The emergence of grid cells: Intelligent design or just adaptation?

Probable nature of higher-dimensional symmetries underlying mammalian grid-cell activity patterns

Lattices abound in nature-from the crystal structure of minerals to the honey-comb organization of ommatidia in the compound eye of insects. These arrangements provide solutions for optimal packings, efficient resource distribution, and cryptographic protocols. Do lattices also play a role in how the brain represents information? We focus on higher-dimensional stimulus domains, with particular emphasis on neural representations of physical space, and derive which neuronal lattice codes maximize spatial resolution. For mammals navigating on a surface, we show that the hexagonal activity patterns of grid cells are optimal. For species that move freely in three dimensions, a face-centered cubic lattice is best. This prediction could be tested experimentally in flying bats, arboreal monkeys, or marine mammals. More generally, our theory suggests that the brain encodes higher-dimensional sensory or cognitive variables with populations of grid-cell-like neurons whose activity patterns exhibit lattice structures at multiple, nested scales.

Environmental boundaries as an error correction mechanism for grid cells

Medial entorhinal grid cells fire in periodic, hexagonally patterned locations and are proposed to support path-integration-based navigation. The recursive nature of path integration results in accumulating error and, without a corrective mechanism, a breakdown in the calculation of location. The observed long-term stability of grid patterns necessitates that the system either performs highly precise internal path integration or implements an external landmark-based error correction mechanism. To distinguish these possibilities, we examined grid cells in behaving rodents as they made long trajectories across an open arena. We found that error accumulates relative to time and distance traveled since the animal last encountered a boundary. This error reflects coherent drift in the grid pattern. Further, interactions with boundaries yield direction-dependent error correction, suggesting that border cells serve as a neural substrate for error correction. These observations, combined with simulations of an attractor network grid cell model, demonstrate that landmarks are crucial to grid stability.

Current questions on space and time encoding

The Nobel Prize in Physiology or Medicine 2014 celebrated the groundbreaking findings on place cells and grid cells by John O'Keefe and May-Britt Moser and Edvard Moser. These findings provided an essential foothold for understanding the cognitive encoding of space and time in episodic memory function. This foothold provides a closer view of a broad new world of important research questions raised by the phenomena of place cells and grid cells. These questions concern the mechanisms of generation of place and grid cell firing, including sensory influences, circuit dynamics and intrinsic properties. Similar questions concern the generation of time cells. In addition, questions concern the functional role of place cells, grid cells and time cells in mediating goal-directed behavior and episodic memory function.

The self-organization of grid cells in 3D

Do we expect periodic grid cells to emerge in bats, or perhaps dolphins, exploring a three-dimensional environment? How long will it take? Our self-organizing model, based on ring-rate adaptation, points at a complex answer. The mathematical analysis leads to asymptotic states resembling face centered cubic (FCC) and hexagonal close packed (HCP) crystal structures, which are calculated to be very close to each other in terms of cost function. The simulation of the full model, however, shows that the approach to such asymptotic states involves several sub-processes over distinct time scales. The smoothing of the initially irregular multiple fields of individual units and their arrangement into hexagonal grids over certain best planes are observed to occur relatively quickly, even in large 3D volumes. The correct mutual orientation of the planes, though, and the coordinated arrangement of different units, take a longer time, with the network showing no sign of convergence towards either a pure FCC or HCP ordering.

Correlations and functional connections in a population of grid cells

We study the statistics of spike trains of simultaneously recorded grid cells in freely behaving rats. We evaluate pairwise correlations between these cells and, using a maximum entropy kinetic pairwise model (kinetic Ising model), study their functional connectivity. Even when we account for the covariations in firing rates due to overlapping fields, both the pairwise correlations and functional connections decay as a function of the shortest distance between the vertices of the spatial firing pattern of pairs of grid cells, i.e. their phase difference. They take positive values between cells with nearby phases and approach zero or negative values for larger phase differences. We find similar results also when, in addition to correlations due to overlapping fields, we account for correlations due to theta oscillations and head directional inputs. The inferred connections between neurons in the same module and those from different modules can be both negative and positive, with a mean close to zero, but with the strongest inferred connections found between cells of the same module. Taken together, our results suggest that grid cells in the same module do indeed form a local network of interconnected neurons with a functional connectivity that supports a role for attractor dynamics in the generation of grid pattern.

The way mammals navigate in space is hypothesized to depend on neural structures in the temporal lobe including the hippocampus and medial entorhinal cortex (MEC). In particular, grid cells, neurons whose firing is mostly restricted to regions of space that form a hexagonal pattern, are believed to be an important part of this circuitry. Despite several years of work, not much is known about the correlated activity of neurons in the MEC and how grid cells are functionally coupled to each other. Here, we have taken a statistical approach to these questions and studied pairwise correlations and functional connections between simultaneously recorded grid cells. Through careful statistical analysis, we demonstrate that grid cells with nearby firing vertices tend to have positive effects on eliciting responses in each other, while those further apart tend to have inhibitory or no effects. Cells that respond similarly to manipulations of the environment are considered to belong to the same module. Cells belonging to a module have stronger interactions with each other than those in different modules. These results are consistent with and shed light on the population-based mechanisms suggested by models for the generation of grid cell firing.

