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

Spatial reasoning via recurrent neural dynamics in mouse retrosplenial cortex

From visual perception to language, sensory stimuli change their meaning depending on previous experience. Recurrent neural dynamics can interpret stimuli based on externally cued context, but it is unknown whether they can compute and employ internal hypotheses to resolve ambiguities. Here we show that mouse retrosplenial cortex (RSC) can form several hypotheses over time and perform spatial reasoning through recurrent dynamics. In our task, mice navigated using ambiguous landmarks that are identified through their mutual spatial relationship, requiring sequential refinement of hypotheses. Neurons in RSC and in artificial neural networks encoded mixtures of hypotheses, location and sensory information, and were constrained by robust low-dimensional dynamics. RSC encoded hypotheses as locations in activity space with divergent trajectories for identical sensory inputs, enabling their correct interpretation. Our results indicate that interactions between internal hypotheses and external sensory data in recurrent circuits can provide a substrate for complex sequential cognitive reasoning.

A memory model of rodent spatial navigation in which place cells are memories arranged in a grid and grid cells are non-spatial

A theory and neurocomputational model are presented that explain grid cell responses as the byproduct of equally dissimilar hippocampal memories. On this account, place and grid cells are best understood as the natural consequence of memory encoding and retrieval; a precise hexagonal grid is the exception rather than the rule, emerging when the animal explores a large surface that is devoid of landmarks and objects. In the proposed memory model, place cells represent memories that are conjunctions of both spatial and non-spatial attributes, and grid cells primarily represent the non-spatial attributes (e.g. sounds, surface texture, etc.) found throughout the two-dimensional recording enclosure. Place cells support memories of the locations where non-spatial attributes can be found (e.g. positions with a particular sound), which are arranged in a hexagonal lattice owing to memory encoding and consolidation processes (pattern separation) as applied to situations in which the non-spatial attributes are found at all locations of a two-dimensional surface. Grid cells exhibit their spatial firing pattern owing to feedback from hippocampal place cells (i.e. a hexagonal pattern of remembered locations for the non-spatial attribute represented by a grid cell). Model simulations explain a wide variety of results in the rodent spatial navigation literature.

Spontaneous alternation of place-cell sequences in the open field through spike frequency adaptation

Spatial sequences encoded by cells in the hippocampal-entorhinal region have been observed to spontaneously alternate across the animal's midline during navigation in the open field, but it is unknown how this occurs. We show that sinusoidal sampling patterns emerge rapidly and robustly in a simple model of the hippocampal place-cell sequences based on spike frequency adaptation that makes no assumptions about sequence direction. We corroborate our findings using hippocampal data from rats performing a spatial memory task in the open field.

Modeled grid cells aligned by a flexible attractor

Entorhinal grid cells implement a spatial code with hexagonal periodicity, signaling the position of the animal within an environment. Grid maps of cells belonging to the same module share spacing and orientation, only differing in relative two-dimensional spatial phase, which could result from being part of a two-dimensional attractor guided by path integration. However, this architecture has the drawbacks of being complex to construct and rigid, path integration allowing for no deviations from the hexagonal pattern such as the ones observed under a variety of experimental manipulations. Here, we show that a simpler one-dimensional attractor is enough to align grid cells equally well. Using topological data analysis, we show that the resulting population activity is a sample of a torus, while the ensemble of maps preserves features of the network architecture. The flexibility of this low dimensional attractor allows it to negotiate the geometry of the representation manifold with the feedforward inputs, rather than imposing it. More generally, our results represent a proof of principle against the intuition that the architecture and the representation manifold of an attractor are topological objects of the same dimensionality, with implications to the study of attractor networks across the brain.

Synaptic interactions between stellate cells and parvalbumin interneurons in layer 2 of the medial entorhinal cortex are organized at the scale of grid cell clusters

Interactions between excitatory and inhibitory neurons are critical to computations in cortical circuits but their organization is difficult to assess with standard electrophysiological approaches. Within the medial entorhinal cortex, representation of location by grid and other spatial cells involves circuits in layer 2 in which excitatory stellate cells interact with each other via inhibitory parvalbumin expressing interneurons. Whether this connectivity is structured to support local circuit computations is unclear. Here, we introduce strategies to address the functional organization of excitatory-inhibitory interactions using crossed Cre- and Flp-driver mouse lines to direct targeted presynaptic optogenetic activation and postsynaptic cell identification. We then use simultaneous patch-clamp recordings from postsynaptic neurons to assess their shared input from optically activated presynaptic populations. We find that extensive axonal projections support spatially organized connectivity between stellate cells and parvalbumin interneurons, such that direct connections are often, but not always, shared by nearby neurons, whereas multisynaptic interactions coordinate inputs to neurons with greater spatial separation. We suggest that direct excitatory-inhibitory synaptic interactions may operate at the scale of grid cell clusters, with local modules defined by excitatory-inhibitory connectivity, while indirect interactions may coordinate activity at the scale of grid cell modules.

