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ScienceDirect
Spatial coding and attractor dynamics of grid cells in the
entorhinal cortex
Yoram Burak
Recent experiments support the theoretical hypothesis that
recurrent connectivity plays a central role within the medial
entorhinal cortex, by shaping activity of large neural
populations, such that their joint activity lies within a continuous
attractor. This conjecture involves dynamics within each
population (module) of cells that share the same grid spacing.
In addition, recent theoretical works raise a hypothesis that,
taken together, grid cells from all modules maintain a
sophisticated representation of position with uniquely large
dynamical range, when compared with other known neural
codes in the brain. To maintain such a code, activity in different
modules must be coupled, within the entorhinal cortex or
through the hippocampus.
Addresses
Edmond and Lily Safra Center for Brain Sciences, and Racah Institute of
Physics, Hebrew University, Jerusalem 91904, Israel
Corresponding author: Burak, Yoram ([email protected])
Current Opinion in Neurobiology 2014, 25:169–175
This review comes from a themed issue on Theoretical and
computational neuroscience
Edited by Adrienne Fairhall and Haim Sompolinsky
For a complete overview see the Issue and the Editorial
Available online 20th February 2014
0959-4388/$ – see front matter, # 2014 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.conb.2014.01.013
Introduction
Grid cells in the dorsocaudal medial entorhinal cortex [1]
(dMEC) provide a captivating glimpse into the inner
workings of neural computation in the brain. Their geometric firing patterns in the animal’s environment, and
their synaptic projections to place cells in the hippocampus, link grid cells to spatial computation and representation. Their discovery in the entorhinal cortices of rats
[1], mice [2], bats [3!], monkeys [4], and most recently in
humans [5], suggests that their computational role was
highly conserved during evolution.
Here I review how theoretical models have recently
contributed to the understanding of activity in dMEC,
focusing on two topics: the properties of grid cell activity
as a code for position, and network dynamics within
modules — with a focus on attractor models. Studies
related to development, and to the precise timing of
spikes relative to the theta oscillation in rodents, are
mentioned only briefly, within these contexts. Several
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salient properties of grid cell activity are summarized in
Box 1 and in Fig. 1.
The grid cell code for position
The periodic spatial activity pattern of grid cells raises
questions about the representation of position in the
entorhinal cortex. Recently, it has become clear [6!!] that
grid cells are functionally grouped into large sub-populations (modules), in which all cells share identical spacing
and orientation, and differ in the phase of their periodic
response (Box 1 and Fig. 1). Because of the periodic
nature of the grid response, the neural activity of all cells
within a module encodes position up to an arbitrary
discrete displacement (Figs. 1 and 2a), and may thus
seem redundant and ambiguous. However, the activity
of several modules, taken together, can encode a large
range of positions without ambiguity [7–10] (Fig. 2b).
Below, we refer to the range of unambiguously
represented locations as the capacity of the code, and
to the accuracy in which position can be inferred from the
neural activity as its resolution. Increasing the number of
neurons by adding modules typically results in exponential increase of the capacity, whereas adding neurons in a
place-cell representation leads only to linear increase of
the capacity [9]. Thus, the grid cell code has a much larger
representable range than a place-cell code with the same
resolution and number of neurons. This range can exceed
by far the behavioral foraging range in rodents [9].
What are the possible benefits of having such a huge
capacity? One possibility is that the huge capacity confers
the neural code with error-correcting capabilities [9,11]:
for example, a severely corrupted phase in one of the
modules will, in general, throw off the represented position to an unrealistic location, allowing for the error to be
identified and corrected based on phases of the other
modules. Another possible benefit of the large capacity is
that grid cells may enable global remapping in the hippocampus [12]: independently shifting the phases in different modules, or rotating the axes of the grids relative to
each other, can produce distinct representations for
different environments by place cells, with negligible
likelihood for overlap between the neural codes — a
property that is inherited by place cells if their firing
field is determined by inputs from the entorhinal cortex
[7–9,13,14].