Grid cell firing patterns may arise from feedback interaction between intrinsic rebound spiking and transverse traveling waves with multiple heading angles

This article presents a model using cellular resonance and rebound properties to model grid cells in medial entorhinal cortex. The model simulates the intrinsic resonance properties of single layer II stellate cells with different frequencies due to the hyperpolarization activated cation current (h current). The stellate cells generate rebound spikes after a delay interval that differs for neurons with different resonance frequency. Stellate cells drive inhibitory interneurons to cause rebound from inhibition in an alternate set of stellate cells that drive interneurons to activate the first set of cells. This allows maintenance of activity with cycle skipping of the spiking of cells that matches recent physiological data on theta cycle skipping. The rebound spiking interacts with subthreshold oscillatory input to stellate cells or interneurons regulated by medial septal input and defined relative to the spatial location coded by neurons. The timing of rebound determines whether the network maintains the activity for the same location or shifts to phases of activity representing a different location. Simulations show that spatial firing patterns similar to grid cells can be generated with a range of different resonance frequencies, indicating how grid cells could be generated with low frequencies present in bats and in mice with knockout of the HCN1 subunit of the h current.

Beyond the bolus: transgenic tools for investigating the neurophysiology of learning and memory

Understanding the neural mechanisms underlying learning and memory in the entorhinal-hippocampal circuit is a central challenge of systems neuroscience. For more than 40 years, electrophysiological recordings in awake, behaving animals have been used to relate the receptive fields of neurons in this circuit to learning and memory. However, the vast majority of such studies are purely observational, as electrical, surgical, and pharmacological circuit manipulations are both challenging and relatively coarse, being unable to distinguish between specific classes of neurons. Recent advances in molecular genetic tools can overcome many of these limitations, enabling unprecedented control over neural activity in behaving animals. Expression of pharmaco- or optogenetic transgenes in cell-type-specific "driver" lines provides unparalleled anatomical and cell-type specificity, especially when delivered by viral complementation. Pharmacogenetic transgenes are specially designed neurotransmitter receptors exclusively activated by otherwise inactive synthetic ligands and have kinetics similar to traditional pharmacology. Optogenetic transgenes use light to control the membrane potential, and thereby operate at the millisecond timescale. Thus, activation of pharmacogenetic transgenes in specific neuronal cell types while recording from other parts of the circuit allows investigation of the role of those neurons in the steady state, whereas optogenetic transgenes allow one to determine the immediate network response.

A model of grid cell development through spatial exploration and spike time-dependent plasticity

Grid cell responses develop gradually after eye opening, but little is known about the rules that govern this process. We present a biologically plausible model for the formation of a grid cell network. An asymmetric spike time-dependent plasticity rule acts upon an initially unstructured network of spiking neurons that receive inputs encoding animal velocity and location. Neurons develop an organized recurrent architecture based on the similarity of their inputs, interacting through inhibitory interneurons. The mature network can convert velocity inputs into estimates of animal location, showing that spatially periodic responses and the capacity of path integration can arise through synaptic plasticity, acting on inputs that display neither. The model provides numerous predictions about the necessity of spatial exploration for grid cell development, network topography, the maturation of velocity tuning and neural correlations, the abrupt transition to stable patterned responses, and possible mechanisms to set grid period across grid modules.

Grid maps for spaceflight, anyone? They are for free!

We show that, given extensive exploration of a three-dimensional volume, grid units can form with the approximate periodicity of a face-centered cubic crystal, as the spontaneous product of a self-organizing process at the single unit level, driven solely by firing rate adaptation.

New and distinct hippocampal place codes are generated in a new environment during septal inactivation

The hippocampus generates distinct neural codes to disambiguate similar experiences, a process thought to underlie episodic memory function. Entorhinal grid cells provide a prominent spatial signal to hippocampus, and changes in their firing pattern could thus generate a distinct spatial code in each context. We examined whether we would preclude the emergence of new spatial representations in a novel environment during muscimol inactivation of the medial septal area, a manipulation known to disrupt theta oscillations and grid cell firing. We found that new, highly distinct configurations of place fields emerged immediately and remained stable during the septal inactivation. The new place code persisted when theta oscillations had recovered. Theta rhythmicity and feedforward input from grid cell networks were thus not required to generate new spatial representations in the hippocampus.

Grid cells and cortical representation

One of the grand challenges in neuroscience is to comprehend neural computation in the association cortices, the parts of the cortex that have shown the largest expansion and differentiation during mammalian evolution and that are thought to contribute profoundly to the emergence of advanced cognition in humans. In this Review, we use grid cells in the medial entorhinal cortex as a gateway to understand network computation at a stage of cortical processing in which firing patterns are shaped not primarily by incoming sensory signals but to a large extent by the intrinsic properties of the local circuit.