Learning to use landmarks for navigation amplifies their representation in retrosplenial cortex

Visual landmarks provide powerful reference signals for efficient navigation by altering the activity of spatially tuned neurons, such as place cells, head direction cells, and grid cells. To understand the neural mechanism by which landmarks exert such strong influence, it is necessary to identify how these visual features gain spatial meaning. In this study, we characterized visual landmark representations in mouse retrosplenial cortex (RSC) using chronic two-photon imaging of the same neuronal ensembles over the course of spatial learning. We found a pronounced increase in landmark-referenced activity in RSC neurons that, once established, remained stable across days. Changing behavioral context by uncoupling treadmill motion from visual feedback systematically altered neuronal responses associated with the coherence between visual scene flow speed and self-motion. To explore potential underlying mechanisms, we modeled how burst firing, mediated by supralinear somatodendritic interactions, could efficiently mediate context- and coherence-dependent integration of landmark information. Our results show that visual encoding shifts to landmark-referenced and context-dependent codes as these cues take on spatial meaning during 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.

Induced neural phase precession through exogenous electric fields

The gradual shifting of preferred neural spiking relative to local field potentials (LFPs), known as phase precession, plays a prominent role in neural coding. Correlations between the phase precession and behavior have been observed throughout various brain regions. As such, phase precession is suggested to be a global neural mechanism that promotes local neuroplasticity. However, causal evidence and neuroplastic mechanisms of phase precession are lacking so far. Here we show a causal link between LFP dynamics and phase precession. In three experiments, we modulated LFPs in humans, a non-human primate, and computational models using alternating current stimulation. We show that continuous stimulation of motor cortex oscillations in humans lead to a gradual phase shift of maximal corticospinal excitability by ~90°. Further, exogenous alternating current stimulation induced phase precession in a subset of entrained neurons (~30%) in the non-human primate. Multiscale modeling of realistic neural circuits suggests that alternating current stimulation-induced phase precession is driven by NMDA-mediated synaptic plasticity. Altogether, the three experiments provide mechanistic and causal evidence for phase precession as a global neocortical process. Alternating current-induced phase precession and consequently synaptic plasticity is crucial for the development of novel therapeutic neuromodulation methods.

Induced neural phase precession through exogeneous electric fields

The gradual shifting of preferred neural spiking relative to local field potentials (LFPs), known as phase precession, plays a prominent role in neural coding. Correlations between the phase precession and behavior have been observed throughout various brain regions. As such, phase precession is suggested to be a global neural mechanism that promotes local neuroplasticity. However, causal evidence and neuroplastic mechanisms of phase precession are lacking so far. Here we show a causal link between LFP dynamics and phase precession. In three experiments, we modulated LFPs in humans, a non-human primate, and computational models using alternating current stimulation. We show that continuous stimulation of motor cortex oscillations in humans lead to a gradual phase shift of maximal corticospinal excitability by ~90°. Further, exogenous alternating current stimulation induced phase precession in a subset of entrained neurons (~30%) in the non-human primate. Multiscale modeling of realistic neural circuits suggests that alternating current stimulation-induced phase precession is driven by NMDA-mediated synaptic plasticity. Altogether, the three experiments provide mechanistic and causal evidence for phase precession as a global neocortical process. Alternating current-induced phase precession and consequently synaptic plasticity is crucial for the development of novel therapeutic neuromodulation methods.

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.

The grid code for ordered experience

Entorhinal cortical grid cells fire in a periodic pattern that tiles space, which is suggestive of a spatial coordinate system. However, irregularities in the grid pattern as well as responses of grid cells in contexts other than spatial navigation have presented a challenge to existing models of entorhinal function. In this Perspective, we propose that hippocampal input provides a key informative drive to the grid network in both spatial and non-spatial circumstances, particularly around salient events. We build on previous models in which neural activity propagates through the entorhinal-hippocampal network in time. This temporal contiguity in network activity points to temporal order as a necessary characteristic of representations generated by the hippocampal formation. We advocate that interactions in the entorhinal-hippocampal loop build a topological representation that is rooted in the temporal order of experience. In this way, the structure of grid cell firing supports a learned topology rather than a rigid coordinate frame that is bound to measurements of the physical world.

