Grid cells: the position code, neural network models of activity, and the problem of learning

Grid Cells in Cognition: Mechanisms and Function

The activity patterns of grid cells form distinctively regular triangular lattices over the explored spatial environment and are largely invariant to visual stimuli, animal movement, and environment geometry. These neurons present numerous fascinating challenges to the curious (neuro)scientist: What are the circuit mechanisms responsible for creating spatially periodic activity patterns from the monotonic input-output responses of single neurons? How and why does the brain encode a local, nonperiodic variable-the allocentric position of the animal-with a periodic, nonlocal code? And, are grid cells truly specialized for spatial computations? Otherwise, what is their role in general cognition more broadly? We review efforts in uncovering the mechanisms and functional properties of grid cells, highlighting recent progress in the experimental validation of mechanistic grid cell models, and discuss the coding properties and functional advantages of the grid code as suggested by continuous attractor network models of grid cells.

Grid cells, border cells, and discrete complex analysis

We propose a mechanism enabling the appearance of border cells-neurons firing at the boundaries of the navigated enclosures. The approach is based on the recent discovery of discrete complex analysis on a triangular lattice, which allows constructing discrete epitomes of complex-analytic functions and making use of their inherent ability to attain maximal values at the boundaries of generic lattice domains. As it turns out, certain elements of the discrete-complex framework readily appear in the oscillatory models of grid cells. We demonstrate that these models can extend further, producing cells that increase their activity toward the frontiers of the navigated environments. We also construct a network model of neurons with border-bound firing that conforms with the oscillatory models.

Grid Cell Percolation

Grid cells play a principal role in enabling cognitive representations of ambient environments. The key property of these cells-the regular arrangement of their firing fields-is commonly viewed as a means for establishing spatial scales or encoding specific locations. However, using grid cells' spiking outputs for deducing geometric orderliness proves to be a strenuous task due to fairly irregular activation patterns triggered by the animal's sporadic visits to the grid fields. This article addresses statistical mechanisms enabling emergent regularity of grid cell firing activity from the perspective of percolation theory. Using percolation phenomena for modeling the effect of the rat's moves through the lattices of firing fields sheds new light on the mechanisms of spatial information processing, spatial learning, path integration, and establishing spatial metrics. It is also shown that physiological parameters required for spiking percolation match the experimental range, including the characteristic 2/3 ratio between the grid fields' size and the grid spacing, pointing at a biological viability of the approach.

Grid Cells, Border Cells and Discrete Complex Analysis

We propose a mechanism enabling the appearance of border cells-neurons firing at the boundaries of the navigated enclosures. The approach is based on the recent discovery of discrete complex analysis on a triangular lattice, which allows constructing discrete epitomes of complex-analytic functions and making use of their inherent ability to attain maximal values at the boundaries of generic lattice domains. As it turns out, certain elements of the discrete-complex framework readily appear in the oscillatory models of grid cells. We demonstrate that these models can extend further, producing cells that increase their activity towards the frontiers of the navigated environments. We also construct a network model of neurons with border-bound firing that conforms with the oscillatory models.

Microcircuits for spatial coding in the medial entorhinal cortex

The hippocampal formation is critically involved in learning and memory and contains a large proportion of neurons encoding aspects of the organism's spatial surroundings. In the medial entorhinal cortex (MEC), this includes grid cells with their distinctive hexagonal firing fields as well as a host of other functionally defined cell types including head direction cells, speed cells, border cells, and object-vector cells. Such spatial coding emerges from the processing of external inputs by local microcircuits. However, it remains unclear exactly how local microcircuits and their dynamics within the MEC contribute to spatial discharge patterns. In this review we focus on recent investigations of intrinsic MEC connectivity, which have started to describe and quantify both excitatory and inhibitory wiring in the superficial layers of the MEC. Although the picture is far from complete, it appears that these layers contain robust recurrent connectivity that could sustain the attractor dynamics posited to underlie grid pattern formation. These findings pave the way to a deeper understanding of the mechanisms underlying spatial navigation and memory.

Grid-cell modules remain coordinated when neural activity is dissociated from external sensory cues

The representation of an animal’s position in the medial entorhinal cortex (MEC) is distributed across several modules of grid cells, each characterized by a distinct spatial scale. The population activity within each module is tightly coordinated and preserved across environments and behavioral states. Little is known, however, about the coordination of activity patterns across modules. We analyzed the joint activity patterns of hundreds of grid cells simultaneously recorded in animals that were foraging either in the light, when sensory cues could stabilize the representation, or in darkness, when such stabilization was disrupted. We found that the states of different modules are tightly coordinated, even in darkness, when the internal representation of position within the MEC deviates substantially from the true position of the animal. These findings suggest that internal brain mechanisms dynamically coordinate the representation of position in different modules, ensuring that they jointly encode a coherent and smooth trajectory.

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Hundreds of grid cells were recorded simultaneously from multiple grid modules

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Coordination between grid modules was assessed in rats that foraged in darkness

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Coordination persists despite relative drift of the represented versus true position

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This suggests that internal network mechanisms maintain inter-module coordination

A Model of Spatial Cell Development in Rat Hippocampus Based on Artificial Neural Network

Physiological studies have shown that the hippocampal structure of rats develops at different stages, in which the place cells continue to develop during the whole juvenile period of rats and mature after the juvenile period. As the main information source of place cells, grid cells should mature earlier than place cells. In order to make better use of the biological information exhibited by the rat brain hippocampus in the environment, we propose a position cognition model based on the spatial cell development mechanism of rat hippocampus. The model uses a recurrent neural network with parametric bias (RNNPB) to simulate changes in the discharge characteristics during the development of a single stripe cell. The oscillatory interference mechanism is able to fuse the developing stripe waves, thus indirectly simulating the developmental process of the grid cells. The output of the grid cells is then used as the information input of the place cells, whose development process is simulated by BP neural network. After the place cells matured, the position matrix generated by the place cell group was used to realize the position cognition of rats in a given spatial region. The experimental results show that this model can simulate the development process of grid cells and place cells, and it can realize high precision positioning in the given space area. Moreover, the experimental effect of cognitive map construction using this model is basically consistent with the effect of RatSLAM, which verifies the validity and accuracy of the model.