Vestibular control of entorhinal cortex activity in spatial navigation

Navigation in rodents depends on both self-motion (idiothetic) and external (allothetic) information. Idiothetic information has a predominant role when allothetic information is absent or irrelevant. The vestibular system is a major source of idiothetic information in mammals. By integrating the signals generated by angular and linear accelerations during exploration, a rat is able to generate and update a vector pointing to its starting place and to perform accurate return. This navigation strategy, called path integration, has been shown to involve a network of brain structures. Among these structures, the entorhinal cortex (EC) may play a pivotal role as suggested by lesion and electrophysiological data. In particular, it has been recently discovered that some neurons in the medial EC display multiple firing fields producing a regular grid-like pattern across the environment. Such regular activity may arise from the integration of idiothetic information. This hypothesis would be strongly strengthened if it was shown that manipulation of vestibular information interferes with grid cell activity. In the present paper we review neuroanatomical and functional evidence indicating that the vestibular system influences the activity of the brain network involved in spatial navigation. We also provide new data on the effects of reversible inactivation of the peripheral vestibular system on the EC theta rhythm. The main result is that tetrodotoxin (TTX) administration abolishes velocity-controlled theta oscillations in the EC, indicating that vestibular information is necessary for EC activity. Since recent data demonstrate that disruption of theta rhythm in the medial EC induces a disorganization of grid cell firing, our findings indicate that the integration of idiothetic information in the EC is essential to form a spatial representation of the environment.

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.

Parvalbumin interneurons provide grid cell-driven recurrent inhibition in the medial entorhinal cortex

Grid cells in the medial entorhinal cortex (MEC) generate metric spatial representations. Recent attractor-network models suggest an essential role for GABAergic interneurons in the emergence of the grid-cell firing pattern through recurrent inhibition dependent on grid-cell phase. To test this hypothesis, we studied identified parvalbumin-expressing (PV(+)) interneurons that are the most likely candidate for providing this recurrent inhibition onto grid cells. Using optogenetics and tetrode recordings in mice, we found that PV(+) interneurons exhibited high firing rates, low spatial sparsity and no spatial periodicity. PV(+) interneurons inhibited all functionally defined cell types in the MEC and were in turn recruited preferentially by grid cells. To our surprise, we found that individual PV(+) interneurons received input from grid cells with various phases, which most likely accounts for the broadly tuned spatial firing activity of PV(+) interneurons. Our data argue against the notion that PV(+) interneurons provide phase-dependent recurrent inhibition and challenge recent attractor-network models of grid cells.

A feedforward model for the formation of a grid field where spatial information is provided solely from place cells

Grid cells (GCs) in the medial entorhinal cortex (mEC) have the property of having their firing activity spatially tuned to a regular triangular lattice. Several theoretical models for grid field formation have been proposed, but most assume that place cells (PCs) are a product of the grid cell system. There is, however, an alternative possibility that is supported by various strands of experimental data. Here we present a novel model for the emergence of gridlike firing patterns that stands on two key hypotheses: (1) spatial information in GCs is provided from PC activity and (2) grid fields result from a combined synaptic plasticity mechanism involving inhibitory and excitatory neurons mediating the connections between PCs and GCs. Depending on the spatial location, each PC can contribute with excitatory or inhibitory inputs to GC activity. The nature and magnitude of the PC input is a function of the distance to the place field center, which is inferred from rate decoding. A biologically plausible learning rule drives the evolution of the connection strengths from PCs to a GC. In this model, PCs compete for GC activation, and the plasticity rule favors efficient packing of the space representation. This leads to gridlike firing patterns. In a new environment, GCs continuously recruit new PCs to cover the entire space. The model described here makes important predictions and can represent the feedforward connections from hippocampus CA1 to deeper mEC layers.

CA3 retrieves coherent representations from degraded input: direct evidence for CA3 pattern completion and dentate gyrus pattern separation

Theories of associative memory suggest that successful memory storage and recall depend on a balance between two complementary processes: pattern separation (to minimize interference) and pattern completion (to retrieve a memory when presented with partial or degraded input cues). Putative attractor circuitry in the hippocampal CA3 region is thought to be the final arbiter between these two processes. Here we present direct, quantitative evidence that CA3 produces an output pattern closer to the originally stored representation than its degraded input patterns from the dentate gyrus (DG). We simultaneously recorded activity from CA3 and DG of behaving rats when local and global reference frames were placed in conflict. CA3 showed a coherent population response to the conflict (pattern completion), even though its DG inputs were severely disrupted (pattern separation). The results thus confirm the hallmark predictions of a longstanding computational model of hippocampal memory processing.

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.

Limbic systems for emotion and for memory, but no single limbic system

The concept of a (single) limbic system is shown to be outmoded. Instead, anatomical, neurophysiological, functional neuroimaging, and neuropsychological evidence is described that anterior limbic and related structures including the orbitofrontal cortex and amygdala are involved in emotion, reward valuation, and reward-related decision-making (but not memory), with the value representations transmitted to the anterior cingulate cortex for action-outcome learning. In this 'emotion limbic system' a computational principle is that feedforward pattern association networks learn associations from visual, olfactory and auditory stimuli, to primary reinforcers such as taste, touch, and pain. In primates including humans this learning can be very rapid and rule-based, with the orbitofrontal cortex overshadowing the amygdala in this learning important for social and emotional behaviour. Complementary evidence is described showing that the hippocampus and limbic structures to which it is connected including the posterior cingulate cortex and the fornix-mammillary body-anterior thalamus-posterior cingulate circuit are involved in episodic or event memory, but not emotion. This 'hippocampal system' receives information from neocortical areas about spatial location, and objects, and can rapidly associate this information together by the different computational principle of autoassociation in the CA3 region of the hippocampus involving feedback. The system can later recall the whole of this information in the CA3 region from any component, a feedback process, and can recall the information back to neocortical areas, again a feedback (to neocortex) recall process. Emotion can enter this memory system from the orbitofrontal cortex etc., and be recalled back to the orbitofrontal cortex etc. during memory recall, but the emotional and hippocampal networks or 'limbic systems' operate by different computational principles, and operate independently of each other except insofar as an emotional state or reward value attribute may be part of an episodic memory.