Biomimetic FPGA-based spatial navigation model with grid cells and place cells

The mammalian spatial navigation system is characterized by an initial divergence of internal representations, with disparate classes of neurons responding to distinct features including location, speed, borders and head direction; an ensuing convergence finally enables navigation and path integration. Here, we report the algorithmic and hardware implementation of biomimetic neural structures encompassing a feed-forward trimodular, multi-layer architecture representing grid-cell, place-cell and decoding modules for navigation. The grid-cell module comprised of neurons that fired in a grid-like pattern, and was built of distinct layers that constituted the dorsoventral span of the medial entorhinal cortex. Each layer was built as an independent continuous attractor network with distinct grid-field spatial scales. The place-cell module comprised of neurons that fired at one or few spatial locations, organized into different clusters based on convergent modular inputs from different grid-cell layers, replicating the gradient in place-field size along the hippocampal dorso-ventral axis. The decoding module, a two-layer neural network that constitutes the convergence of the divergent representations in preceding modules, received inputs from the place-cell module and provided specific coordinates of the navigating object. After vital design optimizations involving all modules, we implemented the tri-modular structure on Zynq Ultrascale+ field-programmable gate array silicon chip, and demonstrated its capacity in precisely estimating the navigational trajectory with minimal overall resource consumption involving a mere 2.92% Look Up Table utilization. Our implementation of a biomimetic, digital spatial navigation system is stable, reliable, reconfigurable, real-time with execution time of about 32 s for 100k input samples (in contrast to 40 minutes on Intel Core i7-7700 CPU with 8 cores clocking at 3.60 GHz) and thus can be deployed for autonomous-robotic navigation without requiring additional sensors.

Reinforcement learning approaches to hippocampus-dependent flexible spatial navigation

Humans and non-human animals show great flexibility in spatial navigation, including the ability to return to specific locations based on as few as one single experience. To study spatial navigation in the laboratory, watermaze tasks, in which rats have to find a hidden platform in a pool of cloudy water surrounded by spatial cues, have long been used. Analogous tasks have been developed for human participants using virtual environments. Spatial learning in the watermaze is facilitated by the hippocampus. In particular, rapid, one-trial, allocentric place learning, as measured in the delayed-matching-to-place variant of the watermaze task, which requires rodents to learn repeatedly new locations in a familiar environment, is hippocampal dependent. In this article, we review some computational principles, embedded within a reinforcement learning framework, that utilise hippocampal spatial representations for navigation in watermaze tasks. We consider which key elements underlie their efficacy, and discuss their limitations in accounting for hippocampus-dependent navigation, both in terms of behavioural performance (i.e. how well do they reproduce behavioural measures of rapid place learning) and neurobiological realism (i.e. how well do they map to neurobiological substrates involved in rapid place learning). We discuss how an actor-critic architecture, enabling simultaneous assessment of the value of the current location and of the optimal direction to follow, can reproduce one-trial place learning performance as shown on watermaze and virtual delayed-matching-to-place tasks by rats and humans, respectively, if complemented with map-like place representations. The contribution of actor-critic mechanisms to delayed-matching-to-place performance is consistent with neurobiological findings implicating the striatum and hippocampo-striatal interaction in delayed-matching-to-place performance, given that the striatum has been associated with actor-critic mechanisms. Moreover, we illustrate that hierarchical computations embedded within an actor-critic architecture may help to account for aspects of flexible spatial navigation. The hierarchical reinforcement learning approach separates trajectory control via a temporal-difference error from goal selection via a goal prediction error and may account for flexible, trial-specific, navigation to familiar goal locations, as required in some arm-maze place memory tasks, although it does not capture one-trial learning of new goal locations, as observed in open field, including watermaze and virtual, delayed-matching-to-place tasks. Future models of one-shot learning of new goal locations, as observed on delayed-matching-to-place tasks, should incorporate hippocampal plasticity mechanisms that integrate new goal information with allocentric place representation, as such mechanisms are supported by substantial empirical evidence.

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.

Inhibitory Plasticity: From Molecules to Computation and Beyond

Synaptic plasticity is the cellular and molecular counterpart of learning and memory and, since its first discovery, the analysis of the mechanisms underlying long-term changes of synaptic strength has been almost exclusively focused on excitatory connections. Conversely, inhibition was considered as a fixed controller of circuit excitability. Only recently, inhibitory networks were shown to be finely regulated by a wide number of mechanisms residing in their synaptic connections. Here, we review recent findings on the forms of inhibitory plasticity (IP) that have been discovered and characterized in different brain areas. In particular, we focus our attention on the molecular pathways involved in the induction and expression mechanisms leading to changes in synaptic efficacy, and we discuss, from the computational perspective, how IP can contribute to the emergence of functional properties of brain circuits.