Inferences on a multidimensional social hierarchy use a grid-like code

Generalizing experiences to guide decision-making in novel situations is a hallmark of flexible behavior. Cognitive maps of an environment or task can theoretically afford such flexibility, but direct evidence has proven elusive. In this study, we found that discretely sampled abstract relationships between entities in an unseen two-dimensional social hierarchy are reconstructed into a unitary two-dimensional cognitive map in the hippocampus and entorhinal cortex. We further show that humans use a grid-like code in entorhinal cortex and medial prefrontal cortex for inferred direct trajectories between entities in the reconstructed abstract space during discrete decisions. These grid-like representations in the entorhinal cortex are associated with decision value computations in the medial prefrontal cortex and temporoparietal junction. Collectively, these findings show that grid-like representations are used by the human brain to infer novel solutions, even in abstract and discrete problems, and suggest a general mechanism underpinning flexible decision-making and generalization.

Place-cell capacity and volatility with grid-like inputs

What factors constrain the arrangement of the multiple fields of a place cell? By modeling place cells as perceptrons that act on multiscale periodic grid-cell inputs, we analytically enumerate a place cell's repertoire - how many field arrangements it can realize without external cues while its grid inputs are unique - and derive its capacity - the spatial range over which it can achieve any field arrangement. We show that the repertoire is very large and relatively noise-robust. However, the repertoire is a vanishing fraction of all arrangements, while capacity scales only as the sum of the grid periods so field arrangements are constrained over larger distances. Thus, grid-driven place field arrangements define a large response scaffold that is strongly constrained by its structured inputs. Finally, we show that altering grid-place weights to generate an arbitrary new place field strongly affects existing arrangements, which could explain the volatility of the place code.

Recurrent amplification of grid-cell activity

High-level cognitive abilities such as navigation and spatial memory are thought to rely on the activity of grid cells in the medial entorhinal cortex (MEC), which encode the animal's position in space with periodic triangular patterns. Yet the neural mechanisms that underlie grid-cell activity are still unknown. Recent in vitro and in vivo experiments indicate that grid cells are embedded in highly structured recurrent networks. But how could recurrent connectivity become structured during development? And what is the functional role of these connections? With mathematical modeling and simulations, we show that recurrent circuits in the MEC could emerge under the supervision of weakly grid-tuned feedforward inputs. We demonstrate that a learned excitatory connectivity could amplify grid patterns when the feedforward sensory inputs are available and sustain attractor states when the sensory cues are lost. Finally, we propose a Fourier-based measure to quantify the spatial periodicity of grid patterns: the grid-tuning index.

A theory of joint attractor dynamics in the hippocampus and the entorhinal cortex accounts for artificial remapping and grid cell field-to-field variability

The representation of position in the mammalian brain is distributed across multiple neural populations. Grid cell modules in the medial entorhinal cortex (MEC) express activity patterns that span a low-dimensional manifold which remains stable across different environments. In contrast, the activity patterns of hippocampal place cells span distinct low-dimensional manifolds in different environments. It is unknown how these multiple representations of position are coordinated. Here, we develop a theory of joint attractor dynamics in the hippocampus and the MEC. We show that the system exhibits a coordinated, joint representation of position across multiple environments, consistent with global remapping in place cells and grid cells. In addition, our model accounts for recent experimental observations that lack a mechanistic explanation: variability in the firing rate of single grid cells across firing fields, and artificial remapping of place cells under depolarization, but not under hyperpolarization, of layer II stellate cells of the MEC.

Efficient and flexible representation of higher-dimensional cognitive variables with grid cells

We shed light on the potential of entorhinal grid cells to efficiently encode variables of dimension greater than two, while remaining faithful to empirical data on their low-dimensional structure. Our model constructs representations of high-dimensional inputs through a combination of low-dimensional random projections and "classical" low-dimensional hexagonal grid cell responses. Without reconfiguration of the recurrent circuit, the same system can flexibly encode multiple variables of different dimensions while maximizing the coding range (per dimension) by automatically trading-off dimension with an exponentially large coding range. It achieves high efficiency and flexibility by combining two powerful concepts, modularity and mixed selectivity, in what we call "mixed modular coding". In contrast to previously proposed schemes, the model does not require the formation of higher-dimensional grid responses, a cell-inefficient and rigid mechanism. The firing fields observed in flying bats or climbing rats can be generated by neurons that combine activity from multiple grid modules, each representing higher-dimensional spaces according to our model. The idea expands our understanding of grid cells, suggesting that they could implement a general circuit that generates on-demand coding and memory states for variables in high-dimensional vector spaces.

Velocity coupling of grid cell modules enables stable embedding of a low dimensional variable in a high dimensional neural attractor

Grid cells in the medial entorhinal cortex (MEC) encode position using a distributed representation across multiple neural populations (modules), each possessing a distinct spatial scale. The modular structure of the representation confers the grid cell neural code with large capacity. Yet, the modularity poses significant challenges for the neural circuitry that maintains the representation, and updates it based on self motion. Small incompatible drifts in different modules, driven by noise, can rapidly lead to large, abrupt shifts in the represented position, resulting in catastrophic readout errors. Here, we propose a theoretical model of coupled modules. The coupling suppresses incompatible drifts, allowing for a stable embedding of a two-dimensional variable (position) in a higher dimensional neural attractor, while preserving the large capacity. We propose that coupling of this type may be implemented by recurrent synaptic connectivity within the MEC with a relatively simple and biologically plausible structure.

