entorhinal grid cells exhibit phase
precession as well. They recorded 174 MEA grid cells from 23 rats while the
rats ran back and forth on two linear tracks (235cm and 320cm, both 10cm
wide). The grid cells were identified in a two-dimensional open field
environment in which grid spacings ranged from 30cm to 70cm.

Grid cells in MEA layer II exhibited clear phase precession. On average grid
cells started to fire at $222\pm 62^{\circ}$ when the animal entered a grid
field and stopped firing at $59\pm 78^{\circ}$ on exit labeling the peak of
the theta phase with $0^{\circ}$ and the trough with $180^{\circ}$. The mean
slope of a best fit linear regression line on the individual firing events of
each cell was $-2.77\pm 0.31^{\circ}\,\mathrm{cm}^{-1}$. In layer III of MEA
25% of all recorded grid cells showed phase precession over the complete theta
cycle, another 25% showed phase precession limited to the trough of the theta-
phase, and 50% exhibited no phase precession at all. On average layer III grid
cells started to fire at $195\pm 81^{\circ}$ and stopped firing at $192\pm
123^{\circ}$ of the theta-phase. The mean slope of the linear regression line
was $-0.078\pm 0.283^{\circ}\,\mathrm{cm}^{-1}$.

In order to test whether the observed phase precession of grid cells is
independent of signals from the hippocampus the firing phases of grid cells
were recorded after inactivation of the hippocampus. This inactivation was
achieved by a local infusion of muscimol3335-aminomethyl-3-hydroxyisoxazole, a
$\mathrm{GABA}_{\mathrm{A}}$ receptor agonist. After inactivation, the firing
fields of layer II grid cells remained spatially confined, albeit the firing
fields of the grid cells became wider and less stable. Theta activity and grid
cell phase precession was unaffected by the inactivation of the hippocampus as
well.

The question whether the phenomenon of phase precession is constitutive of
grid cell firing or not is controversial [47]. A number of computational
models that classify as _oscillatory interference_ models use this property to
explain the hexagonal firing patterns of grid cells [19]. However,
experimental evidence shows that grid cell firing in bats can be observed
without any sign of concurrent theta activity contradicting the prior of
oscillatory interference models [74].

###  5.2 Influence of External Cues

Figure 5: Anchoring of the firing pattern to an external landmark. Firing rate
maps of a single grid cell recorded in a circular environment with a prominent
visual cue (arc) attached to the environment’s wall. Rate maps are shown
before rotation (A), after rotation by 90 degrees (B), and after counter-
rotation by 90 degrees (A’). Peak firing rates are given next to the rate
maps. The shaded rate map depicts the rate map of B artificially counter-
rotated by 90 degrees to ease visual comparison with A and A’. Figure adapted
from Hafting et al. [22].

Hafting et al. [22] were the first to characterize basic functional properties
of grid cells. One of the key questions they address in this work is whether
the locations of grid cell firing fields are determined by external landmarks
(allothetic cues) or internal information about the rat’s movement (idiothetic
cues). They found that the firing fields of grid cells stay constant across
successive exposures to a single environment indicating a strong influence of
allothetic cues. To further investigate a possible anchoring of the firing
fields to external landmarks 24 grid cells in the dorsocaudal region of the
medial entorhinal cortex (dMEC) in three rats were recorded. While the rats
chased food pellets in a circular environment, a prominent visual cue at the
wall of the environment was rotated by 90 degrees. The grid orientation of all
24 grid cells followed this rotation and only returned to their original
alignment when the visual cue was counter-rotated to its original position
(fig. 5).

Despite this strong influence of allothetic cues on the orientation and phase
of the grid cell firing fields external cues are not necessary to maintain the
firing pattern itself. Hafting et al. demonstrated this property by recording
the activity of grid cells (4 rats, 33 cells) in total darkness (30 minutes)
preceded by a period of regular illumination (10 minutes). Besides a weak
dispersal of the firing fields the overall grid pattern stayed intact during
the period without illumination. Hafting et al. conclude that these findings
indicate that the grid pattern itself may result to a large extent from
hardwired network mechanisms, but that the particular alignment of the pattern
is determined by external landmarks. Kraus et al. [31] investigated the
maintenance of the grid pattern based on idiothetic cues and elapsed time
further. They recorded grid cells while rats were running on a motorized tread
mill and could observe that grid cells were able to integrate elapsed time as
well as the distance that the animals had run.

###  5.3 Realignment

A similar influence of external landmarks as described in the previous section
was already known to apply to hippocampal place cells [51], i.e., a rotation
of visual cues results in a corresponding rotation of a place cell’s firing
field. Likewise, the firing pattern of a place cell remains intact during
locomotion in darkness. Further investigation of the influence of distant
visual cues on place cell activity revealed the phenomenon of remapping [49,
48]. For a given environment a certain set of place cells represents this
particular environment. When the environment changes, e.g., when a rat is
placed from one experimental environment into another, a different set of
place cells gets recruited to represent the new environment. This switch from
one set of place cells to another is termed global remapping [38, 26] and it
only occurs if the environment changes significantly. If, in contrast, the
environment changes only slightly, the same set of place cells is used but the
maximum intensity with which individual place cells are active changes. This
change of maximum activity is termed rate remapping [38].

Given the similar behavior of grid cells and place cells with respect to the
influence of external landmarks, Fyhn et al. [15] investigated how grid cells
would behave during environment changes that reliably induce either global
remapping or rate remapping in hippocampal place cells. Global remapping was
induced by three protocols: switching between a square and a circular
environment at a fixed location in one room, switching between similar square
environments in two rooms with different background cues, and switching
between light and darkness in a single, square environment. Rate remapping was
induced by a single protocol in which the colors of the walls of a single,
square environment were changed. Neuronal activity was recorded in dorsocaudal
MEA and/or dorsal CA3 in 19 rats. The protocols reliably induced global and
rate remapping in all trials in which CA3 place cells were recorded.