Inter- and intra-animal variation in the integrative properties of stellate cells in the medial entorhinal cortex

Distinctions between cell types underpin organizational principles for nervous system function. Functional variation also exists between neurons of the same type. This is exemplified by correspondence between grid cell spatial scales and the synaptic integrative properties of stellate cells (SCs) in the medial entorhinal cortex. However, we know little about how functional variability is structured either within or between individuals. Using ex-vivo patch-clamp recordings from up to 55 SCs per mouse, we found that integrative properties vary between mice and, in contrast to the modularity of grid cell spatial scales, have a continuous dorsoventral organization. Our results constrain mechanisms for modular grid firing and provide evidence for inter-animal phenotypic variability among neurons of the same type. We suggest that neuron type properties are tuned to circuit-level set points that vary within and between animals.

The effects of developmental alcohol exposure on the neurobiology of spatial processing

The consumption of alcohol during gestation is detrimental to the developing central nervous system. One functional outcome of this exposure is impaired spatial processing, defined as sensing and integrating information pertaining to spatial navigation and spatial memory. The hippocampus, entorhinal cortex, and anterior thalamus are brain regions implicated in spatial processing and are highly susceptible to the effects of developmental alcohol exposure. Some of the observed effects of alcohol on spatial processing may be attributed to changes at the synaptic to circuit level. In this review, we first describe the impact of developmental alcohol exposure on spatial behavior followed by a summary of the development of brain areas involved in spatial processing. We then provide an examination of the consequences of prenatal and early postnatal alcohol exposure in rodents on hippocampal, anterior thalamus, and entorhinal cortex-dependent spatial processing from the cellular to behavioral level. We conclude by highlighting several unanswered questions which may provide a framework for future investigation.

Replay as wavefronts and theta sequences as bump oscillations in a grid cell attractor network

Grid cells fire in sequences that represent rapid trajectories in space. During locomotion, theta sequences encode sweeps in position starting slightly behind the animal and ending ahead of it. During quiescence and slow wave sleep, bouts of synchronized activity represent long trajectories called replays, which are well-established in place cells and have been recently reported in grid cells. Theta sequences and replay are hypothesized to facilitate many cognitive functions, but their underlying mechanisms are unknown. One mechanism proposed for grid cell formation is the continuous attractor network. We demonstrate that this established architecture naturally produces theta sequences and replay as distinct consequences of modulating external input. Driving inhibitory interneurons at the theta frequency causes attractor bumps to oscillate in speed and size, which gives rise to theta sequences and phase precession, respectively. Decreasing input drive to all neurons produces traveling wavefronts of activity that are decoded as replays.

The Neurobiology of Mammalian Navigation

Mammals have evolved specialized brain systems to support efficient navigation within diverse habitats and over varied distances, but while navigational strategies and sensory mechanisms vary across species, core spatial components appear to be widely shared. This review presents common elements found in mammalian spatial mapping systems, focusing on the cells in the hippocampal formation representing orientational and locational spatial information, and 'core' mammalian hippocampal circuitry. Mammalian spatial mapping systems make use of both allothetic cues (space-defining cues in the external environment) and idiothetic cues (cues derived from self-motion). As examples of each cue type, we discuss: environmental boundaries, which control both orientational and locational neuronal activity and behaviour; and 'path integration', a process that allows the estimation of linear translation from velocity signals, thought to depend upon grid cells in the entorhinal cortex. Building cognitive maps entails sampling environments: we consider how the mapping system controls exploration to acquire spatial information, and how exploratory strategies may integrate idiothetic with allothetic information. We discuss how 'replay' may act to consolidate spatial maps, and simulate trajectories to aid navigational planning. Finally, we discuss grid cell models of vector navigation.

A geometric attractor mechanism for self-organization of entorhinal grid modules

Grid cells in the medial entorhinal cortex (MEC) respond when an animal occupies a periodic lattice of 'grid fields' in the environment. The grids are organized in modules with spatial periods, or scales, clustered around discrete values separated on average by ratios in the range 1.4-1.7. We propose a mechanism that produces this modular structure through dynamical self-organization in the MEC. In attractor network models of grid formation, the grid scale of a single module is set by the distance of recurrent inhibition between neurons. We show that the MEC forms a hierarchy of discrete modules if a smooth increase in inhibition distance along its dorso-ventral axis is accompanied by excitatory interactions along this axis. Moreover, constant scale ratios between successive modules arise through geometric relationships between triangular grids and have values that fall within the observed range. We discuss how interactions required by our model might be tested experimentally.