Robust computation with rhythmic spike patterns

This work makes 2 contributions. First, we present a neural network model of associative memory that stores and retrieves sparse patterns of complex variables. This network can store analog information as fixed-point attractors in the complex domain; it is governed by an energy function and has increased memory capacity compared to early models. Second, we translate complex attractor networks into spiking networks, where the timing of the spike indicates the phase of a complex number. We show that complex fixed points correspond to stable periodic spike patterns. It is demonstrated that such networks can be constructed with resonate-and-fire or integrate-and-fire neurons with biologically plausible mechanisms and be used for robust computations, such as image retrieval.

Information coding by precise timing of spikes can be faster and more energy efficient than traditional rate coding. However, spike-timing codes are often brittle, which has limited their use in theoretical neuroscience and computing applications. Here, we propose a type of attractor neural network in complex state space and show how it can be leveraged to construct spiking neural networks with robust computational properties through a phase-to-timing mapping. Building on Hebbian neural associative memories, like Hopfield networks, we first propose threshold phasor associative memory (TPAM) networks. Complex phasor patterns whose components can assume continuous-valued phase angles and binary magnitudes can be stored and retrieved as stable fixed points in the network dynamics. TPAM achieves high memory capacity when storing sparse phasor patterns, and we derive the energy function that governs its fixed-point attractor dynamics. Second, we construct 2 spiking neural networks to approximate the complex algebraic computations in TPAM, a reductionist model with resonate-and-fire neurons and a biologically plausible network of integrate-and-fire neurons with synaptic delays and recurrently connected inhibitory interneurons. The fixed points of TPAM correspond to stable periodic states of precisely timed spiking activity that are robust to perturbation. The link established between rhythmic firing patterns and complex attractor dynamics has implications for the interpretation of spike patterns seen in neuroscience and can serve as a framework for computation in emerging neuromorphic devices.

Grid cell co-activity patterns during sleep reflect spatial overlap of grid fields during active behaviors

Continuous-attractor network models of grid formation posit that recurrent connectivity between grid cells controls their patterns of co-activation. Grid cells from a common module exhibit stable offsets in their periodic spatial tuning curves across environments, and this may reflect recurrent connectivity or correlated sensory inputs. Here we explore whether cell-cell relationships predicted by attractor models persist during sleep states in which spatially informative sensory inputs are absent. We recorded ensembles of grid cells in superficial layers of medial entorhinal cortex during active exploratory behaviors and overnight sleep. Per grid cell pair and collectively, and across waking, rapid eye movement sleep and non-rapid eye movement sleep, we found preserved patterns of spike-time correlations that reflected the spatial tuning offsets between these grid cells during active exploration. The preservation of cell-cell relationships across waking and sleep states was not explained by theta oscillations or activity in hippocampal subregion CA1. These results indicate that recurrent connectivity within the grid cell network drives grid cell activity across behavioral states.

Spatial synchronization codes from coupled rate-phase neurons

During spatial navigation, the frequency and timing of spikes from spatial neurons including place cells in hippocampus and grid cells in medial entorhinal cortex are temporally organized by continuous theta oscillations (6-11 Hz). The theta rhythm is regulated by subcortical structures including the medial septum, but it is unclear how spatial information from place cells may reciprocally organize subcortical theta-rhythmic activity. Here we recorded single-unit spiking from a constellation of subcortical and hippocampal sites to study spatial modulation of rhythmic spike timing in rats freely exploring an open environment. Our analysis revealed a novel class of neurons that we termed 'phaser cells,' characterized by a symmetric coupling between firing rate and spike theta-phase. Phaser cells encoded space by assigning distinct phases to allocentric isocontour levels of each cell's spatial firing pattern. In our dataset, phaser cells were predominantly located in the lateral septum, but also the hippocampus, anteroventral thalamus, lateral hypothalamus, and nucleus accumbens. Unlike the unidirectional late-to-early phase precession of place cells, bidirectional phase modulation acted to return phaser cells to the same theta-phase along a given spatial isocontour, including cells that characteristically shifted to later phases at higher firing rates. Our dynamical models of intrinsic theta-bursting neurons demonstrated that experience-independent temporal coding mechanisms can qualitatively explain (1) the spatial rate-phase relationships of phaser cells and (2) the observed temporal segregation of phaser cells according to phase-shift direction. In open-field phaser cell simulations, competitive learning embedded phase-code entrainment maps into the weights of downstream targets, including path integration networks. Bayesian phase decoding revealed error correction capable of resetting path integration at subsecond timescales. Our findings suggest that phaser cells may instantiate a subcortical theta-rhythmic loop of spatial feedback. We outline a framework in which location-dependent synchrony reconciles internal idiothetic processes with the allothetic reference points of sensory experience.

Inferring circuit mechanisms from sparse neural recording and global perturbation in grid cells

A goal of systems neuroscience is to discover the circuit mechanisms underlying brain function. Despite experimental advances that enable circuit-wide neural recording, the problem remains open in part because solving the 'inverse problem' of inferring circuity and mechanism by merely observing activity is hard. In the grid cell system, we show through modeling that a technique based on global circuit perturbation and examination of a novel theoretical object called the distribution of relative phase shifts (DRPS) could reveal the mechanisms of a cortical circuit at unprecedented detail using extremely sparse neural recordings. We establish feasibility, showing that the method can discriminate between recurrent versus feedforward mechanisms and amongst various recurrent mechanisms using recordings from a handful of cells. The proposed strategy demonstrates that sparse recording coupled with simple perturbation can reveal more about circuit mechanism than can full knowledge of network activity or the synaptic connectivity matrix.