Network mechanisms of grid cells

One of the major breakthroughs in neuroscience is the emerging understanding of how signals from the external environment are extracted and represented in the primary sensory cortices of the mammalian brain. The operational principles of the rest of the cortex, however, have essentially remained in the dark. The discovery of grid cells, and their functional organization, opens the door to some of the first insights into the workings of the association cortices, at a stage of neural processing where firing properties are shaped not primarily by the nature of incoming sensory signals but rather by internal self-organizing principles. Grid cells are place-modulated neurons whose firing locations define a periodic triangular array overlaid on the entire space available to a moving animal. The unclouded firing pattern of these cells is rare within the association cortices. In this paper, we shall review recent advances in our understanding of the mechanisms of grid-cell formation which suggest that the pattern originates by competitive network interactions, and we shall relate these ideas to new insights regarding the organization of grid cells into functionally segregated modules.

An isomorphic mapping hypothesis of the grid representation

We introduce a grid cell microcircuit hypothesis. We propose the ‘grid in the world’ (evident in grid cell discharges) is generated by a ‘grid in the cortex’. This cortical grid is formed by patches of calbindin-positive pyramidal neurons in layer 2 of medial entorhinal cortex (MEC). Our isomorphic mapping hypothesis assumes three types of isomorphism: (i) metric correspondence of neural space (the two-dimensional cortical sheet) and the external two-dimensional space within patches; (ii) isomorphism between cellular connectivity matrix and firing field; (iii) isomorphism between single cell and population activity. Each patch is a grid cell lattice arranged in a two-dimensional map of space with a neural : external scale of approximately 1 : 2000 in the dorsal part of rat MEC. The lattice behaves like an excitable medium with neighbouring grid cells exciting each other. Spatial scale is implemented as an intrinsic scaling factor for neural propagation speed. This factor varies along the dorsoventral cortical axis. A connectivity scheme of the grid system is described. Head direction input specifies the direction of activity propagation. We extend the theory to neurons between grid patches and predict a rare discharge pattern (inverted grid cells) and the relative location and proportion of grid cells and spatial band cells.

Coordinated learning of grid cell and place cell spatial and temporal properties: multiple scales, attention and oscillations

A neural model proposes how entorhinal grid cells and hippocampal place cells may develop as spatial categories in a hierarchy of self-organizing maps (SOMs). The model responds to realistic rat navigational trajectories by learning both grid cells with hexagonal grid firing fields of multiple spatial scales, and place cells with one or more firing fields, that match neurophysiological data about their development in juvenile rats. Both grid and place cells can develop by detecting, learning and remembering the most frequent and energetic co-occurrences of their inputs. The model's parsimonious properties include: similar ring attractor mechanisms process linear and angular path integration inputs that drive map learning; the same SOM mechanisms can learn grid cell and place cell receptive fields; and the learning of the dorsoventral organization of multiple spatial scale modules through medial entorhinal cortex to hippocampus (HC) may use mechanisms homologous to those for temporal learning through lateral entorhinal cortex to HC ('neural relativity'). The model clarifies how top-down HC-to-entorhinal attentional mechanisms may stabilize map learning, simulates how hippocampal inactivation may disrupt grid cells, and explains data about theta, beta and gamma oscillations. The article also compares the three main types of grid cell models in the light of recent data.

How environment geometry affects grid cell symmetry and what we can learn from it

The mammalian hippocampal formation provides neuronal representations of environmental location but the underlying mechanisms are unclear. The majority of cells in medial entorhinal cortex and parasubiculum show spatially periodic firing patterns. Grid cells exhibit hexagonal symmetry and form an important subset of this more general class. Occasional changes between hexagonal and non-hexagonal firing patterns imply a common underlying mechanism. Importantly, the symmetrical properties are strongly affected by the geometry of the environment. Here, we introduce a field–boundary interaction model where we demonstrate that the grid cell pattern can be formed from competing place-like and boundary inputs. We show that the modelling results can accurately capture our current experimental observations.

Neuronal rebound spiking, resonance frequency and theta cycle skipping may contribute to grid cell firing in medial entorhinal cortex

Data show a relationship of cellular resonance and network oscillations in the entorhinal cortex to the spatial periodicity of grid cells. This paper presents a model that simulates the resonance and rebound spiking properties of entorhinal neurons to generate spatial periodicity dependent upon phasic input from medial septum. The model shows that a difference in spatial periodicity can result from a difference in neuronal resonance frequency that replicates data from several experiments. The model also demonstrates a functional role for the phenomenon of theta cycle skipping in the medial entorhinal cortex.

Architecture of spatial circuits in the hippocampal region

The hippocampal region contains several principal neuron types, some of which show distinct spatial firing patterns. The region is also known for its diversity in neural circuits and many have attempted to causally relate network architecture within and between these unique circuits to functional outcome. Still, much is unknown about the mechanisms or network properties by which the functionally specific spatial firing profiles of neurons are generated, let alone how they are integrated into a coherently functioning meta-network. In this review, we explore the architecture of local networks and address how they may interact within the context of an overarching space circuit, aiming to provide directions for future successful explorations.

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.