Modeling grid fields instead of modeling grid cells : An effective model at the macroscopic level and its relationship with the underlying microscopic neural system

A neuron's firing correlates are defined as the features of the external world to which its activity is correlated. In many parts of the brain, neurons have quite simple such firing correlates. A striking example are grid cells in the rodent medial entorhinal cortex: their activity correlates with the animal's position in space, defining 'grid fields' arranged with a remarkable periodicity. Here, we show that the organization and evolution of grid fields relate very simply to physical space. To do so, we use an effective model and consider grid fields as point objects (particles) moving around in space under the influence of forces. We reproduce several observations on the geometry of grid patterns. This particle-like behavior is particularly salient in a recent experiment in which two separate grid patterns merge. We discuss pattern formation in the light of known results from physics of two-dimensional colloidal systems. Notably, we study the limitations of the widely used 'gridness score' and show how physics of 2d systems could be a source of inspiration, both for data analysis and computational modeling. Finally, we draw the relationship between our 'macroscopic' model for grid fields and existing 'microscopic' models of grid cell activity and discuss how a description at the level of grid fields allows to put constraints on the underlying grid cell network.

The Role of Hierarchical Dynamical Functions in Coding for Episodic Memory and Cognition

Behavioral research in human verbal memory function led to the initial definition of episodic memory and semantic memory. A complete model of the neural mechanisms of episodic memory must include the capacity to encode and mentally reconstruct everything that humans can recall from their experience. This article proposes new model features necessary to address the complexity of episodic memory encoding and recall in the context of broader cognition and the functional properties of neurons that could contribute to this broader scope of memory. Many episodic memory models represent individual snapshots of the world with a sequence of vectors, but a full model must represent complex functions encoding and retrieving the relations between multiple stimulus features across space and time on multiple hierarchical scales. Episodic memory involves not only the space and time of an agent experiencing events within an episode but also features shown in neurophysiological data such as coding of speed, direction, boundaries, and objects. Episodic memory includes not only a spatio-temporal trajectory of a single agent but also segments of spatio-temporal trajectories for other agents and objects encountered in the environment consistent with data on encoding the position and angle of sensory features of objects and boundaries. We will discuss potential interactions of episodic memory circuits in the hippocampus and entorhinal cortex with distributed neocortical circuits that must represent all features of human cognition.

The Firing Rate Speed Code of Entorhinal Speed Cells Differs across Behaviorally Relevant Time Scales and Does Not Depend on Medial Septum Inputs

The firing rate of speed cells, a dedicated subpopulation of neurons in the medial entorhinal cortex (MEC), is correlated with running speed. This correlation has been interpreted as a speed code used in various computational models for path integration. These models consider firing rate to be linearly tuned by running speed in real-time. However, estimation of firing rates requires integration of spiking events over time, setting constraints on the temporal accuracy of the proposed speed code. We therefore tested whether the proposed speed code by firing rate is accurate at short time scales using data obtained from open-field recordings in male rats and mice. We applied a novel filtering approach differentiating between speed codes at multiple time scales ranging from deciseconds to minutes. In addition, we determined the optimal integration time window for firing-rate estimation using a general likelihood framework and calculated the integration time window that maximizes the correlation between firing rate and running speed. Data show that these time windows are on the order of seconds, setting constraints on real-time speed coding by firing rate. We further show that optogenetic inhibition of either cholinergic, GABAergic, or glutamatergic neurons in the medial septum/diagonal band of Broca does not affect modulation of firing rates by running speed at each time scale tested. These results are relevant for models of path integration and for our understanding of how behavioral activity states may modulate firing rates and likely information processing in the MEC.SIGNIFICANCE STATEMENT Path integration is the most basic form of navigation relying on self-motion cues. Models of path integration use medial septum/diagonal band of Broca (MSDB)-dependent MEC grid-cell firing patterns as the neurophysiological substrate of path integration. These models use a linear speed code by firing rate, but do not consider temporal constraints of integration over time for firing-rate estimation. We show that firing-rate estimation for speed cells requires integration over seconds. Using optogenetics, we show that modulation of firing rates by running speed is independent of MSDB inputs. These results enhance our understanding of path integration mechanisms and the role of the MSDB for information processing in the MEC.

A Map-like Micro-Organization of Grid Cells in the Medial Entorhinal Cortex

How the topography of neural circuits relates to their function remains unclear. Although topographic maps exist for sensory and motor variables, they are rarely observed for cognitive variables. Using calcium imaging during virtual navigation, we investigated the relationship between the anatomical organization and functional properties of grid cells, which represent a cognitive code for location during navigation. We found a substantial degree of grid cell micro-organization in mouse medial entorhinal cortex: grid cells and modules all clustered anatomically. Within a module, the layout of grid cells was a noisy two-dimensional lattice in which the anatomical distribution of grid cells largely matched their spatial tuning phases. This micro-arrangement of phases demonstrates the existence of a topographical map encoding a cognitive variable in rodents. It contributes to a foundation for evaluating circuit models of the grid cell network and is consistent with continuous attractor models as the mechanism of grid formation.