Figure 6: Illustration of grid cell realignment. The maps show cross-
correlations of firing rate maps for small populations of grid cells (5 to 9
cells) in individual animals (5-digit numbers). A: Switching between a square
and circular environment causes a shift of the grid pattern while spacing,
orientation, and relative phases are preserved. B: Switching between square
environments located in different rooms causes not only a shift of the grid
pattern but also a rotation of the pattern. The angle of rotation was
determined by successively rotating the firing rate maps of environment B
until the resulting pattern in the cross-correlogram had maximal grid
structure. Figure adapted from Fyhn et al. [15].

Global remapping induced by the first protocol, i.e., alternating between
square and circular environments, led to an absolute shift of the grid up to
one-half grid spacing while the distributions of grid spacing, grid
orientation, and relative phases were preserved (fig. 6A). The directions of
the grid shifts were uniformly distributed across experiments. However, for
individual cells that were recorded over several days the direction and
magnitude of the grid shift remained constant. The second protocol,
constituting a stronger environment change, led not only to a shift of the
grid pattern but also to a grid rotation (fig. 6B). Additionally, in some
cases (three out of seven for the first protocol, five out of eight for the
second protocol) the overall grid spacing slightly scaled between
environments. These results indicate that grid cell ensembles do not change
during global remapping and instead perform a realignment of grid orientation
and position. They further suggest, that the relative phases of grid cells
within a local ensemble may be in a rigid relationship (this hypothesis was
recently supported by results of Yoon et al. [75]). The third global remapping
protocol was used with two rats that had electrodes in MEA as well as CA3 to
determine whether grid cell realignment and place cell remapping are
coincident. In one animal global remapping of place cells coincided with an
equally fast grid cell realignment, whereas in the second rat place cells and
grid cells maintained their firing fields after a switch from the dark to the
light condition. In the latter case, global remapping and realignment could be
triggered by temporarily interrupting the movement of the rat by placing it
for one minute on a pedestal. Fyhn et al. argue that this delayed remapping
reflects a continued influence of self-motion information on the location of
place cell and grid cell firing fields.

Contrary to the global remapping condition no change of grid cell activity
could be observed during rate remapping. The results obtained by Fyhn et al.
suggest two main implications. First, as the ensemble of grid cells and their
relative, spatial relationship does not change across environments, similar
paths of an animal in different environments are mapped onto similar sequences
of grid cells supporting the hypothesis that grid cells are part of an
universal metric for path-integration-based navigation. Second, as hippocampal
CA3 receives both direct and indirect input from the MEA the realignment of
grid cells could serve as a basis for the remapping occurring in hippocampal
place cells [44].

A number of recent publications provide further insight into the relations
between MEC and hippocampus. Zhang et al. [76] used a combined optogenetic-
electrophysiological strategy to map out the functional inputs to hippocampal
place cells that originate in the entorhinal cortex. Their main finding
suggests that hippocampal place cells receive input from a wide variety of
functional cells in the MEC including grid cells, border cells, head direction
cells, irregular spatial cells, and non-spatial cells. Miao et al. [43]
investigated the influence of partial MEC inactivation on hippocampal place
cells. They induced the partial inactivation by a pharmacogenetic approach and
could observe hippocampal remapping in CA3 as response. The inactivation did
not influence the shape or size of the observed place fields, but their
distribution at the ensemble level changed significantly. Importantly, this
remapping took place almost instantaneously. In contrast, Rueckemann et al.
[57] performed a similar experiment, but observed a slow and lasting change in
the place cell ensemble after a partial inactivation of MEC. However,
Rueckemann et al. used an optogenetic instead of a pharmacogenetic strategy to
partially silence the MEC and, more importantly, they observed hippocampal
place cells in CA1 instead of CA3. These differences in the experimental setup
may explain the diverging results regarding the time course of the observed
remapping. Marozzi et al. [40] explored the effects of non-metric context
changes, i.e., changes in color and smell of the environment on grid cell
firing patterns and could observe a realignment of the grid phase as response.
Finally, Olafsdottir et al. [53] addressed the question if and how the
activity of grid cells is involved in the well known phenomenon of hippocampal
replay. They recorded grid cells in layers V and VI of the MEC and could show
that these cells were spatially coherent with hippocampal place cells during
replay. The grid cells encoded the particular locations with a 11ms delay,
indicating monosynaptic feedback connections from hippocampus to layers V and
VI of MEC.

###  5.4 Rescaling

The slight scaling effects observed by Fyhn et al. [15] during realignment can
have a much larger magnitude under certain circumstances. Barry et al.
discovered and investigated these stronger rescaling phenomena in two
different contexts [2, 1].

Figure 7: Illustration of grid cell rescaling. A,B: Firing events (green and
red dots) of single grid cells superimposed on the path (black lines) of two
rats in scaled environments. Trials 1 and 5 took place in the environment to
which the rats were accustomed to (red outline). C,E: Color-coded (blue: low,
red: high) firing rate maps. Grid scaling for each dimension indicated by
numeric labels. D,F: Color-coded autocorrelograms of the firing rate maps.
Peaks of the central hexagons are marked with black crosses. Figure adapted
from Barry et al. [2].

The first context in which a significant rescaling of grid cell firing
patterns can be observed is the geometric deformation of a known environment
[2]. Barry et al. trained six rats to be accustomed to either a square
($1m\times 1m$, 3 rats) or a rectangular ($0.7m\times 1m$, 3 rats)
environment. Subsequently, the trained rats were exposed to scaled versions of
these environments while the activity of MEA grid cells was recorded. The
square (s) and rectangular (r) environments were scaled horizontally
($s:0.7m\times 1m\;|\;r:1m\times 1m$), vertically ($s:1m\times
0.7m\;|\;r:0.7m\times 0.7m$), and in both directions ($s:0.7m\times
0.7m\;|\;r:1m\times 0.7m$). In all cases the recorded firing patterns of the
grid cells showed strong rescaling in the direction in which the particular
environment was scaled, though with a lesser magnitude (fig. 7). Re-exposure
to the unscaled, familiar environment (Trial 5) showed, that the rescaling
effect is not permanent. On average grid patterns rescaled by 47.9% with
respect to the change of the environment. In addition, cases of uni-
directional scaling, i.e., in either horizontal or vertical direction, were
accompanied by a small, opposite rescaling (7.9% on average) of the grid
patterns in a direction orthogonal to the change of the environment.