Transgenically targeted rabies virus demonstrates a major monosynaptic projection from hippocampal area CA2 to medial entorhinal layer II neurons

The enormous potential of modern molecular neuroanatomical tools lies in their ability to determine the precise connectivity of the neuronal cell types comprising the innate circuitry of the brain. We used transgenically targeted viral tracing to identify the monosynaptic inputs to the projection neurons of layer II of medial entorhinal cortex (MEC-LII) in mice. These neurons are not only major inputs to the hippocampus, the structure most clearly implicated in learning and memory, they also are "grid cells." Here we address the question of what kinds of inputs are specifically targeting these MEC-LII cells. Cell-specific infection of MEC-LII with recombinant rabies virus results in unambiguous labeling of monosynaptic inputs. Furthermore, ratios of labeled neurons in different regions are largely consistent between animals, suggesting that label reflects density of innervation. While the results mostly confirm prior anatomical work, they also reveal a novel major direct input to MEC-LII from hippocampal pyramidal neurons. Interestingly, the vast majority of these direct hippocampal inputs arise not from the major hippocampal subfields of CA1 and CA3, but from area CA2, a region that has historically been thought to merely be a transitional zone between CA3 and CA1. We confirmed this unexpected result using conventional tracing techniques in both rats and mice.

The mechanisms for pattern completion and pattern separation in the hippocampus

The mechanisms for pattern completion and pattern separation are described in the context of a theory of hippocampal function in which the hippocampal CA3 system operates as a single attractor or autoassociation network to enable rapid, one-trial, associations between any spatial location (place in rodents, or spatial view in primates) and an object or reward, and to provide for completion of the whole memory during recall from any part. The factors important in the pattern completion in CA3 together with a large number of independent memories stored in CA3 include a sparse distributed representation which is enhanced by the graded firing rates of CA3 neurons, representations that are independent due to the randomizing effect of the mossy fibers, heterosynaptic long-term depression as well as long-term potentiation in the recurrent collateral synapses, and diluted connectivity to minimize the number of multiple synapses between any pair of CA3 neurons which otherwise distort the basins of attraction. Recall of information from CA3 is implemented by the entorhinal cortex perforant path synapses to CA3 cells, which in acting as a pattern associator allow some pattern generalization. Pattern separation is performed in the dentate granule cells using competitive learning to convert grid-like entorhinal cortex firing to place-like fields. Pattern separation in CA3, which is important for completion of any one of the stored patterns from a fragment, is provided for by the randomizing effect of the mossy fiber synapses to which neurogenesis may contribute, by the large number of dentate granule cells each with a sparse representation, and by the sparse independent representations in CA3. Recall to the neocortex is achieved by a reverse hierarchical series of pattern association networks implemented by the hippocampo-cortical backprojections, each one of which performs some pattern generalization, to retrieve a complete pattern of cortical firing in higher-order cortical areas.

Comparison of properties of medial entorhinal cortex layer II neurons in two anatomical dimensions with and without cholinergic activation

Mechanisms underlying grid cell firing in the medial entorhinal cortex (MEC) still remain unknown. Computational modeling studies have suggested that cellular properties such as spike frequency adaptation and persistent firing might underlie the grid cell firing. Recent in vivo studies also suggest that cholinergic activation influences grid cell firing. Here we investigated the anatomical distribution of firing frequency adaptation, the medium spike after hyperpolarization potential (mAHP), subthreshold membrane potential oscillations, sag potential, input resistance and persistent firing, in MEC layer II principal cells using in vitro whole-cell patch clamp recordings in rats. Anatomical distributions of these properties were compared along both the dorso-ventral and medio-lateral axes, both with and without the cholinergic receptor agonist carbachol. We found that spike frequency adaptation is significantly stronger in ventral than in dorsal neurons both with and without carbachol. Spike frequency adaptation was significantly correlated with the duration of the mAHP, which also showed a gradient along the dorso-ventral axis. In carbachol, we found that about 50% of MEC layer II neurons show persistent firing which lasted more than 30 seconds. Persistent firing of MEC layer II neurons might contribute to grid cell firing by providing the excitatory drive. Dorso-ventral differences in spike frequency adaptation we report here are opposite from previous predictions by a computational model. We discuss an alternative mechanism as to how dorso-ventral differences in spike frequency adaptation could contribute to different scales of grid spacing.

The spatial representations acquired in CA3 by self-organizing recurrent connections

Neural computation models have hypothesized that the dentate gyrus (DG) drives the storage in the CA3 network of new memories including, e.g., in rodents, spatial memories. Can recurrent CA3 connections self-organize, during storage, and form what have been called continuous attractors, or charts—so that they express spatial information later, when aside from a partial cue the information may not be available in the inputs? We use a simplified mathematical network model to contrast the properties of spatial representations self-organized through simulated Hebbian plasticity with those of charts pre-wired in the synaptic matrix, a control case closer to the ideal notion of continuous attractors. Both models form granular quasi-attractors, characterized by drift, which approach continuous ones only in the limit of an infinitely large network. The two models are comparable in terms of precision, but not of accuracy: with self-organized connections, the metric of space remains distorted, ill-adequate for accurate path integration, even when scaled up to the real hippocampus. While prolonged self-organization makes charts somewhat more informative about position in the environment, some positional information is surprisingly present also about environments never learned, borrowed, as it were, from unrelated charts. In contrast, context discrimination decreases with more learning, as different charts tend to collapse onto each other. These observations challenge the feasibility of the idealized CA3 continuous chart concept, and are consistent with a CA3 specialization for episodic memory rather than path integration.