Coupling between modules
The neural representation in each module is probably
prone not only to occasional severe errors, but also to
Current Opinion in Neurobiology 2014, 25:169–175
170 Theoretical and computational neuroscience
Box 1 Essential properties of grid cell activity
1. Grid cells fire in multiple locations within the animal’s environment.
These locations are approximately arranged on vertices of a
triangular lattice that tiles the plane [1]. Grid spacing of a given cell
remains similar in different environments [1] (Fig. 1a).
2. Grid cells in dMEC are grouped in discrete modules [6!!,29]. Cells
in the same module share the same grid spacing, orientation, and
precise structure of the lattice [1,6!!,28!!], but differ in phase
(Fig. 1b). All phases are (probably) densely represented.
3. Cells from modules with increasing spacing are found in increasingly ventral locations along the dorsoventral axis of dMEC [1,6!!]
(Fig. 1c). In rats, spacings range from approximately 30 cm up to (at
least) several meters [21]. The discrete spacings form, approximately, a geometric series with a ratio #1.4 between successive
modules, averaged across module pairs and animals [6!!].
4. Conjunctive cells: A sub-population of grid cells is selective also
to head direction [31]. The activity of many cells in the entorhinal
cortex is also modulated by the animal’s velocity.
5. Phase precession: In rodents, spike timing is modulated temporally, in synchrony with the LFP theta oscillation. The timing of
spikes gradually recedes relative to the LFP theta oscillation as
the animal crosses a firing field [22,24], similarly to place cells in
the hippocampus.
6. Grid phases and orientations are closely matched when repeatedly placing the same animal within a given environment. Different
animals often express similar grid orientations within the same
environment (T. Stensola et al., abstract 769.15/JJJ34, Neuroscience 2013, San Diego 2013).
gradual diffusive drift, induced by intrinsic noise. Such
errors are bound to accumulate and gradually corrupt the
estimate of position [15]. This problem is of special
significance in the absence of salient sensory cues, since
the phases represented in different modules might drift
independently of each other under these conditions.
Possible perturbations of M phases occupy a 2M-dimensional space, but when the animal moves in the presence
of sensory cues, all the phases update in unison, in
response to the motion in two dimensions (2D). Thus,
perturbations that match valid positions (in the local
vicinity of the animal) lie in a restricted, 2D subspace.
It is unknown whether drift within the larger 2M dimensional space occurs in the absence of sensory cues
(Fig. 2c). Such independent drift would rapidly lead to
combinations of phases that do not match any position in
the local environment of the animal [9,16!!].
The neural circuitry can potentially solve this problem if
the modules are coupled to each other, in a way that
prevents them from drifting independently, thus suppressing the independent components of the noise (Fig. 2c).
Such an interlocking mechanism might be implemented
through feedback from place cells in the hippocampus, as
recently proposed in [16!!], or perhaps by connectivity
within dMEC. Hippocampal inputs may be instrumental
also in resetting the grid cell representation based on
sensory inputs [17], and in correcting large errors in
individual phases. Overall, coupling the modules through
the hippocampus may result in a neural encoding of
Current Opinion in Neurobiology 2014, 25:169–175
Figure 1
(a)
(b)
cell 1
cell 2
(c)
Same module
1m
Different modules
Current Opinion in Neurobiology
Schematic and idealized firing fields of grid cells. (a) Firing rate of a grid
cell (right), as a function of position within a 1 m " 1 m environment (left).
(b) Firing fields of two cells (blue, red) belonging to the same module
share the same grid spacing and orientation, but differ in phase. (c)
Firing fields of two cells from different modules (blue, red) differ in
spacing, and possibly also in orientation.
position that achieves exquisite stability, alongside an
exponentially large capacity [16!!].
Optimization of the grid spacings
Several recent works [18!,19,20!] considered how the
resolution of the grid cell code depends on the number
of neurons and the allocation of grid spacings to modules,
while keeping the maximal grid spacing fixed. In Refs.