Barry et al. hypothesize that environmental features like boundaries become
associated with the grid pattern over time, such that a sudden deformation of
the environment causes a corresponding deformation of the grid pattern.
However, with an increasing number of trials, grid patterns in the changed
environment rescaled with lower and lower magnitude indicating that rescaling
is not solely a reflection of the environment’s changed geometry but also a
reflection of the animal’s increasing familiarity with the modified
environment. Barry et al. suggest that this reduction in rescaling reflects a
tendency of the grid cells to revert to an intrinsic grid scale.

The second context in which rescaling of grid cell firing patterns was
observed is exposure to novel environments [1]. In the corresponding study MEA
grid cells of eight rats were recorded while the animals foraged in $1m\times
1m$ environments. The rats underwent five trials (20 minutes each) per day for
up to seven consecutive days. Each day, the rats were first exposed to a
familiar444familiar := a minimum of 100 minutes of prior exposure environment,
followed by three novel environments, and finally the familiar environment
again. Novel environments differed in texture, visual appearance, and odor.
Exposure to the first novel environment on the first day caused realignment
(rotation and shift) as well as rescaling of all recorded grid patterns. On
average the grid patterns scaled up by 37.3% (min.: 10.5%; max.: 71.1%). In
addition, the average gridness score dropped from 0.65 in the familiar
environment to 0.04 in the novel environment as the hexagons of the grid
patterns were less circular in the novel environment. Exposure to the second
and third novel environment resulted in rescaling with less magnitude, i.e.,
grid patterns in the fourth trial scaled up 21.3% on average. This trend
continued over subsequent days. On the second day the average increase was
16.2% and on day five no grid cell showed any discernable rescaling. In some
cases the firing pattern of individual grid cells already showed no apparent
rescaling after three days.

In order to examine if place cell remapping co-occurs with grid cell rescaling
another seven rats were implanted with electrodes recording simultaneously in
MEA and CA1. The rats were tested by a similar protocol as described before,
yet the environments used were not identical. Upon first exposure to a novel
environment the firing patterns of the recorded grid cells scaled up by 33.9%
on average, and the average gridness score decreased from 0.96 to 0.31.
Simultaneously recorded place cells showed an immediate and complete remapping
and the average size of place fields in the novel environment increased by
28.8%. Furthermore place fields in the novel environment appeared to be less
stable and more fraying. A subsequent weakening of grid pattern rescaling in
the second and third novel environment could be observed as well (trial 3:
14.6%; trial 4: 16.8%) and was accompanied by a similar, though smaller
decrease of place field scaling (trial 3: 11.3%; trial 4: 7.6%555This
measurement did not reach a significance level of 0.05 (0.07).).

In summary these results show that in novel environments the firing patterns
of grid cells as well as place cells expand and become less regular. With
increasing exposure to the new environments the firing fields of grid cells
and place cells reacquire the properties seen in familiar environments, i.e.,
they become more regular and smaller in spatial scale. As a further mechanism
that may underlie these rescaling phenomena Barry et al. propose a possible
influence of the neuromodulator acetylcholin (ACh) which is implicated in
novelty detection. This hypothesis is supported by the fact that co-recorded
grid cell patterns scale up by similar amounts.

###  5.5 Influence of Environmental Geometry

The phenomena described in the previous sections were further investigated by
Krupic et al. [33]. In contrast to previous reports they could show that
environmental geometry can have a lasting influence on the firing patterns of
grid cells. In a first experiment they demonstrate that the geometry of the
experimental arena can control grid orientation and override the influence of
prominent distal cues if the geometry is polarized as, e.g., in the case of a
square environment. If such an experimental enclosure is rotated $45^{\circ}$,
the grid orientation follows this rotation (mean grid rotation
$42.5^{\circ}\pm 2.9^{\circ}$) despite the presence of stationary distal cues.
However, if the enclosure is rotated $90^{\circ}$, the grid orientation
remains unchanged (mean grid rotation $1.1^{\circ}\pm 0.9^{\circ}$). The
latter case indicates that other local cues like smells or textures do not
influence grid orientation. Further investigation (275 grid cells, 41 rats)
showed that grid orientation aligns to the walls of square enclosures at a
mean angle of $8.8^{\circ}\pm 0.6^{\circ}$ while the results for unpolarized
environments were less clustered. As one consequence of this alignment the
relative orientation between different grid cell modules should be either
$0^{\circ}$ or $30^{\circ}$ in square environments due to the $60^{\circ}$
symmetry of the grid pattern in relation to the $90^{\circ}$ angles of the
environment. Apart from a few intermediate values the observed relative
orientations confirmed this hypothesis. Yet, these relative orientations
remained stable in circular and hexagonal environments as well indicating that
anatomically close grid cell modules can act coherently across different
environments. The findings of Krupic et al. regarding the alignment of grid
orientation to enclosure walls matches the findings of a similar study by
Stensola et al. [62]. The latter hypothesize that the alignment might be a way
to minimize symmetry with the borders of the environment.

Krupic et al. also repeated the rescaling experiments of Barry et al. [2]
described above, but they switched from a square to a trapezoid environment
instead of a rectangular one. Their observations matched those of Barry et al.
as firing fields expanded in response to the novel environment. Yet, contrary
to the observations of Barry et al. this expansion reduced only slightly
afterwards and firing fields remained 20% to 30% larger compared with the
firing field sizes in a concurrently recorded square environment. In addition,
grid patterns in the trapezoidal enclosure showed a permanent decrease in
gridness score due to more elliptical, hexagonal patterns and less evenly
distributed firing fields. Based on these results Krupic et al. conclude “that
most assumptions about the invariant nature of grid cell firing are invalid”
and that “in particular the role of environmental boundaries has been
underestimated”.

###  5.6 Fragmentation

Figure 8:  Firing rate maps of three grid cells (rows A, B, C). Firing rates
are represented by blue (low) to red (high) colors. Peak firing rates are
given below the maps. The first and last columns show the firing rate maps in
the open-field, while the second and third columns show the firing rate maps
in the hairpin maze according to the animal’s running direction (indicated by
black arrows). Figure adapted from Derdikman et al. [11].