Degeneracy in the robust expression of spectral selectivity, subthreshold oscillations, and intrinsic excitability of entorhinal stellate cells

Biological heterogeneities are ubiquitous and play critical roles in the emergence of physiology at multiple scales. Although neurons in layer II (LII) of the medial entorhinal cortex (MEC) express heterogeneities in channel properties, the impact of such heterogeneities on the robustness of their cellular-scale physiology has not been assessed. Here, we performed a 55-parameter stochastic search spanning nine voltage- or calcium-activated channels to assess the impact of channel heterogeneities on the concomitant emergence of 10 in vitro electrophysiological characteristics of LII stellate cells (SCs). We generated 150,000 models and found a heterogeneous subpopulation of 449 valid models to robustly match all electrophysiological signatures. We employed this heterogeneous population to demonstrate the emergence of cellular-scale degeneracy in SCs, whereby disparate parametric combinations expressing weak pairwise correlations resulted in similar models. We then assessed the impact of virtually knocking out each channel from all valid models and demonstrate that the mapping between channels and measurements was many-to-many, a critical requirement for the expression of degeneracy. Finally, we quantitatively predict that the spike-triggered average of SCs should be endowed with theta-frequency spectral selectivity and coincidence detection capabilities in the fast gamma-band. We postulate this fast gamma-band coincidence detection as an instance of cellular-scale-efficient coding, whereby SC response characteristics match the dominant oscillatory signals in LII MEC. The heterogeneous population of valid SC models built here unveils the robust emergence of cellular-scale physiology despite significant channel heterogeneities, and forms an efficacious substrate for evaluating the impact of biological heterogeneities on entorhinal network function.

NEW & NOTEWORTHY We assessed the impact of heterogeneities in channel properties on the robustness of cellular-scale physiology of medial entorhinal cortical stellate neurons. We demonstrate that neuronal models with disparate channel combinations were endowed with similar physiological characteristics, as a consequence of the many-to-many mapping between channel properties and the physiological characteristics that they modulate. We predict that the spike-triggered average of stellate cells should be endowed with theta-frequency spectral selectivity and fast gamma-band coincidence detection capabilities.

Spatial representation in the hippocampal formation: a history

Since the first place cell was recorded and the cognitive-map theory was subsequently formulated, investigation of spatial representation in the hippocampal formation has evolved in stages. Early studies sought to verify the spatial nature of place cell activity and determine its sensory origin. A new epoch started with the discovery of head direction cells and the realization of the importance of angular and linear movement-integration in generating spatial maps. A third epoch began when investigators turned their attention to the entorhinal cortex, which led to the discovery of grid cells and border cells. This review will show how ideas about integration of self-motion cues have shaped our understanding of spatial representation in hippocampal-entorhinal systems from the 1970s until today. It is now possible to investigate how specialized cell types of these systems work together, and spatial mapping may become one of the first cognitive functions to be understood in mechanistic detail.

From grid cells to place cells with realistic field sizes

While grid cells in the medial entorhinal cortex (MEC) of rodents have multiple, regularly arranged firing fields, place cells in the cornu ammonis (CA) regions of the hippocampus mostly have single spatial firing fields. Since there are extensive projections from MEC to the CA regions, many models have suggested that a feedforward network can transform grid cell firing into robust place cell firing. However, these models generate place fields that are consistently too small compared to those recorded in experiments. Here, we argue that it is implausible that grid cell activity alone can be transformed into place cells with robust place fields of realistic size in a feedforward network. We propose two solutions to this problem. Firstly, weakly spatially modulated cells, which are abundant throughout EC, provide input to downstream place cells along with grid cells. This simple model reproduces many place cell characteristics as well as results from lesion studies. Secondly, the recurrent connections between place cells in the CA3 network generate robust and realistic place fields. Both mechanisms could work in parallel in the hippocampal formation and this redundancy might account for the robustness of place cell responses to a range of disruptions of the hippocampal circuitry.

Bio-inspired homogeneous multi-scale place recognition

Robotic mapping and localization systems typically operate at either one fixed spatial scale, or over two, combining a local metric map and a global topological map. In contrast, recent high profile discoveries in neuroscience have indicated that animals such as rodents navigate the world using multiple parallel maps, with each map encoding the world at a specific spatial scale. While a number of theoretical-only investigations have hypothesized several possible benefits of such a multi-scale mapping system, no one has comprehensively investigated the potential mapping and place recognition performance benefits for navigating robots in large real world environments, especially using more than two homogeneous map scales. In this paper we present a biologically-inspired multi-scale mapping system mimicking the rodent multi-scale map. Unlike hybrid metric-topological multi-scale robot mapping systems, this new system is homogeneous, distinguishable only by scale, like rodent neural maps. We present methods for training each network to learn and recognize places at a specific spatial scale, and techniques for combining the output from each of these parallel networks. This approach differs from traditional probabilistic robotic methods, where place recognition spatial specificity is passively driven by models of sensor uncertainty. Instead we intentionally create parallel learning systems that learn associations between sensory input and the environment at different spatial scales. We also conduct a systematic series of experiments and parameter studies that determine the effect on performance of using different neural map scaling ratios and different numbers of discrete map scales. The results demonstrate that a multi-scale approach universally improves place recognition performance and is capable of producing better than state of the art performance compared to existing robotic navigation algorithms. We analyze the results and discuss the implications with respect to several recent discoveries and theories regarding how multi-scale neural maps are learnt and used in the mammalian brain.

Bridging the gap between spatial and mnemonic views of the hippocampal formation

While it has long been recognized that medial temporal lobe structures are important for memory formation, studies in rodents have also identified exquisite spatial representations in these regions in the form of place cells in the hippocampus and grid cells in the entorhinal cortex. Spatial representations entail neural activity that is observed when the rat is in a given physical location, and these representations are thought to form the basis of navigation via path integration. Recent studies in nonhuman primates have suggested that similar kinds of spatial representations can be identified, even in the absence of physical movement through an environment. Here, I will highlight some recent work that addresses similarities and differences between spatial responses as identified in rodents and primates. I will also discuss areas of opportunity for future research to further our understanding of the function of the hippocampal formation.