[18!,19] the resolution was estimated by evaluating the
Fisher information (a lower bound on precision in which
position can be decoded from the spiking activity), as well
as the performance of maximum-likelihood decoders,
while assuming that grid cells fire independently as
Poisson spike generators. The Fisher information was
maximized by choosing grid spacings that form a geometric series. Under these conditions, the resolution of
the code is related to the smallest grid spacing, and is thus
exponentially smaller than the maximal spacing. Tiling
an area comparable to the largest grid spacing with a place
cell code of the same resolution as the grid cell code,
would require an exponentially large number of neurons.
Recent characterization of grid spacings in rats [6!!]
points to an approximately fixed ratio between successive
grid spacings, in accord with the above optimization
principle. Moreover, simple scaling arguments [20!],
somewhat different from those proposed in [18!,19],
further support the optimality of a geometric ratio of grid
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Coding and attractor dynamics in grid cells Burak 171
Figure 2
(a)
(c)
?
?
Independent drift
?
1
?
?
?
2
?
3
?
?
?
?
?
1
?
?
‘Interlocking’
2
?
3
(b)
Module 1
Module 2
Module 3
Grid spacing
Inferred
position
Current Opinion in Neurobiology
(a) All neurons in a single module respond to position with the same grid
periodicity. Hence, their joint activity represents position only up to an
arbitrary discrete displacement. (b) In a one dimensional (1D) analog of
the grid cell code, all neurons in a single module represent position as a
1D phase, relative to the grid spacing. The thin black bars represent the
encoded phase in each module (1st, 2nd, and 3rd), and the red, blue,
and green bars represent the possible locations compatible with the
activity in each module. The width of the colored bars represents
schematically the accuracy of readout. Combining the activity from all
neurons in the three modules can greatly reduce the ambiguity. Here, the
joint activity from the three modules is compatible with a single location
(yellow bar) within the range shown. The full range over which positions
can be unambiguously inferred is the capacity of the code, whereas the
width of the yellow bar roughly represents the resolution of the code. (c)
In the absence of sensory cues, the phase represented in each module
will probably drift gradually, relative to the value that matches the true
position of the animal. These drifts may occur independently in different
modules (top). Alternatively, an interlocking mechanism might constrain
these drifts to occur in unison, in compatibility with motion within the
local environment of the animal (bottom).
range of the grid-cell code can be exponential in the
number of neurons. However, optimizing the dynamic
range is a different goal from the one considered in
[18!,20!], where resolution was minimized while keeping
the largest grid spacing fixed — and these two objectives
may perhaps lead to different predictions for the optimal
sequence of grid spacings.
Within all theoretical treatments discussed above, the
resolution of the grid-cell code is essentially set by the
lowest grid spacing, which is known to be approximately
30 cm in rats (further reduction of the resolution is only
linear in the number of neurons). It is less clear how the
capacity depends on the precise grid spacings and orientations.pIfffiffiffi the ratio between successive spacings is precisely 2 in rats [6!!] then many of the spacings are
integer multiples of each other, which in 1D would
severely limit the capacity of the code [9]. However,
small deviations from this ratio might be sufficient to
restore the large capacity of the code. Moreover, in 2D the
capacity depends also on the relative grid orientations in
different modules. Thus, obtaining a well grounded estimate of capacity in rats requires further experimental and
theoretical analysis. Even if the grid cell code in rats can
represent an exponentially large range of positions
relative to the maximal grid spacing (#10 m [6!!,21]), it
is unknown whether the brain makes full use of this large
capacity — an important question from the biological and
behavioral perspectives.
Two recent works addressed spike timing of grid cells,
relative to the LFP theta cycle in rodents. Spike timing of
individual cells, recorded in rats running through linear
tracks [22], were found to contribute more to Fisher
information on position than spike counts [23!]. Phase
precession was recently demonstrated also in open
environments [24]. These observations should be taken
into account when assessing the resolution of the code.
Qualitatively, it is unlikely that they affect conclusions
about the exponential capacity of the representation
[9,16!!,18!], because the noise observed in the temporal
coding is largely independent in subsequently visited
firing fields [23!].
Attractor dynamics of grid cell activity
spacings, and predict that to minimize the number of
encoding neurons, while keeping the resolution and
grid spacing fixed, this ratio should be #1.4 or
maximal
pffiffi
e ’ 1:65 in 2D (depending on details of the decoding
scheme used to extract phase from the neural activity) —
in close agreement with the observed value [6!!].