The phenomena of realignment and rescaling of grid cell firing patterns were
observed and examined in rectangular or circular environments in which the
rats could move around freely without encountering any obstacles. However,
real environments are more likely to consist of multiple, connected
subenvironments. To investigate whether grid cells exhibit a continuous or
fragmented firing pattern across such subdivided environments Derdikman et al.
[11] recorded MEA grid cells of rats that ran through a hairpin maze
consisting of several corridor-like passages connected by sharp, $180^{\circ}$
turns (white bars in fig. 8). The experimental protocol consisted of four
consecutive, twenty minute trials per day. In the first and last trial each
rat foraged in an open-field box ($1.5m\times 1.5m$). In between, i.e., in the
second and third trial, each rat ran back and forth in a hairpin maze which
was inserted into the open-field box. A total of 105 MEA grid cells in 16 rats
were recorded.

Figure 9:  Modified hairpin maze with two arms shortened by $50cm$. A,B:
Firing rate maps of two grid cells recorded in a modified hairpin maze. Firing
rates are represented by blue (low) to red (high) colors. Peak firing rates
are given below the maps. The animal’s running direction is indicated by black
arrows. C: Correlation between the activity (10cm bins) in a short arm with
the mean activity (10cm bins) in the corresponding long arms. Figure adapted
from Derdikman et al. [11].

Within this experimental setup grid cells showed the expected triangular
firing pattern in the open-field box and lost this pattern in the hairpin
environment (fig. 8). In the hairpin environment the firing fields of grid
cells were located at similar positions in individual maze arms relative to
the rats running direction, i.e., firing fields in maze arms with equal
running directions had similar positions while firing fields in maze arms with
opposite running directions had differing positions in general. This result
suggests that grid cell firing patterns reset at the turning points of the
hairpin maze. To test whether the firing fields were anchored to the preceding
or upcoming turning point, two arms of the hairpin maze were shortened by
$50cm$ (fig. 9). The resulting firing activity of each shortened arm was
correlated with the activity of all corresponding long arms using $10cm$ wide
bins and a progressive shift of the short arm to cover any possible alignment.
Figure 9C shows exemplarly the correlogram of one short arm with the mean
activity of all corresponding long arms. In general, the short arm correlates
more strongly with the longer arms if it is aligned to the preceding wall,
i.e., the activity pattern within a short arm is more similar to the start of
a long arm than to its end with respect to the shared wall. However, the end
of the short arm also correlates to some degree with the end of the long arms,
i.e., when aligned to the upcoming wall. Together, these results indicate that
the location of the firing fields is possibly determined by two mechanisms: in
part by a form of path integration after a turning point, and in part by some
form of environment-based alignment towards the upcoming turning point.

To exclude the possibility that the observed firing patterns could be caused
by the running pattern of the animals, the rats were trained to run the path
of the hairpin maze in the open-field environment without the presence of the
maze walls. The observed grid cell firing pattern matched the firing pattern
during the random foraging task in the open-field box. Thus, excluding a
behavioral cause for the fragmentation seen in the hairpin maze.

Derdikman et al. examined also whether the observed resetting of the grid cell
firing fields would influence hippocampal place fields. They recorded 111
place cells in CA3 (4 rats) and 47 place cells in CA1 (3 rats) and recorded
their activity using the same experimental protocol as above. Place cells
recorded while the animal was in the hairpin environment showed a
fragmentation of their firing fields similar to that of grid cells. Place
fields in arms with the same running direction were highly correlated, whereas
place fields in arms with different running direction were only weakly
correlated. This co-occurring fragmentation of MEA grid cell firing fields and
CA1/CA3 place fields is another indication for a possible coupling between the
spatial representations in the MEA and the hippocampus.

Figure 10: Prolonged exposure to an environment can lead to a global
representation in grid cells. (a) Schematic of the experimental setup. Two
perceptually identical environments A and B were connected by a corridor. (b)
Example of two firing rate maps (rows) fitted to a local and a global model
(columns). The rate map from an early session (top row) was better explained
by the local model, whereas the rate map from a late session (bottom row) was
better explained by the global model. Figures adapted from Carpenter et al.
[10].

The observations made by Derdikman et al. [11] and Krupic et al. [33] suggest,
that contrary to prior assumptions grid cell firing may primarily be
determined by local environmental cues instead of forming a single, global
representation of the environment. To test this hypothesis Carpenter et al.
[10] devised an experiment where rats foraged in two perceptually identical
enclosures connected by a corridor (fig. 10a). The walls of the environment
were painted black and distal cues were hidden by black curtains. Lighting was
provided by two single lights on the south wall of each enclosure. To control
for other sensory cues such as odor, which could help the animal to
distinguish between the compartments, the floor was rotated, the compartments
swapped, and the environment cleaned in the middle of each recording session,
i.e., each session consisted of a 40 minute trial (Trial 1), then the
reconfiguration and cleaning of the environment, and a second 40 minute trial
(Trial 2).

To determine if the firing pattern of a grid cell would locally replicate in
each compartment or if it would globally cover both compartments, Carpenter et
al. fitted a local and a global model to the firing rate map. In case of the
local model the fitted grid had the same orientation, scale, and phase in both
compartments. In case of the global model the fitted grid covered both
compartments, i.e., the phase remained continuous across compartments. During
the first recording sessions, the firing rate maps of the recorded grid cells
(85 cells, 8 rats) were best described by the local model confirming the
observations of Derdikman et al. and Krupic et al. described above (top row in
fig. 10b). However, with prolonged exposure to the environment, the fit of the
local model steadily decreased while the fit of the global model increased,
i.e., the observed grid cells formed a global instead of a local
representation of the environment with experience (bottom row in fig. 10b).
The transition from a local to a global representation appeared to occur
gradually and continuously. In particular, Carpenter et al. were able to track
the adjustment of the firing fields of a single grid cell over a prolonged
period of time observing that the firing fields “appeared to shift
continuously, rather than undergo a sudden transformation”.