A hybrid oscillatory interference/continuous attractor network model of grid cell firing

Grid cells in the rodent medial entorhinal cortex exhibit remarkably regular spatial firing patterns that tessellate all environments visited by the animal. Two theoretical mechanisms that could generate this spatially periodic activity pattern have been proposed: oscillatory interference and continuous attractor dynamics. Although a variety of evidence has been cited in support of each, some aspects of the two mechanisms are complementary, suggesting that a combined model may best account for experimental data. The oscillatory interference model proposes that the grid pattern is formed from linear interference patterns or "periodic bands" in which velocity-controlled oscillators integrate self-motion to code displacement along preferred directions. However, it also allows the use of symmetric recurrent connectivity between grid cells to provide relative stability and continuous attractor dynamics. Here, we present simulations of this type of hybrid model, demonstrate that it generates intracellular membrane potential profiles that closely match those observed in vivo, addresses several criticisms aimed at pure oscillatory interference and continuous attractor models, and provides testable predictions for future empirical studies.

Spatial coding and attractor dynamics of grid cells in the entorhinal cortex

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.

The development of spatial behaviour and the hippocampal neural representation of space

The role of the hippocampal formation in spatial cognition is thought to be supported by distinct classes of neurons whose firing is tuned to an organism's position and orientation in space. In this article, we review recent research focused on how and when this neural representation of space emerges during development: each class of spatially tuned neurons appears at a different age, and matures at a different rate, but all the main spatial responses tested so far are present by three weeks of age in the rat. We also summarize the development of spatial behaviour in the rat, describing how active exploration of space emerges during the third week of life, the first evidence of learning in formal tests of hippocampus-dependent spatial cognition is observed in the fourth week, whereas fully adult-like spatial cognitive abilities require another few weeks to be achieved. We argue that the development of spatially tuned neurons needs to be considered within the context of the development of spatial behaviour in order to achieve an integrated understanding of the emergence of hippocampal function and spatial cognition.

How to build a grid cell

Neurons in the medial entorhinal cortex fire action potentials at regular spatial intervals, creating a striking grid-like pattern of spike rates spanning the whole environment of a navigating animal. This remarkable spatial code may represent a neural map for path integration. Recent advances using patch-clamp recordings from entorhinal cortex neurons in vitro and in vivo have revealed how the microcircuitry in the medial entorhinal cortex may contribute to grid cell firing patterns, and how grid cells may transform synaptic inputs into spike output during firing field crossings. These new findings provide key insights into the ingredients necessary to build a grid cell.

The development of the hippocampal neural representation of space

The hippocampal formation (HF) contains a neural representation of the environment, based on the activity of several classes of neurons whose firing is tuned to an animal's position and orientation in space. Recently, work has begun on understanding when and how this neural map of space emerges during development. Different classes of spatially tuned neurons emerge at different ages, some of them very early during development, before animals have started exploring their environment. The developmental timeline thus far uncovered has yielded insights into both the mechanisms of the ontogeny of the neural code for space, as well as how this system functions in the adult.

Specific evidence of low-dimensional continuous attractor dynamics in grid cells

We examined simultaneously recorded spikes from multiple rat grid cells, to explain mechanisms underlying their activity. Among grid cells with similar spatial periods, the population activity was confined to lie close to a two-dimensional (2D) manifold: grid cells differed only along two dimensions of their responses and otherwise were nearly identical. Relationships between cell pairs were conserved despite extensive deformations of single-neuron responses. Results from novel environments suggest such structure is not inherited from hippocampal or external sensory inputs. Across conditions, cell-cell relationships are better conserved than responses of single cells. Finally, the system is continually subject to perturbations that, were the 2D manifold not attractive, would drive the system to inhabit a different region of state space than observed. These findings have strong implications for theories of grid-cell activity and substantiate the general hypothesis that the brain computes using low-dimensional continuous attractors.

Feedback inhibition enables θ-nested γ oscillations and grid firing fields

Cortical circuits are thought to multiplex firing rate codes with temporal codes that rely on oscillatory network activity, but the circuit mechanisms that combine these coding schemes are unclear. We establish with optogenetic activation of layer II of the medial entorhinal cortex that theta frequency drive to this circuit is sufficient to generate nested gamma frequency oscillations in synaptic activity. These nested gamma oscillations closely resemble activity during spatial exploration, are generated by local feedback inhibition without recurrent excitation, and have clock-like features suitable as reference signals for multiplexing temporal codes within rate-coded grid firing fields. In network models deduced from our data, feedback inhibition supports coexistence of theta-nested gamma oscillations with attractor states that generate grid firing fields. These results indicate that grid cells communicate primarily via inhibitory interneurons. This circuit mechanism enables multiplexing of oscillation-based temporal codes with rate-coded attractor states.

Membrane potential dynamics of grid cells

During navigation, grid cells increase their spike rates in firing fields arranged on a strikingly regular triangular lattice, while their spike timing is often modulated by theta oscillations. Oscillatory interference models of grid cells predict theta amplitude modulations of membrane potential during firing field traversals, while competing attractor network models predict slow depolarizing ramps. Here, using in-vivo whole-cell recordings, we tested these models by directly measuring grid cell intracellular potentials in mice running along linear tracks in virtual reality. Grid cells had large and reproducible ramps of membrane potential depolarization that were the characteristic signature tightly correlated with firing fields. Grid cells also exhibited intracellular theta oscillations that influenced their spike timing. However, the properties of theta amplitude modulations were not consistent with the view that they determine firing field locations. Our results support cellular and network mechanisms in which grid fields are produced by slow ramps, as in attractor models, while theta oscillations control spike timing.