A quantitative theory of the functions of the hippocampal CA3 network in memory

A quantitative computational theory of the operation of the hippocampal CA3 system as an autoassociation or attractor network used in episodic memory system is described. In this theory, the CA3 system operates as a single attractor or autoassociation network to enable rapid, one-trial, associations between any spatial location (place in rodents, or spatial view in primates) and an object or reward, and to provide for completion of the whole memory during recall from any part. The theory is extended to associations between time and object or reward to implement temporal order memory, also important in episodic memory. The dentate gyrus (DG) performs pattern separation by competitive learning to produce sparse representations suitable for setting up new representations in CA3 during learning, producing for example neurons with place-like fields from entorhinal cortex grid cells. The dentate granule cells produce by the very small number of mossy fiber (MF) connections to CA3 a randomizing pattern separation effect important during learning but not recall that separates out the patterns represented by CA3 firing to be very different from each other, which is optimal for an unstructured episodic memory system in which each memory must be kept distinct from other memories. The direct perforant path (pp) input to CA3 is quantitatively appropriate to provide the cue for recall in CA3, but not for learning. Tests of the theory including hippocampal subregion analyses and hippocampal NMDA receptor knockouts are described, and support the theory.

Theta rhythm and the encoding and retrieval of space and time

Physiological data demonstrates theta frequency oscillations associated with memory function and spatial behavior. Modeling and data from animals provide a perspective on the functional role of theta rhythm, including correlations with behavioral performance and coding by timing of spikes relative to phase of oscillations. Data supports a theorized role of theta rhythm in setting the dynamics for encoding and retrieval within cortical circuits. Recent data also supports models showing how network and cellular theta rhythmicity allows neurons in the entorhinal cortex and hippocampus to code time and space as a possible substrate for encoding events in episodic memory. Here we discuss these models and relate them to current physiological and behavioral data.

Unveiling the metric structure of internal representations of space

How are neuronal representations of space organized in the hippocampus? The self-organization of such representations, thought to be driven in the CA3 network by the strong randomizing input from the Dentate Gyrus, appears to run against preserving the topology and even less the exact metric of physical space. We present a way to assess this issue quantitatively, and find that in a simple neural network model of CA3, the average topology is largely preserved, but the local metric is loose, retaining e.g., 10% of the optimal spatial resolution.

Neurogenesis in an early protostome relative: progenitor cells in the ventral nerve center of chaetognath hatchlings are arranged in a highly organized geometrical pattern

Emerging evidence suggests that Chaetognatha represent an evolutionary lineage that is the sister group to all other Protostomia thus promoting these animals as a pivotal model for our understanding of bilaterian evolutionary history. We have analyzed the proliferation of neuronal progenitor cells in the developing ventral nerve center (VNC) of Spadella cephaloptera hatchlings. To that end, for the first time in Chaetognatha, we performed in vivo incorporation experiments with the S-phase specific mitosis marker bromodeoxyuridine (BrdU). Our experiments provide evidence for a high level of mitotic activity in the VNC for ca. 3 days after hatching. Neurogenesis is carried by presumptive neuronal progenitor cells that cycle rapidly and most likely divide asymmetrically. These progenitors are arranged in a distinct grid-like geometrical pattern including about 35 transverse rows. Considering Chaetognaths to be an early offshoot of the protostome lineage we conclude that the presence of neuronal progenitor cells with asymmetric division seems to be a feature that is rooted deeply in the Metazoa. In the light of previous evidence indicating the presence of serially iterated peptidergic neurons with individual identities in the chaetognath VNC, we discuss if these neuronal progenitor cells give rise to distinct lineages. Furthermore, we evaluate the serially iterated arrangement of the progenitor cells in the light of evolution of segmentation.

Cellular mechanisms of spatial navigation in the medial entorhinal cortex

Neurons in the medial entorhinal cortex exhibit a grid-like spatial pattern of spike rates that has been proposed to represent a neural code for path integration. To understand how grid cell firing arises from the combination of intrinsic conductances and synaptic input in medial entorhinal stellate cells, we performed patch-clamp recordings in mice navigating in a virtual-reality environment. We found that the membrane potential signature of stellate cells during firing field crossings consisted of a slow depolarization driving spike output. This was best predicted by network models in which neurons receive sustained depolarizing synaptic input during a field crossing, such as continuous attractor network models of grid cell firing. Another key feature of the data, phase precession of intracellular theta oscillations and spiking with respect to extracellular theta oscillations, was best captured by an oscillatory interference model. Thus, these findings provide crucial new information for a quantitative understanding of the cellular basis of spatial navigation in the entorhinal cortex.

The entorhinal grid map is discretized

The medial entorhinal cortex (MEC) is part of the brain's circuit for dynamic representation of self-location. The metric of this representation is provided by grid cells, cells with spatial firing fields that tile environments in a periodic hexagonal pattern. Limited anatomical sampling has obscured whether the grid system operates as a unified system or a conglomerate of independent modules. Here we show with recordings from up to 186 grid cells in individual rats that grid cells cluster into a small number of layer-spanning anatomically overlapping modules with distinct scale, orientation, asymmetry and theta-frequency modulation. These modules can respond independently to changes in the geometry of the environment. The discrete topography of the grid-map, and the apparent autonomy of the modules, differ from the graded topography of maps for continuous variables in several sensory systems, raising the possibility that the modularity of the grid map is a product of local self-organizing network dynamics.