Wernle et al. [68] conducted a similar experiment in which rats were first
trained to forage for food in two adjacent environments A and B that shared
common distal cues and were separated by a central wall. After training the
rats were then exposed to the joint environment AB by removing the separating
wall. Wernle et al. recorded 128 grid cells (10 rats) that were active in all
three environments (A, B, and AB). In agreement with the observations of
Carpenter et al. the majority of the grid cells represented the environments A
and B with separate grid patterns that differed in phase, and occasionally
also in orientation and scale. After removal of the central wall the separate
grid patterns merged into a single, locally coherent grid pattern. In contrast
to the observations of Carpenter et al. this reorganization of the grid
pattern happened on a fast time scale, i.e., within minutes during the first
trial in the joint environment. Furthermore, Wernle et al. report that firing
fields near the outer borders of the joint environment remained anchored to
their positions during reorganization while firing fields near the removed
central wall moved away from their original locations in order to self-
organize into a locally coherent grid pattern.

In summary, these results indicate that the early hypothesis of stable grid
patterns that span the entire environment and serve as a metric for space [46,
45] has to be revised to reflect these more recent observations of dynamic
changes in grid cell firing patterns and their anchoring to environmental
cues. Carpenter and Barry [9] provide a first discussion in this regard.

##  6 Neuronal Structure

Neurons with grid-like firing patterns were found in all populated layers of
the MEA [22, 58] as well as the PrS and PaS [3] (see Tang et al. [64] for a
recent contradictory report on layer III). Interestingly, the subsequently
described morphology of neurons found in these layers is not homogeneous,
i.e., grid-like firing patterns are exhibited by cells with varying
morphologies [56, 20]. It is an open question whether grid patterns are
generated independently by various types of cells or by just a single cell
type which then projects a grid signal to other cells that merely forward the
signal.

###  6.1 Neuron Morphology in the Entorhinal Cortex

The entorhinal cortex is the origin of a prominent pathway, the perforant
path, that connects the EC with all regions of the hippocampus. In addition,
the EC receives inputs from various regions of the brain including the
neocortex [72]. As a consequence, the EC is commonly thought of as the gateway
to the hippocampus. As such the neuronal structure of the EC has been
extensively studied even before the discovery of grid cells. If not noted
otherwise the neuronal structures reported below refer to the rat brain. As
the PHR-HF is constantly present in all mammalian species with little
variation during its phylogenetic development, insights into the neuronal
structures of rat EC can be generalized to other mammalian species, e.g.,
humans to a certain degree [25]. The following paragraphs will briefly
summarize the key morphological properties of principal neurons found in the
EC layer by layer and provide an overview of other structural properties like
local microcircuits and indications of a possible columnar organization of the
EC.

The overall structure of the EC comprises six layers. Layers I to III are
referred to as superficial layers whereas layers IV to VI are referred to as
deep layers. Layer I contains only few neuron soma and is mainly occupied by
dendritic and axonal processes originating from cells located in lower layers.
Layer II is densely populated by stellate cells and small pyramidal cells as
its principal neurons [39]. In LEA layer II splits into Layers IIa and IIb.
Layer IIb is a continuation of MEA layer II whereas layer IIa is located more
superficially and consists of stellate cells forming local clusters or
“islands” [35]. The principal cells of layer III are mostly pyramidal cells.
Layer III is relatively thick and is followed by cell poor layer IV which
contains only sparsely scattered pyramidal cells. The relatively thick layer V
hosts mostly small pyramidal neurons [39]. Both layer IV and layer V are
thicker in LEA than in MEA due to a higher neuron count and a less dense
packaging of the cells in LEA [35]. The principal cells of layer VI are mainly
globular and polygonal cells [39].

####  6.1.1 Layer II

The principal neurons in EC layer II are of stellate and pyramidal morphology.
In addition, small numbers of neurons with fusiform, horizontal tripolar, and
bipolar shapes were found. The distribution of the two main neuron types,
stellate cells and pyramidal cells, is not uniform between LEA and MEA. In the
latter stellate cells are more abundant and pyramidal cells are mostly found
near the layer III border [30].

Figure 11: Photomicrograph of a stellate cell in layer II of the MEA. (left)
The numerous primary dendrites taper gently and remain relatively thick far
from the soma. (right) Image section showing thick dendrites covered evenly
with spines. Figure adapted from Klink and Alonso [30].

The stellate neurons in layer II have triangular, rectangular, or trapezoidal
cell bodies [39]. They possess several thick, primary dendrites which decrease
in numbers dorsoventrally from an average of ten down to seven (max. 14, min.
5) [17]. The average ratio of primary dendrites to the number of dendritic
endpoints is about 1:15 (min. 1:10.8, max. 1:24.7) and measurements of the
overall dendritic length range from 12.8mm to 18.1mm [39]. Reports on the
shape of the dendritic domain vary. Lingenhöhl and Finch [39] report a
spectrum of shapes from circular domains with centrally located soma to broad,
ellipsoidal domains with marginally located soma. Klink and Alonso [30]
describe double V-shaped, bi-triangular dendritic domains. Others refrain from
characterizing the overall, general shape of the dendritic domain [18, 54, 7].
In general the apical dendrites of stellate cells branch profusely in layer II
and layer I and reach up to the pial surface [18, 39, 30, 54]. The basal
dendrites extend within layer II and superficial regions of layer III [30,
54]. In some cases basal dendrites extend down to layer IV [39]. The entire
dendritic tree of the stellate cells in layer II is evenly covered by
dendritic spines (fig. 11) with an estimated 0.5 to 1 spines per 1µm [39].
Considering an overall dendritic length of about 15mm leads to an estimate of
7500 to 15000 spines per stellate cell where each spine can host one or more
synaptic connections. The stellate cell’s axon gives rise to about three to
five axon collaterals within its first 400µm. The axon collaterals are
oriented towards the superficial layers, branch repeatedly, and form a
“delicate net over the entire dendritic domain” [30]. A number of long axons
branches extend parallel within layer I in mediolateral direction beyond the
cell’s dendritic field [30, 54]. Additionally, axon collaterals spawning in
deep portions of layer III and in layers IV to VI were also observed [30].