Associative memory of phase-coded spatiotemporal patterns in leaky Integrate and Fire networks

We study the collective dynamics of a Leaky Integrate and Fire network in which precise relative phase relationship of spikes among neurons are stored, as attractors of the dynamics, and selectively replayed at different time scales. Using an STDP-based learning process, we store in the connectivity several phase-coded spike patterns, and we find that, depending on the excitability of the network, different working regimes are possible, with transient or persistent replay activity induced by a brief signal. We introduce an order parameter to evaluate the similarity between stored and recalled phase-coded pattern, and measure the storage capacity. Modulation of spiking thresholds during replay changes the frequency of the collective oscillation or the number of spikes per cycle, keeping preserved the phases relationship. This allows a coding scheme in which phase, rate and frequency are dissociable. Robustness with respect to noise and heterogeneity of neurons parameters is studied, showing that, since dynamics is a retrieval process, neurons preserve stable precise phase relationship among units, keeping a unique frequency of oscillation, even in noisy conditions and with heterogeneity of internal parameters of the units.

Voltage dependence of subthreshold resonance frequency in layer II of medial entorhinal cortex

The resonance properties of individual neurons in entorhinal cortex (EC) may contribute to their functional properties in awake, behaving rats. Models propose that entorhinal grid cells could arise from shifts in the intrinsic frequency of neurons caused by changes in membrane potential owing to depolarizing input from neurons coding velocity. To test for potential changes in intrinsic frequency, we measured the resonance properties of neurons at different membrane potentials in neurons in medial and lateral EC. In medial entorhinal neurons, the resonant frequency of individual neurons decreased in a linear manner as the membrane potential was depolarized between -70 and -55 mV. At more hyperpolarized membrane potentials, cells asymptotically approached a maximum resonance frequency. Consistent with the previous studies, near resting potential, the cells of the medial EC possessed a decreasing gradient of resonance frequency along the dorsal to ventral axis, and cells of the lateral EC lacked resonant properties, regardless of membrane potential or position along the medial to lateral axis within lateral EC. Application of 10 μM ZD7288, the H-channel blocker, abolished all resonant properties in MEC cells, and resulted in physiological properties very similar to lateral EC cells. These results on resonant properties show a clear change in frequency response with depolarization that could contribute to the generation of grid cell firing properties in the medial EC.

The abrupt development of adult-like grid cell firing in the medial entorhinal cortex

Understanding the development of the neural circuits subserving specific cognitive functions such as navigation remains a central problem in neuroscience. Here, we characterize the development of grid cells in the medial entorhinal cortex, which, by nature of their regularly spaced firing fields, are thought to provide a distance metric to the hippocampal neural representation of space. Grid cells emerge at the time of weaning in the rat, at around 3 weeks of age. We investigated whether grid cells in young rats are functionally equivalent to those observed in the adult as soon as they appear, or if instead they follow a gradual developmental trajectory. We find that, from the very youngest ages at which reproducible grid firing is observed (postnatal day 19): grid cells display adult-like firing fields that tessellate to form a coherent map of the local environment; that this map is universal, maintaining its internal structure across different environments; and that grid cells in young rats, as in adults, also encode a representation of direction and speed. To further investigate the developmental processes leading up to the appearance of grid cells, we present data from individual medial entorhinal cortex cells recorded across more than 1 day, spanning the period before and after the grid firing pattern emerged. We find that increasing spatial stability of firing was correlated with increasing gridness.

Models of grid cell spatial firing published 2005-2011

Since the discovery of grid cells in rat entorhinal cortex, many models of their hexagonally arrayed spatial firing fields have been suggested. We review the models and organize them according to the mechanisms they use to encode position, update the positional code, read it out in the spatial grid pattern, and learn any patterned synaptic connections needed. We mention biological implementations of the models, but focus on the models on Marr’s algorithmic level, where they are not things to individually prove or disprove, but rather are a valuable collection of metaphors of the grid cell system for guiding research that are all likely true to some degree, with each simply emphasizing different aspects of the system. For the convenience of interested researchers, MATLAB implementations of the discussed grid cell models are provided at ModelDB accession 144006 or http://people.bu.edu/zilli/gridmodels.html.

Attractor dynamics of spatially correlated neural activity in the limbic system

Attractor networks are a popular computational construct used to model different brain systems. These networks allow elegant computations that are thought to represent a number of aspects of brain function. Although there is good reason to believe that the brain displays attractor dynamics, it has proven difficult to test experimentally whether any particular attractor architecture resides in any particular brain circuit. We review models and experimental evidence for three systems in the rat brain that are presumed to be components of the rat's navigational and memory system. Head-direction cells have been modeled as a ring attractor, grid cells as a plane attractor, and place cells both as a plane attractor and as a point attractor. Whereas the models have proven to be extremely useful conceptual tools, the experimental evidence in their favor, although intriguing, is still mostly circumstantial.

Computational models of grid cells

Grid cells are space-modulated neurons with periodic firing fields. In moving animals, the multiple firing fields of an individual grid cell form a triangular pattern tiling the entire space available to the animal. Collectively, grid cells are thought to provide a context-independent metric representation of the local environment. Since the discovery of grid cells in 2005, a number of models have been proposed to explain the formation of spatially repetitive firing patterns as well as the conversion of these signals to place signals one synapse downstream in the hippocampus. The present article reviews the most recent developments in our understanding of how grid patterns are generated, maintained, and transformed, with particular emphasis on second-generation computational models that have emerged during the past 2-3 years in response to criticism and new data.