Recurrent inhibitory circuitry as a mechanism for grid formation

Grid cells in layer II of the medial entorhinal cortex form a principal component of the mammalian neural representation of space. The firing pattern of a single grid cell has been hypothesized to be generated through attractor dynamics in a network with a specific local connectivity including both excitatory and inhibitory connections. However, experimental evidence supporting the presence of such connectivity among grid cells in layer II is limited. Here we report recordings from more than 600 neuron pairs in rat entorhinal slices, demonstrating that stellate cells, the principal cell type in the layer II grid network, are mainly interconnected via inhibitory interneurons. Using a model attractor network, we demonstrate that stable grid firing can emerge from a simple recurrent inhibitory network. Our findings thus suggest that the observed inhibitory microcircuitry between stellate cells is sufficient to generate grid-cell firing patterns in layer II of the medial entorhinal cortex.

Grid alignment in entorhinal cortex

The spatial responses of many of the cells recorded in all layers of rodent medial entorhinal cortex (mEC) show mutually aligned grid patterns. Recent experimental findings have shown that grids can often be better described as elliptical rather than purely circular and that, beyond the mutual alignment of their grid axes, ellipses tend to also orient their long axis along preferred directions. Are grid alignment and ellipse orientation aspects of the same phenomenon? Does the grid alignment result from single-unit mechanisms or does it require network interactions? We address these issues by refining a single-unit adaptation model of grid formation, to describe specifically the spontaneous emergence of conjunctive grid-by-head-direction cells in layers III, V, and VI of mEC. We find that tight alignment can be produced by recurrent collateral interactions, but this requires head-direction (HD) modulation. Through a competitive learning process driven by spatial inputs, grid fields then form already aligned, and with randomly distributed spatial phases. In addition, we find that the self-organization process is influenced by any anisotropy in the behavior of the simulated rat. The common grid alignment often orients along preferred running directions (RDs), as induced in a square environment. When speed anisotropy is present in exploration behavior, the shape of individual grids is distorted toward an ellipsoid arrangement. Speed anisotropy orients the long ellipse axis along the fast direction. Speed anisotropy on its own also tends to align grids, even without collaterals, but the alignment is seen to be loose. Finally, the alignment of spatial grid fields in multiple environments shows that the network expresses the same set of grid fields across environments, modulo a coherent rotation and translation. Thus, an efficient metric encoding of space may emerge through spontaneous pattern formation at the single-unit level, but it is coherent, hence context-invariant, if aided by collateral interactions.

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.

Optimal population codes for space: grid cells outperform place cells

Rodents use two distinct neuronal coordinate systems to estimate their position: place fields in the hippocampus and grid fields in the entorhinal cortex. Whereas place cells spike at only one particular spatial location, grid cells fire at multiple sites that correspond to the points of an imaginary hexagonal lattice. We study how to best construct place and grid codes, taking the probabilistic nature of neural spiking into account. Which spatial encoding properties of individual neurons confer the highest resolution when decoding the animal's position from the neuronal population response? A priori, estimating a spatial position from a grid code could be ambiguous, as regular periodic lattices possess translational symmetry. The solution to this problem requires lattices for grid cells with different spacings; the spatial resolution crucially depends on choosing the right ratios of these spacings across the population. We compute the expected error in estimating the position in both the asymptotic limit, using Fisher information, and for low spike counts, using maximum likelihood estimation. Achieving high spatial resolution and covering a large range of space in a grid code leads to a trade-off: the best grid code for spatial resolution is built of nested modules with different spatial periods, one inside the other, whereas maximizing the spatial range requires distinct spatial periods that are pairwisely incommensurate. Optimizing the spatial resolution predicts two grid cell properties that have been experimentally observed. First, short lattice spacings should outnumber long lattice spacings. Second, the grid code should be self-similar across different lattice spacings, so that the grid field always covers a fixed fraction of the lattice period. If these conditions are satisfied and the spatial "tuning curves" for each neuron span the same range of firing rates, then the resolution of the grid code easily exceeds that of the best possible place code with the same number of neurons.

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.

Possible role of acetylcholine in regulating spatial novelty effects on theta rhythm and grid cells

Existing pharmacological and lesion data indicate that acetylcholine plays an important role in memory formation. For example, increased levels of acetylcholine in the hippocampal formation are known to be associated with successful encoding while disruption of the cholinergic system leads to impairments on a range of mnemonic tasks. However, cholinergic signaling from the medial septum also plays a central role in generating and pacing theta-band oscillations throughout the hippocampal formation. Recent experimental results suggest a potential link between these distinct phenomena. Environmental novelty, a condition associated with strong cholinergic drive, has been shown to induce an expansion in the firing pattern of entorhinal grid cells and a reduction in the frequency of theta measured from the LFP. Computational modeling suggests the spatial activity of grid cells is produced by interference between neuronal oscillators; scale being determined by theta-band oscillations impinging on entorhinal stellate cells, the frequency of which is modulated by acetylcholine. Here we propose that increased cholinergic signaling in response to environmental novelty triggers grid expansion by reducing the frequency of the oscillations. Furthermore, we argue that cholinergic induced grid expansion may enhance, or even induce, encoding by producing a mismatch between expanded grid cells and other spatial inputs to the hippocampus, such as boundary vector cells. Indeed, a further source of mismatch is likely to occur between grid cells of different native scales which may expand by different relative amounts.