A hierarchical anti-Hebbian network model for the formation of spatial cells in three-dimensional space

Three-dimensional (3D) spatial cells in the mammalian hippocampal formation are believed to support the existence of 3D cognitive maps. Modeling studies are crucial to comprehend the neural principles governing the formation of these maps, yet to date very few have addressed this topic in 3D space. Here we present a hierarchical network model for the formation of 3D spatial cells using anti-Hebbian network. Built on empirical data, the model accounts for the natural emergence of 3D place, border, and grid cells, as well as a new type of previously undescribed spatial cell type which we call plane cells. It further explains the plausible reason behind the place and grid-cell anisotropic coding that has been observed in rodents and the potential discrepancy with the predicted periodic coding during 3D volumetric navigation. Lastly, it provides evidence for the importance of unsupervised learning rules in guiding the formation of higher-dimensional cognitive maps.

Neurons in the hippocampal formation encode diverse spatial properties. Here, the authors present a hierarchical network model for 3D spatial navigation that accounts for the observed neuronal representations and predict as yet unreported cell types with planar selectivity.

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.

Analysis of an Attractor Neural Network's Response to Conflicting External Inputs

The theory of attractor neural networks has been influential in our understanding of the neural processes underlying spatial, declarative, and episodic memory. Many theoretical studies focus on the inherent properties of an attractor, such as its structure and capacity. Relatively little is known about how an attractor neural network responds to external inputs, which often carry conflicting information about a stimulus. In this paper we analyze the behavior of an attractor neural network driven by two conflicting external inputs. Our focus is on analyzing the emergent properties of the megamap model, a quasi-continuous attractor network in which place cells are flexibly recombined to represent a large spatial environment. In this model, the system shows a sharp transition from the winner-take-all mode, which is characteristic of standard continuous attractor neural networks, to a combinatorial mode in which the equilibrium activity pattern combines embedded attractor states in response to conflicting external inputs. We derive a numerical test for determining the operational mode of the system a priori. We then derive a linear transformation from the full megamap model with thousands of neurons to a reduced 2-unit model that has similar qualitative behavior. Our analysis of the reduced model and explicit expressions relating the parameters of the reduced model to the megamap elucidate the conditions under which the combinatorial mode emerges and the dynamics in each mode given the relative strength of the attractor network and the relative strength of the two conflicting inputs. Although we focus on a particular attractor network model, we describe a set of conditions under which our analysis can be applied to more general attractor neural networks.

A unified hierarchical oscillatory network model of head direction cells, spatially periodic cells, and place cells

Spatial cells in the hippocampal complex play a pivotal role in the navigation of an animal. Exact neural principles behind these spatial cell responses have not been completely unraveled yet. Here we present two models for spatial cells, namely the Velocity Driven Oscillatory Network (VDON) and Locomotor Driven Oscillatory Network. Both models have basically three stages in common such as direction encoding stage, path integration (PI) stage, and a stage of unsupervised learning of PI values. In the first model, the following three stages are implemented: head direction layer, frequency modulation by a layer of oscillatory neurons, and an unsupervised stage that extracts the principal components from the oscillator outputs. In the second model, a refined version of the first model, the stages are extraction of velocity representation from the locomotor input, frequency modulation by a layer of oscillators, and two cascaded unsupervised stages consisting of the lateral anti-hebbian network. The principal component stage of VDON exhibits grid cell-like spatially periodic responses including hexagonal firing fields. Locomotor Driven Oscillatory Network shows the emergence of spatially periodic grid cells and periodically active border-like cells in its lower layer; place cell responses are found in its higher layer. This model shows the inheritance of phase precession from grid cell to place cell in both one- and two-dimensional spaces. It also shows a novel result on the influence of locomotion rhythms on the grid cell activity. The study thus presents a comprehensive, unifying hierarchical model for hippocampal spatial cells.

Neural mechanisms of navigation involving interactions of cortical and subcortical structures

Animals must perform spatial navigation for a range of different behaviors, including selection of trajectories toward goal locations and foraging for food sources. To serve this function, a number of different brain regions play a role in coding different dimensions of sensory input important for spatial behavior, including the entorhinal cortex, the retrosplenial cortex, the hippocampus, and the medial septum. This article will review data concerning the coding of the spatial aspects of animal behavior, including location of the animal within an environment, the speed of movement, the trajectory of movement, the direction of the head in the environment, and the position of barriers and objects both relative to the animal's head direction (egocentric) and relative to the layout of the environment (allocentric). The mechanisms for coding these important spatial representations are not yet fully understood but could involve mechanisms including integration of self-motion information or coding of location based on the angle of sensory features in the environment. We will review available data and theories about the mechanisms for coding of spatial representations. The computation of different aspects of spatial representation from available sensory input requires complex cortical processing mechanisms for transformation from egocentric to allocentric coordinates that will only be understood through a combination of neurophysiological studies and computational modeling.