Pyramidal neurons in layer II have one or two apical dendrites that start to
branch at the border between layer II and layer I. Secondary dendrites appear
to branch off almost perpendicular resulting in a bitufted appearance of the
dendritic arbor [18, 30]. The basal dendrites of layer II pyramidal cells
branch extensively around the soma and are confined to layer II and the upper
parts of layer III. The density of dendritic spines is higher in comparison to
stellate cells, especially in case of the apical dendrites [30]. The ratio of
primary dendrites to the number of dendritic end points was measured in one
case to be 1:9.4 [39]. The axon of layer II pyramidal cells is thin and
meanders through layer II,III, and IV giving off several, mainly horizontal
collaterals that branch multiple times. Some collaterals ascend to upper
layers [18, 30]. The main axon continues on a radial path towards the angular
bundle666The angular bundle lies below layer VI. (AB) [30].

####  6.1.2 Layer III

The principal neurons of EC layer III are pyramidal cells with a prominent,
triangular cell body and radially extending primary dendrites. The apical
dendrites of large pyramidal cells bifurcate in layer III and extend further
into layer II and layer I where they continue to branch. Smaller pyramidal
cells do not bifurcate but still extend into layer II and layer I. Basal
dendrites branch extensively in layer III, but extend to deep layers as well.
All dendrites of layer III pyramidal cells are densely covered with spines
[39, 54]. Lingenhöhl and Finch report of up to five primary dendrites with an
average ratio between primary dendrites and the number of dendritic endpoints
of about 1:12 (min. 1:11.2, max. 1:15.5) [39]. In the same study the total
dendritic length of a big pyramidal neuron was measured to be 11.3mm and the
corresponding length of a small pyramidal neuron was measured to be 6.6mm. The
axon of layer III pyramidal cells gives off several collaterals already within
layer III which either extend parallel to the layer along the anteposterior
axis or extend towards the superficial layers where they stay restricted to
the region occupied by the dendritic domain of the cell. Further collaterals
of the axon branch off in layer V. Compared to the axons of layer II neurons,
the axonal branches of layer III pyramidal cells appear to be distributed more
evenly [54].

####  6.1.3 Layer IV

Layer IV of the EC is commonly regarded as virtually devoid of neurons.
However, Lingenhöhl and Finch [39] managed to sample a number of sparsely
scattered regular-sized and large-sized pyramidal cells as well as one spindle
cell in this layer. The larger pyramidal cells and the spindle cell were
located at the border to layer V. The regular-sized pyramidal cells (n = 3)
had between six and eleven primary dendrites with an average ratio between
primary dendrites and the number of dendritic endpoints of about 1:9 (min.
1:7.4, max. 1:10.1). The total dendritic length of one cell was estimated to
be 8.3mm. The basal dendrites branched predominantly in layer IV and layer V
with a few dendrites reaching up to deep portions of layer III. The apical
dendrites extended into layer II and layer I where they ramified. On their way
up they bifurcated already in layer IV without any further bifurcation in
layer III, leaving the latter essentially free of layer IV dendrites. Spines
were present on all parts of the dendritic tree. The larger pyramidal cells (n
= 2) at the border to layer V had five and eight primary dendrites,
respectively with ratios between primary dendrites and the number of dendritic
endpoints of 1:11.2 and 1:8.8. The total dendritic length was measured as
13.0mm and 13.3mm. In general the dendritic domains were similar to that of
the regular-sized pyramidal cells. Yet, spines were present only after the
first or second dendritic bifurcation. The spindle cell had only 3 primary
dendrites and a ratio between primary dendrites and the number of dendritic
endpoints of 1:13.6. The total length of the dendritic domain was estimated to
be 8.7mm. A single apical dendrite extended into layer II and layer I where it
branched extensively. The two basal dendrites extended towards the subiculum.

####  6.1.4 Layer V/VI

The principal neurons of EC layer V and layer VI are pyramidal cells with an
average of 5.4 primary dendrites (min. 3, max. 7) and an average ratio between
primary dendrites and the number of dendritic endpoints of 1:5.6 (min. 1:4,
max. 1:8.7). This ratio is significantly lower compared to the ratios of
neurons in the upper layers [39]. The basal dendrites distribute horizontally
in layer V and layer VI where they branch sparsely or not at all. In one cell
the basal dendrites extended into layer IV. The apical dendrites either
bifurcate in layer IV and deep layer III or they bifurcate in layer III. In
the former case the dendrites branch extensively in layer IIa and layer I. In
the latter case the dendrites branch only sparsely until they reach layer I.
Spines were present on all segments of the dendritic domain. The average
dendritic length was estimated to be 6.2mm (min. 4.8mm, max. 7.2mm) [39, 54].
Axonal branches were evenly distributed in layer V (66.1%) as well as in layer
III (33.3%) [54].

####  6.1.5 Interneurons

In a recent publication Buetfering et al. [6] were able to shed some light on
the properties of interneurons in layer II of the MEA with the help of
optogenetics. They used an adeno-associated virus to deliver and selectively
incorporate channelrhodopsin-2, i.e., a light-activated ion channel into
approximately 50% of the population of GABAergic777inhibitory interneurons in
the MEA of mice. This enables the external activation of the interneurons by
locally applying pulses of blue laser light through an optic fiber implanted
parallel to the common set of tetrodes used for recording the electrical
activity of the neurons.

The targeted interneurons control the activity of principal neurons in layer
II by local GABAergic connections. The axons of the interneurons extend and
branch widely within layer II where they form basket-like complexes around
other neurons. Their dendritic trees are mainly located in layer I and only
sparsely covered with spines receiving excitatory (fast AMPA-mediated and slow
NMDA-mediated) as well as inhibitory (GABAergic) input [27]. A part of this
input comes from grid cells with various phases. As a consequence the firing
pattern of interneurons is not grid-like. Like many other cells of the MEA the
firing rate of interneurons is modulated by running speed [6].

Collective activation of the interneurons via the light-activated ion channels
silenced neurons of all cell types reliably. The delay ($<$ 5ms) between the
onset of the laser pulse and the following inhibition of postsynaptic cells is
indicative of a monosynaptic connection between interneurons and principal
cells. The recovery of inhibited, postsynaptic cells after the laser was
switched off took 25ms. This duration is consistent with the dynamic
properties of $\mathrm{GABA}_{A}$ receptors [6].

Based on their results Buetfering et al. [6] speculate that interneurons may
control the gain of the grid cells rather than being essential for the
generation of the grid pattern.