Self-organization of multiple spatial and context memories in the hippocampus

One obstacle to understanding the exact processes unfolding inside the hippocampus is that it is still difficult to clearly define what the hippocampus actually does, at the system level. Associated for a long time with the formation of episodic and semantic memories, and with their temporary storage, the hippocampus is also regarded as a structure involved in spatial navigation. These two independent perspectives on the hippocampus are not necessarily exclusive: proposals have been put forward to make them fit into the same conceptual frame. We review both approaches and argue that three critical developments need consideration: (a) recordings of neuronal activity in rodents, revealing beautiful spatial codes expressed in entorhinal cortex, upstream of the hippocampus; (b) comparative behavioral results suggesting, in an evolutionary perspective, qualitative similarity of function across homologous structures with a distinct internal organization; (c) quantitative measures of information, shifting the focus from who does what to how much each neuronal population expresses each code. These developments take the hippocampus away from philosophical discussions of all-or-none cause-effect relations, and into the quantitative mainstream of modern neural science.

Framing spatial cognition: neural representations of proximal and distal frames of reference and their roles in navigation

The most common behavioral test of hippocampus-dependent, spatial learning and memory is the Morris water task, and the most commonly studied behavioral correlate of hippocampal neurons is the spatial specificity of place cells. Despite decades of intensive research, it is not completely understood how animals solve the water task and how place cells generate their spatially specific firing fields. Based on early work, it has become the accepted wisdom in the general neuroscience community that distal spatial cues are the primary sources of information used by animals to solve the water task (and similar spatial tasks) and by place cells to generate their spatial specificity. More recent research, along with earlier studies that were overshadowed by the emphasis on distal cues, put this common view into question by demonstrating primary influences of local cues and local boundaries on spatial behavior and place-cell firing. This paper first reviews the historical underpinnings of the "standard" view from a behavioral perspective, and then reviews newer results demonstrating that an animal's behavior in such spatial tasks is more strongly controlled by a local-apparatus frame of reference than by distal landmarks. The paper then reviews similar findings from the literature on the neurophysiological correlates of place cells and other spatially correlated cells from related brain areas. A model is proposed by which distal cues primarily set the orientation of the animal's internal spatial coordinate system, via the head direction cell system, whereas local cues and apparatus boundaries primarily set the translation and scale of that coordinate system.

Spatial representation: maps in a temporal void

It has been suggested that the matrix-like firing structure of entorhinal grid cells is caused by interference between membrane oscillations at slightly different theta frequencies. A recent report suggests that grid signals can be generated in the absence of theta oscillations.

Lateral entorhinal neurons are not spatially selective in cue-rich environments

The hippocampus is a brain region that is critical for spatial learning, context-dependent memory, and episodic memory. It receives major inputs from the medial entorhinal cortex (MEC) and the lateral EC (LEC). MEC neurons show much greater spatial firing than LEC neurons in a recording chamber with a single, salient landmark. The MEC cells are thought to derive their spatial tuning through path integration, which permits spatially selective firing in such a cue-deprived environment. In accordance with theories that postulate two spatial mapping systems that provide input to the hippocampus-an internal, path-integration system and an external, landmark-based system-it was possible that LEC neurons can also convey a spatial signal, but that the signal requires multiple landmarks to define locations, rather than movement integration. To test this hypothesis, neurons from the MEC and LEC were recorded as rats foraged for food in cue-rich environments. In both environments, LEC neurons showed little spatial specificity, whereas many MEC neurons showed a robust spatial signal. These data strongly support the notion that the MEC and LEC convey fundamentally different types of information to the hippocampus, in terms of their spatial firing characteristics, under various environmental and behavioral conditions.

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.

The structure of networks that produce the transformation from grid cells to place cells

Since grid cells were discovered in the medial entorhinal cortex, several models have been proposed for the transformation from periodic grids to the punctate place fields of hippocampal place cells. These prior studies have each focused primarily on a particular model structure. By contrast, the goal of this study is to understand the general nature of the solutions that generate the grids-to-places transformation, and to exploit this insight to solve problems that were previously unsolved. First, we derive a family of feedforward networks that generate the grids-to-places transformations. These networks have in common an inverse relationship between the synaptic weights and a grid property that we call the normalized offset. Second, we analyze the solutions of prior models in terms of this novel measure and found to our surprise that almost all prior models yield solutions that can be described by this family of networks. The one exception is a model that is unrealistically sensitive to noise. Third, with this insight into the structure of the solutions, we then construct explicitly solutions for the grids-to-places transformation with multiple spatial maps, that is, with place fields in arbitrary locations either within the same (multiple place fields) or in different (global remapping) enclosures. These multiple maps are possible because the weights are learned or assigned in such a way that a group of weights contributes to spatial specificity in one context but remains spatially unstructured in another context. Fourth, we find parameters such that global remapping solutions can be found by synaptic learning in spiking neurons, despite previous suggestions that this might not be possible. In conclusion, our results demonstrate the power of understanding the structure of the solutions and suggest that we may have identified the structure that is common to all robust solutions of the grids-to-places transformation.