Self-Organized Attractor Dynamics in the Developing Head Direction Circuit

Head direction (HD) cells are neurons found in an extended cortical and subcortical network that signal the orientation of an animal’s head relative to its environment [1, 2, 3]. They are a fundamental component of the wider circuit of spatially responsive hippocampal formation neurons that make up the neural cognitive map of space [4]. During post-natal development, HD cells are the first among spatially modulated neurons in the hippocampal circuit to exhibit mature firing properties [5, 6], but before eye opening, HD cell responses in rat pups have low directional information and are directionally unstable [7, 8]. Using Bayesian decoding of HD cell ensemble activity recorded in the anterodorsal thalamic nucleus (ADN), we characterize this instability and identify its source: under-signaling of angular head velocity, which incompletely shifts the directional signal in proportion to head turns. We find evidence that geometric cues (the corners of a square environment) can be used to mitigate this under-signaling and, thereby, stabilize the directional signal even before eye opening. Crucially, even when directional firing cannot be stabilized, ensembles of unstable HD cells show short-timescale (1–10 s) temporal and spatial couplings consistent with an adult-like HD network. The HD network is widely modeled as a continuous attractor whose output is one coherent activity peak, updated during movement by angular head velocity signals and anchored by landmark cues [9, 10, 11]. Our findings present strong evidence for this model, and they demonstrate that the required network circuitry is in place and functional early during development, independent of reference to landmark information.

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Non-visual cues can anchor head direction (HD) cells in pre-eye-opening rat pups

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Internal network dynamics are preserved even when the HD representation is unstable

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Angular velocity under-signaling drives instability, which is mitigated by corners

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Circuit architecture develops even before any landmarks can stabilize HD responses

Learning place cells, grid cells and invariances with excitatory and inhibitory plasticity

Neurons in the hippocampus and adjacent brain areas show a large diversity in their tuning to location and head direction, and the underlying circuit mechanisms are not yet resolved. In particular, it is unclear why certain cell types are selective to one spatial variable, but invariant to another. For example, place cells are typically invariant to head direction. We propose that all observed spatial tuning patterns - in both their selectivity and their invariance - arise from the same mechanism: Excitatory and inhibitory synaptic plasticity driven by the spatial tuning statistics of synaptic inputs. Using simulations and a mathematical analysis, we show that combined excitatory and inhibitory plasticity can lead to localized, grid-like or invariant activity. Combinations of different input statistics along different spatial dimensions reproduce all major spatial tuning patterns observed in rodents. Our proposed model is robust to changes in parameters, develops patterns on behavioral timescales and makes distinctive experimental predictions.

A single-cell spiking model for the origin of grid-cell patterns

Spatial cognition in mammals is thought to rely on the activity of grid cells in the entorhinal cortex, yet the fundamental principles underlying the origin of grid-cell firing are still debated. Grid-like patterns could emerge via Hebbian learning and neuronal adaptation, but current computational models remained too abstract to allow direct confrontation with experimental data. Here, we propose a single-cell spiking model that generates grid firing fields via spike-rate adaptation and spike-timing dependent plasticity. Through rigorous mathematical analysis applicable in the linear limit, we quantitatively predict the requirements for grid-pattern formation, and we establish a direct link to classical pattern-forming systems of the Turing type. Our study lays the groundwork for biophysically-realistic models of grid-cell activity.

Models of spatial and temporal dimensions of memory

Episodic memory involves coding of the spatial location and time of individual events. Coding of space and time is also relevant to working memory, spatial navigation, and the disambiguation of overlapping memory representations. Neurophysiological data demonstrate that neuronal activity codes the current, past and future location of an animal as well as temporal intervals within a task. Models have addressed how neural coding of space and time for memory function could arise, with both dimensions coded by the same neurons. Neural coding could depend upon network oscillatory and attractor dynamics as well as modulation of neuronal intrinsic properties. These models are relevant to the coding of space and time involving structures including the hippocampus, entorhinal cortex, retrosplenial cortex, striatum and parahippocampal gyrus, which have been implicated in both animal and human studies.

A Multiplexed, Heterogeneous, and Adaptive Code for Navigation in Medial Entorhinal Cortex

Medial entorhinal grid cells display strikingly symmetric spatial firing patterns. The clarity of these patterns motivated the use of specific activity pattern shapes to classify entorhinal cell types. While this approach successfully revealed cells that encode boundaries, head direction, and running speed, it left a majority of cells unclassified, and its pre-defined nature may have missed unconventional, yet important coding properties. Here, we apply an unbiased statistical approach to search for cells that encode navigationally relevant variables. This approach successfully classifies the majority of entorhinal cells and reveals unsuspected entorhinal coding principles. First, we find a high degree of mixed selectivity and heterogeneity in superficial entorhinal neurons. Second, we discover a dynamic and remarkably adaptive code for space that enables entorhinal cells to rapidly encode navigational information accurately at high running speeds. Combined, these observations advance our current understanding of the mechanistic origins and functional implications of the entorhinal code for navigation. VIDEO ABSTRACT.