###  6.2 Topographical Organization and Microcircuits

The topographical and modular organization (section 4) of grid cells on a
functional level raises the question whether this organization is reflected in
the underlying neuronal structure. Within the last three decades a number of
studies [35, 24, 39, 23, 71, 17, 54, 8, 4, 7, 20] have identified several
anatomical properties of neurons in the MEA that exhibit a dorsoventral
gradient and/or a modular organization. However, it is yet unknown if these
anatomical properties are causally related to the observed functional
organization of grid cells [7]. A very recent study by Gu et al. [20] provides
first tentative evidence that grid cells with similar grid phases form local
clusters in the MEC that are arranged in a noisy two-dimensional lattice.

Many of the anatomical properties reported below were identified by
histochemical methods, i.e., the selective staining of cell parts by
exploiting unique chemical properties of those parts [36]. This includes
direct binding of dyes to negatively charged nucleic acids (e.g., Nissl
staining using cresyl violet [8]), incorporating dyes in the process of enzyme
catalysis (e.g., cytochrome oxidase [73, 23, 8]), or utilizing the selectivity
of exogenous antibodies towards antigens present in the cells (e.g., reelin
immunoreactivity [7]).

Figure 12: Two reconstructions of neuronal morphologies in the MEA. (a)
Reconstruction of axons (blue) and dendrites (red) of a layer II stellate
cell. Light brown regions indicate small layer II patches of cytochrome
oxidase activity. Dark brown regions indicate large patches of cytochrome
oxidase activity at the border of the MEA. D: dorsal, V: ventral, P:
posterior, A: anterior, RS ctx: retrosplenial cortex. (b) Reconstruction of
axons (green) and dendrites (red) of a cell located in a border patch
identified by cytochrome oxidase activity. Left: parasagittal section, right:
tangential section. L: lateral, M: medial, other labels as in (a). Figure
adapted from Burgalossi et al. [8].

Figure 13: Large-scale overview of MEA layer I axons. (a) Reconstruction of
MEA based on tangential sections. The sections were stained for myelin. Light
brown patches are cell clusters in layer II identified in the sections as
clusters of white somata surrounded by myelin. Dark brown patches are
identified indirectly by extensive axon bundles on the dorsal surface of each
patch. Bundles of putative centrifugal axons (blue) densely populate layer I
with about twice as many bundles as small patches. Axons of cells within one
large patch are drawn in green. D: dorsal, V: ventral, L: lateral, M: medial.
(b). Reconstruction of the MEA patches superimposed on a posteriorlateral view
of the rat brain. Figure adapted from Burgalossi et al. [8].

The layers of the MEA contain regularly distributed clusters of neurons, which
are sometimes referred to as islands. Staining for the enzymes glycogen
phosphorylase and cytochrome oxidase reveals these clusters in layer I and
layer III as well as in layer II (fig. 12), respectively. Both enzymes are
part of metabolic processes and the intensity of the resulting stain reflects
the metabolic activity in the particular region [23, 8, 7]. The presence of
cell clusters in layer II is also indicated by staining for myelin (fig. 13)
[8] and by staining for immunoreactivity to the calcium-binding phosphoprotein
R2D5. In the latter case layer III cells are predominantly R2D5 positive while
layer II cells are R2D5 negative resulting in a pattern of interweaved,
separate columns [24]. Within these columns the dendritic fields of layer II
and layer III neurons receive separate projections from different brain areas
[23]. The cell clusters in layer II are locally uniform in size and are
surrounded by myelinated fibers. Along the dorsoventral axis the average size
of the clusters increases while the average cell size and the degree of
myelination decreases. Adjacent to these cell clusters lies a series of larger
clusters along the dorsal and medial borders of the MEA (fig. 12 and 13) [8].

Figure 14: Variation of morphological properties of MEA layer II stellate
cells along the dorsoventral axis. The plots show (from left to right) the
cross sectional soma area, total dendritic surface area, average dendritic
surface area, and number of primary dendrites of reconstructed stellate cells
plotted against the soma’s distance from the dorsal pole. Open and closed
circles refer to two different recording conditions used to measure the cell’s
electrotonic properties. Both conditions did not influence the measurement of
the morphological properties shown here. Figure adapted from Garden et al.
[17].

The variation of cluster size, cell size, and degree of myelination along the
dorsoventral axis is accompanied by further variations on the level of
individual cell properties. Garden et al. [17] studied these variations in MEA
layer II stellate cells in detail. Morphologically layer II stellate cells
show a dorsoventral decrease in cell body perimeter, cell body cross sectional
area, and dendritic surface area. The latter is not caused by any changes to
the surface area of individual dendritic branches, but by an overall reduction
in the number of primary dendrites from an average of ten primary dendrites at
the dorsal pole to an average of seven primary dendrites at more ventral
locations (fig. 14). Furthermore, the total number of dendritic branch points
also decreases dorsoventrally. In addition to these morphological changes the
electrotonic properties of layer II stellate cells change too along the
dorsoventral axis. In particular, the input resistance and the membrane time
constant increase dorsoventrally. As a consequence the current threshold to
trigger action potentials (APs) decreases 4-fold along the dorsoventral axis,
i.e., the amplitude of positive current required to trigger an AP is much
larger at dorsal neuron locations. Moreover, the time window for the detection
of coincident inputs also varies in dorsoventral direction. It is about three
times wider in the most ventral locations than it is in the most dorsal ones.

Information about the microcircuitry within the EC is incomplete and commonly
based just on information about the coarse morphology of principal neurons.
Potential connections between neurons are often extrapolated based on
overlapping input and output regions which are deduced from the neurons’
dendritic and axonal domains. Although such piecewise and tentative
information does only provide limited insight into the actual neuronal
circuitry it does provide a set of constraints to which possible models and
explanations of grid cell behavior have to adhere. To this end, the following
paragraphs summarize the available, but rather fragmentary information about
EC microcircuitry.