From a theoretical perspective it is interesting to consider, as an optimization goal for a neural code of a
continuous variable, its dynamic range: the ratio between
the capacity and the resolution. The works discussed
above [9,16!!,18!,19,20!] demonstrate that the dynamic
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The main goal of this section is to review the interplay
between theoretical ideas and experiment, since much of
the progress in the past two years has been at this interface. Theoretical models for emergence of grid-like firing
were recently reviewed in [25!,26] (see also [11]). In this
section we focus on populations of cells that share the
same grid spacing and orientation.
Does grid-cell activity lie within a two-dimensional
attractor manifold?
Attractor models of grid cell activity [7,8,15,27] propose
that the activity of multiple grid cells is tightly
Current Opinion in Neurobiology 2014, 25:169–175
172 Theoretical and computational neuroscience
Figure 3
(a) Neural population
(b)
Single neuron
50 cm
constraints, since they persist even in novel environments, and under hippocampal remapping [28!!]. Each
of the above predictions, if refuted, would have posed a
critical challenge for the attractor hypothesis, whereas
models based on feed-forward inputs to dMEC alone do
not explain or predict these features.
Recurrent synaptic connectivity in dMEC
Current Opinion in Neurobiology
An attractor model of grid cell activity (adapted from Ref. [15]). (a)
Recurrent synaptic connectivity within a 2D cortical sheet leads to stable
activity patterns at the population level with periodic grid-like structure.
All translations of this pattern are possible stable states of the dynamics
(red curve: distance dependence of the synaptic connectivity). (b) A
separate mechanism (not shown) shifts the population activity pattern in
exact correspondence with motion of the animal in its environment. As a
result, the activity of a particular neuron displays a grid pattern as a
function of position.
coordinated by recurrent connectivity, such that it lies
within a two-dimensional attractor (a continuous manifold
of stable steady states). An example is sketched in Fig. 3:
cells are arranged on a two-dimensional layer, and ‘mexican-hat’ connectivity between them leads to activity
patterns with grid-like structure within the neural sheet
(Fig. 3a). All translations of the pattern correspond to
possible steady states of the dynamics. Sensory inputs, in
conjunction with inputs representing self-motion, select
one of these phases in correspondence with motion in the
environment. As a result, single-neuron responses form a
grid pattern as a function of the animal’s position
(Fig. 3b). Different locations in space, whose separation
matches the discrete periodicity of the grid are
represented by the same population activity pattern.
Attractor models were partly motivated by measurements from nearby grid cells, recorded from the same
tetrode, which display identical grid spacing and orientation, but different spatial phases [1,3!] — a feature that
arises naturally in these models [7,8,15]. Recent analysis
and experiments confirmed a number of other salient
predictions of these models [15]: grid spacings and orientations are discretized in modules [6!!]; deviations from a
precise triangular-lattice are common to all cells within a
module [6!!,15,28!!]; transient modifications in the grid
structure (for example, during rescaling of the environment [29]) are tightly correlated in cells from the same
module [6!!,28!!]; and the relative spatial phases in cell
pairs are remarkably stable [28!!]. Inputs from the hippocampus are not a likely source for these precise
Current Opinion in Neurobiology 2014, 25:169–175
The evidence in support of a low-dimensional attractor
suggests a central role for recurrent connectivity in dMEC.
An early study in slice preparations [30] found extensive
excitatory transmission within layers III and V of the
entorhinal cortex. However, no monosynaptic transmission
was found between excitatory neurons in layer II, where
pure grid cells are the most prevalent [26,31]. More recent
work [32!,33!] confirmed the latter observation, but found
extensive disynaptic connectivity between stellate cells in
layer II, mediated by inhibitory interneurons.