Sensory feedback, error correction, and remapping in a multiple oscillator model of place-cell activity

Mammals navigate by integrating self-motion signals ("path integration") and occasionally fixing on familiar environmental landmarks. The rat hippocampus is a model system of spatial representation in which place cells are thought to integrate both sensory and spatial information from entorhinal cortex. The localized firing fields of hippocampal place cells and entorhinal grid-cells demonstrate a phase relationship with the local theta (6-10 Hz) rhythm that may be a temporal signature of path integration. However, encoding self-motion in the phase of theta oscillations requires high temporal precision and is susceptible to idiothetic noise, neuronal variability, and a changing environment. We present a model based on oscillatory interference theory, previously studied in the context of grid cells, in which transient temporal synchronization among a pool of path-integrating theta oscillators produces hippocampal-like place fields. We hypothesize that a spatiotemporally extended sensory interaction with external cues modulates feedback to the theta oscillators. We implement a form of this cue-driven feedback and show that it can retrieve fixed points in the phase code of position. A single cue can smoothly reset oscillator phases to correct for both systematic errors and continuous noise in path integration. Further, simulations in which local and global cues are rotated against each other reveal a phase-code mechanism in which conflicting cue arrangements can reproduce experimentally observed distributions of "partial remapping" responses. This abstract model demonstrates that phase-code feedback can provide stability to the temporal coding of position during navigation and may contribute to the context-dependence of hippocampal spatial representations. While the anatomical substrates of these processes have not been fully characterized, our findings suggest several signatures that can be evaluated in future experiments.

Dorsal-ventral organization of theta-like activity intrinsic to entorhinal stellate neurons is mediated by differences in stochastic current fluctuations

The membrane potential dynamics of stellate neurons in layer II of the medial entorhinal cortex are important for neural encoding of location. Previous studies suggest that these neurons generate intrinsic theta-frequency membrane potential oscillations, with a period that depends on neuronal location on the dorsal–ventral axis of themedial entorhinal cortex, and which in behaving animals could support generation of grid-like spatial firing fields. To address the nature and organization of this theta-like activity, we adopt the Lombmethod of least-squares spectral analysis. We demonstrate that peaks in frequency spectra that differ significantly from Gaussian noise do not necessarily imply the existence of a periodic oscillator, but can instead arise from filtered stochastic noise or a stochastic random walk. We show that theta-like membrane potential activity recorded fromstellate neurons in mature brain slices is consistentwith stochastic mechanisms, but not with generation by a periodic oscillator. The dorsal–ventral organization of intrinsic theta-likemembrane potential activity, and themodification of this activity during block of HCN channels, both reflect altered frequency distributions of stochastic spectral peaks, rather than tuning of a periodic oscillator. Our results demonstrate the importance of distinguishing periodic oscillations from stochastic processes.We suggest that dorsal–ventral tuning of theta-like membrane potential activity is due to differences in stochastic current fluctuations resulting from organization of ion channels that also control synaptic integration.

Coupled noisy spiking neurons as velocity-controlled oscillators in a model of grid cell spatial firing

One of the two primary classes of models of grid cell spatial firing uses interference between oscillators at dynamically modulated frequencies. Generally, these models are presented in terms of idealized oscillators (modeled as sinusoids), which differ from biological oscillators in multiple important ways. Here we show that two more realistic, noisy neural models (Izhikevich's simple model and a biophysical model of an entorhinal cortex stellate cell) can be successfully used as oscillators in a model of this type. When additive noise is included in the models such that uncoupled or sparsely coupled cells show realistic interspike interval variance, both synaptic and gap-junction coupling can synchronize networks of cells to produce comparatively less variable network-level oscillations. We show that the frequency of these oscillatory networks can be controlled sufficiently well to produce stable grid cell spatial firing on the order of at least 2-5 min, despite the high noise level. Our results suggest that the basic principles of oscillatory interference models work with more realistic models of noisy neurons. Nevertheless, a number of simplifications were still made and future work should examine increasingly realistic models.

Losing phase

In this issue of Neuron, Remme and colleagues examine the biophysics of synchronization between oscillating dendrites and soma. Their findings suggest that oscillators will quickly phase-lock when weakly coupled. These findings are at odds with assumptions of an influential model of grid cell response generation and have implications for grid cell response mechanisms.

Cellular dynamical mechanisms for encoding the time and place of events along spatiotemporal trajectories in episodic memory

Understanding the mechanisms of episodic memory requires linking behavioral data and lesion effects to data on the dynamics of cellular membrane potentials and population interactions within brain regions. Linking behavior to specific membrane channels and neurochemicals has implications for therapeutic applications. Lesions of the hippocampus, entorhinal cortex and subcortical nuclei impair episodic memory function in humans and animals, and unit recording data from these regions in behaving animals indicate episodic memory processes. Intracellular recording in these regions demonstrates specific cellular properties including resonance, membrane potential oscillations and bistable persistent spiking that could underlie the encoding and retrieval of episodic trajectories. A model presented here shows how intrinsic dynamical properties of neurons could mediate the encoding of episodic memories as complex spatiotemporal trajectories. The dynamics of neurons allow encoding and retrieval of unique episodic trajectories in multiple continuous dimensions including temporal intervals, personal location, the spatial coordinates and sensory features of perceived objects and generated actions, and associations between these elements. The model also addresses how cellular dynamics could underlie unit firing data suggesting mechanisms for coding continuous dimensions of space, time, sensation and action.

Evaluation of the oscillatory interference model of grid cell firing through analysis and measured period variance of some biological oscillators

Models of the hexagonally arrayed spatial activity pattern of grid cell firing in the literature generally fall into two main categories: continuous attractor models or oscillatory interference models. Burak and Fiete (2009, PLoS Comput Biol) recently examined noise in two continuous attractor models, but did not consider oscillatory interference models in detail. Here we analyze an oscillatory interference model to examine the effects of noise on its stability and spatial firing properties. We show analytically that the square of the drift in encoded position due to noise is proportional to time and inversely proportional to the number of oscillators. We also show there is a relatively fixed breakdown point, independent of many parameters of the model, past which noise overwhelms the spatial signal. Based on this result, we show that a pair of oscillators are expected to maintain a stable grid for approximately t = 5µ³/(4πσ)² seconds where µ is the mean period of an oscillator in seconds and σ² its variance in seconds². We apply this criterion to recordings of individual persistent spiking neurons in postsubiculum (dorsal presubiculum) and layers III and V of entorhinal cortex, to subthreshold membrane potential oscillation recordings in layer II stellate cells of medial entorhinal cortex and to values from the literature regarding medial septum theta bursting cells. All oscillators examined have expected stability times far below those seen in experimental recordings of grid cells, suggesting the examined biological oscillators are unfit as a substrate for current implementations of oscillatory interference models. However, oscillatory interference models can tolerate small amounts of noise, suggesting the utility of circuit level effects which might reduce oscillator variability. Further implications for grid cell models are discussed.