The intra-EC projections of principal neurons can be described by their
horizontal extent parallel to the layers of the EC and their longitudinal
extent in the orthogonal direction. In general, neurons in layer III are more
restricted in their horizontal and longitudinal extent than neurons in layer
II. Together they are both more restricted in their extent than neurons in
layers IV to VI [35]. In all layers, the majority of projections are oriented
longitudinally and distribute between the respective layer of origin and the
layers above, i.e., cells in layer V project to layers V, IV, and III; cells
in layer III project to layers III, II, and I; and cells in layer II project
within layer II as well as adjacent parts of layers I and III [71]. In
addition, minor projections in the opposite direction were also observed,
i.e., from layer II cells to layers IV, V, and VI [30, 71]; and from layer III
cells to layer V [54]. By comparison layer V neurons contribute three times
more intrinsic connections than neurons in layer II and five times more than
neurons in layer III. The axonal arbor of layer V neurons is cone shaped with
its base in layer V and its peak in layer II. This suggests that layer V cells
predominantly interact with each other via their basal dendrites. In contrast,
the axonal domain of layer II neurons forms an inverted cone with its base in
layer I and its peak between layers III and V. This allows layer II cells to
communicate with a wide range of neurons in layers II, III, and V. Layer III
neurons may represent some form of bidirectional link between deep and
superficial layers as they can receive input from all layers and possess axons
that converge on smaller subsets of layer II and/or layer V cells [54]. Local
excitatory connections between principal cells were observed in layer III [70,
71] and layer V [71], but none [71, 7] or few [54] in layer II. The latter
results are called into question in a recent study by Winterer et al. [70] who
measured (in vitro) intra-layer excitatory connections between layer II
stellate cells at a rate of $2.5\%$. In the same study Winterer et al. also
report on predominantly unidirectional excitatory connections between layer
II/III pyramidal cells and layer II stellate cells ($13.5\%$ and $7.0\%$,
respectively).

Neurons in the larger clusters at the EC border are targeted by axons of layer
II stellate cells (fig. 12a) as well as layer III pyramidal cells. The so
called centrifugal axons appear to be bundled and oriented in dorsomedial
direction (fig. 13a). The targeted cells in the large clusters differ in their
morphology from all other neurons in the MEA. Their dendritic trees are small
and extend not beyond their home cluster. They possess three main axons: one
axon descending towards the presubiculum, one circumcurrent axon targeting
many other large clusters along the EC border, and one centripetal axon
specifically targeting one to two small layer II clusters in which it
arborizes. In addition, a number of axons branch locally within the home
cluster (fig. 12b). The firing characteristics of large cluster cells range
from spatially multi-peaked firing to spatially broad tuning, all with a high
degree of modulation by head direction [8].

A more detailed view on the microcircuitry of MEA layer II was recently
provided by Varga et al. [67]. They identified two major, non-overlapping cell
groups in layer II by utilizing the cells’ immunoreactivity to either reelin
($53\pm 2.6\%$ of all cells) or calbindin ($44\pm 2.2\%$ of all cells). Only
$2.8\pm 1.1\%$ of all cells were reelin and calbindin double positive888Total
number of cells analyzed in 3 rats: 1152. More importantly, injection of the
retrograde tracer biotinylated dextrane amine (BDA) into the ipsilateral
dentate gyrus labeled predominantly ($98.5\pm 0.5\%$) the reelin positive
principal cells indicating that only these cells project to the dentate gyrus.
In contrast, the calbindin-expressing principal cells were found to project
extra-hippocampally to the contralateral entorhinal cortex. In addition, Varga
et al. were able to show that reelin positive and calbindin positive cells
each interact with a different population of inhibitory interneurons. Reelin-
expressing principal cells interact with fast-spiking interneurons, whereas
calbindin-expressing cells interact with interneurons that form basket-like
axonal domains surrounding the principle cells providing perisomatic
inhibition. The observations of Varga et al. are confirmed by a more recent
study by Ray et al. [55]. In addition, Ray et al. were able to show that
calbindin negative cells were primarily stellate cells whereas calbindin
positive cells were primarily pyramidal cells. The latter were observed to
form hexagonal patches near the layer I/II border. This hexagonal structure
led Ray et al. to hypothesize that the cells’ arrangement “might be an
isomorphism to hexagonal grid activity”. However, they were not able to
measure the spatial modulation in a sufficient number of identified cells to
assess the validity of this hypothesis. In a related study Tang et al. [63]
were able to find potential evidence that grid cells are “preferentially
recruited” from the population of calbindin positive cells, i.e., pyramidal
cells, and that border cells stem preferentially from calbindin negative
cells, i.e., stellate cells.

Regarding projections into the EC Lingenhöhl and Finch [39] as well as
Bonnevie et al. [4] could make interesting, and to some degree contradicting
observations. Lingenhöhl and Finch [39] investigated projections from the
hippocampal area to the EC in vitro. In response to electrical stimulation of
either the DG, CA1 or CA3 all cells that were observed in the EC showed
synaptic responses in form of inhibitory postsynaptic potentials (IPSPs). No
clear excitatory postsynaptic potentials (EPSPs) could be identified
indicating that projections from hippocampal areas to the EC are predominantly
inhibitory. However, Bonnevie et al. [4] investigated projections from the
hippocampus to the MEA in vivo on a functional level. They observed grid cells
in the MEA before and after the inactivation of the hippocampus. In the latter
case grid cells lost their grid-like firing pattern and became direction
sensitive. Based on these results Bonnevie et al. concluded that grid cells
require excitatory drive from the hippocampus. The results of Lingenhöhl and
Finch and Bonnevie et al. do not necessarily contradict each other. The
postulated excitatory drive could reach the EC on an indirect route, e.g., via
the subiculum or the pre- and parasubiculum.

##  7 Conclusions

Since their discovery a decade ago grid cells have attracted considerable
attention resulting in a better and more detailed understanding of their
functional characteristics as well as a better understanding of the
parahippocampal-hippocampal region their embedded in. The observation of new
grid cell phenomena like the strong influence of environmental geometry or the
fragmentation of grid patterns question early hypothesis on grid cell function
and call for revised models of grid cell behavior that allow to formulate new
hypothesis and direct further experimental investigation. Considering the
presence of related cell types like border cells, head direction cells,
irregular spatial cells, and non-spatial cells that all converge on
hippocampal neurons and may receive feedback from these in turn, it appears
essential that new models should aim to integrate these different types of
signals into a single, conceptual framework that explains the processing of
information within a wider context that extends beyond isolated descriptions
of individual cell types and their behavior.

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