Attractor models [7,8,15,27] can be formulated with
purely inhibitory synaptic transmission, as shown explicitly in Ref. [15]. The developmental course of the
transmission between stellate cells in layer II and inhibitory interneurons roughly matches the maturation of
adult-like grid cell activity [32!,34,35], in agreement with
the hypothesis that these recurrent connections contribute to the grid-like response. However, the strengths of
effective inhibitory connections between stellate cells are
probably more narrowly distributed than assumed previously in theoretical models (Fig. 3c in [32!]). Attractor
models [15,27] can be adjusted to explicitly account for
the dynamics of inhibitory and excitatory cells [32!,33!],
while complying with the latter observation [32!].
Topographical and functional organization of the
synaptic connectivity
Attractor models have mostly assumed synaptic connectivity that depends on the distance between neurons
within a 2D surface, leading to smooth variation of the
grid phase within this surface. This prediction is challenged by observations of widely variable grid phases in
different cells recorded from the same tetrode [1].
A recent model of grid cell development (Widloski et al.,
unpublished) proposes that the two-dimensional metric is
inherited during development of the grid-cell network
from the hippocampal representation, which at this phase
is driven by sensory inputs. Within this model, spike time
dependent plasticity leads to radially symmetric synaptic
connectivity within dMEC, in terms of a 2D metric which
is unrelated to the location of grid cells on the cortical sheet
(see also Refs. [7,36,37]). Another possibility is that an
underlying topographical organization does exist, but is not
clearly reflected in the location of individual cell bodies.
Low dimensional attractor dynamics might arise from
connectivity that differs significantly from the radially
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Coding and attractor dynamics in grid cells Burak 173
symmetric structure (within a topographical or functional
metric) proposed in Refs. [8,15,27] (see, for example Refs.
[37,38]). However, to support attractor dynamics, the
connectivity must be related to the functional properties
of the cells: their grid phases, and possibly their inputs
from other brain areas.
source of the slowly varying currents is not yet identified.
In principle, these currents could arise from feed-forward
inputs to dMEC (e.g., from the hippocampus [36] or from
cells with stripe-like responses [50,51]). Within the attractor picture, they are expected to arise primarily from
recurrent connectivity.
In general, it is difficult to map the detailed connectivity in
recurrent neural networks [39]. Nonetheless, characterization of functional and anatomical cell types in dMEC and
their connectivity [40] will be crucial for constraining the
space of possible theoretical models. For example, if
inhibitory interneurons mediate the recurrent connections
responsible for the attractor dynamics [32!], their synaptic
connections with stellate cells are expected to depend on
the grid phases of the synaptic partners. In addition, the
synaptic inputs from inhibitory cells to excitatory grid cells
should contribute significantly to the grid response of
excitatory cells, and the inhibitory neurons are expected
to display a clear spatial response of their own [33!]. These
aspects of the organization in layer II (and similar aspects of
the excitatory connectivity in deeper layers [30]) are not yet
characterized.
Theta oscillations and phase precession are salient features in the rodent dMEC [22,24,44!!,45!!]. Several
recent works [38,45!!,52,53] considered how phase precession can be explained within the attractor picture, in
the presence of intrinsic or extrinsic theta drive in dMEC
(see also Ref. [33!]).
The functional coupling between cells within a module is
likely reflected in differences between inter-module and
intra-module connectivity, another aspect of anatomy in
dMEC that has not yet been characterized experimentally (an issue of importance also for the global properties
discussed in the previous section, ‘The grid cell code for
position’).
Is grid-cell activity driven by slowly varying synaptic
inputs, or by membrane potential oscillations?
Most attractor models rely on slow, asynchronous modulation of synaptic inputs for maintenance of the attractor
state, whereas other models hypothesize that grid cell
activity is modulated by interference of temporally oscillating currents ([41] and references therein; [42,43]). To
distinguish between these scenarios, whole cell recordings were recently collected from grid cells in freely
behaving mice, navigating through a virtual linear track
[44!!,45!!]. Both studies concluded that grid cell activity
is primarily related to the slow modulation of the membrane potential, as expected within attractor models,
while observing only weak correlation with the amplitude
of sub-threshold membrane oscillations. It is conceivable,
however, that interference takes part in the distal dendritic tree, and is not apparent in the membrane potential
at the soma. A separate challenge to oscillatory interference models has arisen from the discovery of grid cells in
bats [3!], without continuous theta oscillations [3!,46]
(but see [25!,47,48,49] and the author’s response).