For many animals, including rats, accurate spatial memory over relatively large areas is important in order to find food and shelter. Just as unique points in time can be efficiently represented by combinations of repeating elements like hours, days, and months, points in space can be represented as combinations of elements that repeat at different spatial scales. Just such a code has been identified in the brains of rats and it shows an intriguing triangular spacing of encoded locations. Two different explanations have been developed as to what general mechanism in the brain might be able to generate this unusual code. However, to date there is not conclusive experimental evidence indicating whether either of the two explanations is correct. Here we show in detail that one of the explanations, called oscillatory interference, has specific requirements regarding the amount of variability in the system that implements it. We then report data experimentally examining candidate systems to evaluate their levels of noise. The large amount of noise that we find presents a challenge to the currently suggested biological implementations of oscillatory interference, but it does not provide support for the alternative explanation.

A phase code for memory could arise from circuit mechanisms in entorhinal cortex

Neurophysiological data reveals intrinsic cellular properties that suggest how entorhinal cortical neurons could code memory by the phase of their firing. Potential cellular mechanisms for this phase coding in models of entorhinal function are reviewed. This mechanism for phase coding provides a substrate for modeling the responses of entorhinal grid cells, as well as the replay of neural spiking activity during waking and sleep. Efforts to implement these abstract models in more detailed biophysical compartmental simulations raise specific issues that could be addressed in larger scale population models incorporating mechanisms of inhibition.

Distance coding strategies based on the entorhinal grid cell system

Estimating and keeping track of the distance from salient points of the environment are important constituents of the spatial awareness and navigation. In rodents, the majority of principal cells in the hippocampus are known to be correlated with the position of the animal. However, the lack of topography in the hippocampal cognitive map does not support the assumption that connections between these cells are able to store and recall distances between coded positions. In contrast, the firing fields of the grid cells in the medial entorhinal cortex form triangular grids and are organized on metrical principles. We suggest a model in which a hypothesized 'distance cell' population is able to extract metrics from the activity of grid cells. We show that storing the momentary activity pattern of the grid cell system in a freely chosen position by one-shot learning and comparing it to the actual grid activity at other positions results in a distance dependent activity of these cells. The actual distance of the animal from the origin can be decoded directly by selecting the distance cell receiving the largest excitation or indirectly via transmission of local interneurons. We found that direct decoding works up to the longest grid spacing, but fails on smaller scales, while the indirect way provides precise distance determination up to the half of the longest grid spacing. In both cases, simulated distance cells have a multi-peaked, patchy spatial activity pattern consistent with the experimentally observed behavior of granule cells in the dentate gyrus.

Computation by oscillations: implications of experimental data for theoretical models of grid cells

Recordings in awake, behaving animals demonstrate that cells in medial entorhinal cortex (mEC) show "grid cell" firing activity when a rat explores an open environment. Intracellular recording in slices from different positions along the dorsal to ventral axis show differences in intrinsic properties such as subthreshold membrane potential oscillations (MPO), resonant frequency, and the presence of the hyperpolarization-activated cation current (h-current). The differences in intrinsic properties correlate with differences in grid cell spatial scale along the dorsal-ventral axis of mEC. Two sets of computational models have been proposed to explain the grid cell firing phenomena: oscillatory interference models and attractor-dynamic models. Both types of computational models are briefly reviewed, and cellular experimental evidence is interpreted and presented in the context of both models. The oscillatory interference model has variations that include an additive model and a multiplicative model. Experimental data on the voltage-dependence of oscillations presented here support the additive model. The additive model also simulates data from ventral neurons showing large spacing between grid firing fields within the limits of observed MPO frequencies. The interactions of h-current with synaptic modification suggest that the difference in intrinsic properties could also contribute to differences in grid cell properties due to attractor dynamics along the dorsal to ventral axis of mEC. Mechanisms of oscillatory interference and attractor dynamics may make complementary contributions to the properties of grid cell firing in entorhinal cortex.

Grid cell mechanisms and function: contributions of entorhinal persistent spiking and phase resetting

This article presents a model of grid cell firing based on the intrinsic persistent firing shown experimentally in neurons of entorhinal cortex. In this model, the mechanism of persistent firing allows individual neurons to hold a stable baseline firing frequency. Depolarizing input from speed-modulated head direction cells transiently shifts the frequency of firing from baseline, resulting in a shift in spiking phase in proportion to the integral of velocity. The convergence of input from different persistent firing neurons causes spiking in a grid cell only when the persistent firing neurons are within similar phase ranges. This model effectively simulates the two-dimensional firing of grid cells in open field environments, as well as the properties of theta phase precession. This model provides an alternate implementation of oscillatory interference models. The persistent firing could also interact on a circuit level with rhythmic inhibition and neurons showing membrane potential oscillations to code position with spiking phase. These mechanisms could operate in parallel with computation of position from visual angle and distance of stimuli. In addition to simulating two-dimensional grid patterns, models of phase interference can account for context-dependent firing in other tasks. In network simulations of entorhinal cortex, hippocampus, and postsubiculum, the reset of phase effectively replicates context-dependent firing by entorhinal and hippocampal neurons during performance of a continuous spatial alternation task, a delayed spatial alternation task with running in a wheel during the delay period (Pastalkova et al., Science, 2008), and a hairpin maze task.