Conclusions
Experiments in the past two years tested key theoretical
hypotheses on the mechanisms responsible for grid cell
dynamics. Many of these studies resulted in support for
the attractor hypothesis, based on the dynamics of neural
activity in pairs of cells [6!!,28!!], as well as the dynamics
of the membrane potential in slice preparations [33!] and
in freely behaving mice [44!!,45!!]. The latter experiments suggest that the attractor state is maintained (at
least, primarily) through slowly varying synaptic inputs
(see also Ref. [3!]). Another line of evidence in support of
the attractor models comes from the existence of extensive recurrent connectivity between stellate cells in layer
II [32!,33!]. However, the role of recurrent connections in
shaping the grid cell response is not yet established.
While several experimental studies reported on agreement
with specific attractor models [6!!,33!,44!!,45!!], the main
support so far is for the broader notion, that network
dynamics within modules is restricted to lie (anywhere)
within a two-dimensional attractor [6!!,28!!], likely
through recurrent connectivity within dMEC. On the other
hand, many details in existing models may or may not be
valid: For example, very little is known from experiments
about mechanisms responsible for updating the attractor
state based on sensory inputs and self motion. In the
foreseeable future, new techniques will perhaps enable
direct tests for some of the theoretical ideas, for example by
selectively manipulating the inputs to dMEC that encode
head direction and velocity, or the activity of conjunctive
cells within dMEC.
The interplay between theoretical and experimental
work demonstrates that the entorhinal cortex, together
with the hippocampus, provides a rich arena for mechanistic studies of neural computation in the association
cortex, at a detailed level that was previously seen mostly
in sensory and motor areas.
The grid cell code
The studies discussed above [44!!,45!!] do not unequivocally establish that the confinement of grid cell firing
activity to a 2D attractor arises within dMEC, because the
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Recent theoretical studies [9,16!!,18!,20!] have proposed
that the entorhinal cortex, perhaps jointly with the hippocampus, maintains a sophisticated, highly precise system
Current Opinion in Neurobiology 2014, 25:169–175
174 Theoretical and computational neuroscience
for encoding position, with uniquely large dynamical
range in comparison with other known coding schemes
in the brain. Important theoretical questions remain
unanswered, for example: if the entorhinal cortex is
responsible for idiothetic path integration (update of
the animal’s estimate of position based on self motion),
it should be possible to extract from the grid cell code a
distance and angle to a known landmark such as the
animal’s initial position (a ‘homing vector’). A biologically
plausible scheme for implementing this calculation in
neural circuitry is not known.
Regardless of the role of grid cells in path integration, the
theoretical models surveyed here rely on key assumptions, and raise important questions that can be clarified
only by experiments: Do the grid responses, observed in
relatively small enclosures, persist also in much larger
open environments? (advances in wireless recording
[46,54] and virtual environments [5,44!!,45!!] may aid
in addressing this question). To what extent do insights
obtained from open environments, carry through to
environments with internal borders [55], or to environments with varying topology or metric [56]; to motion in
three dimensions [46,57]; and to the encoding of a position other than the animal’s location [4]? Are the phases
in different modules restricted, within a particular
environment, to co-vary in a single 2D sub-space, even
in the absence of sensory cues? If so, is this functional
restriction dependent on the hippocampal representation,
as proposed in [16!!]? Existence of an such an interlocking mechanism would pose strong constraints on the
dynamics of simultaneously recorded cells from different
modules in the presence and absence of sensory cues, as
well as mechanistic constraints on the neural circuitry.
Acknowledgements
I thank Dori Derdikman, Ila Fiete, Neta Ravid, Haim Sompolinsky, and
Nachum Ulanovsky for many helpful comments on the manuscript, and
Hannah Monyer for a helpful discussion. Support from the Gatsby
Charitable Foundation is gratefully acknowledged.
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Resolution of the grid cell code is analyzed via the Fisher information,
under an assumption that grid cells fire as Poisson spike generators. An
optimization principle predicts that grid spacings should form a geometric
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Current Opinion in Neurobiology 2014, 25:169–175