Layer-Specific Organization of Local Excitatory and Inhibitory Synaptic Connectivity in the Rat Presubiculum

The presubiculum is part of the parahippocampal spatial navigation system and contains head direction and grid cells upstream of the medial entorhinal cortex. This position within the parahippocampal cortex renders the presubiculum uniquely suited for analyzing the circuit requirements underlying the emergence of spatially tuned neuronal activity. To identify the local circuit properties, we analyzed the topology of synaptic connections between pyramidal cells and interneurons in all layers of the presubiculum by testing 4250 potential synaptic connections using multiple whole-cell recordings of up to 8 cells simultaneously. Network topology showed layer-specific organization of microcircuits consistent with the prevailing distinction of superficial and deep layers. While connections among pyramidal cells were almost absent in superficial layers, deep layers exhibited an excitatory connectivity of 3.9%. In contrast, synaptic connectivity for inhibition was higher in superficial layers though markedly lower than in other cortical areas. Finally, synaptic amplitudes of both excitatory and inhibitory connections showed log-normal distributions suggesting a nonrandom functional connectivity. In summary, our study provides new insights into the microcircuit organization of the presubiculum by revealing area- and layer-specific connectivity rules and sets new constraints for future models of the parahippocampal navigation system.

Inhibitory suppression of heterogeneously tuned excitation enhances spatial coding in CA1 place cells

Place cells in the CA1 region of the hippocampus express location-specific firing despite receiving a steady barrage of heterogeneously tuned excitatory inputs that should compromise output dynamic range and timing. We examined the role of synaptic inhibition in countering the deleterious effects of off-target excitation. Intracellular recordings in behaving mice demonstrate that bimodal excitation drives place cells, while unimodal excitation drives weaker or no spatial tuning in interneurons. Optogenetic hyperpolarization of interneurons had spatially uniform effects on place cell membrane potential dynamics, substantially reducing spatial selectivity. These data and a computational model suggest that spatially uniform inhibitory conductance enhances rate coding in place cells by suppressing out-of-field excitation and by limiting dendritic amplification. Similarly, we observed that inhibitory suppression of phasic noise generated by out-of-field excitation enhances temporal coding by expanding the range of theta phase precession. Thus, spatially uniform inhibition allows proficient and flexible coding in hippocampal CA1 by suppressing heterogeneously tuned excitation.

Computational principles of memory

The ability to store and later use information is essential for a variety of adaptive behaviors, including integration, learning, generalization, prediction and inference. In this Review, we survey theoretical principles that can allow the brain to construct persistent states for memory. We identify requirements that a memory system must satisfy and analyze existing models and hypothesized biological substrates in light of these requirements. We also highlight open questions, theoretical puzzles and problems shared with computer science and information theory.

The Preferred Directions of Conjunctive Grid X Head Direction Cells in the Medial Entorhinal Cortex Are Periodically Organized

The discovery of speed-modulated grid, head direction, and conjunctive grid x head direction cells in the medial entorhinal cortex has led to the hypothesis that path integration, the updating of one’s spatial representation based on movement, may be carried out within this region. This hypothesis has been formalized by many computational models, including a class known as attractor network models. While many of these models propose specific mechanisms by which path integration might occur, predictions of these specific mechanisms have not been tested. Here I derive and test a key prediction of one attractor network path integration mechanism. Specifically, I first demonstrate that this mechanism predicts a periodic distribution of conjunctive cell preferred directions in order to minimize drift. Next, I test whether conjunctive cell preferred directions are in fact periodically organized. Results indicate that conjunctive cells are preferentially tuned to increments of 36°, consistent with drift minimization in this path integration mechanism. By contrast, no periodicity was observed in the preferred directions of either pure grid or pure head direction cells. These results provide the first neural evidence of a nonuniform structure in the directional preferences of any head direction representation found in the brain.

Continuous attractor network models of grid cell firing based on excitatory-inhibitory interactions

Neurons in the medial entorhinal cortex encode location through spatial firing fields that have a grid‐like organisation. The challenge of identifying mechanisms for grid firing has been addressed through experimental and theoretical investigations of medial entorhinal circuits. Here, we discuss evidence for continuous attractor network models that account for grid firing by synaptic interactions between excitatory and inhibitory cells. These models assume that grid‐like firing patterns are the result of computation of location from velocity inputs, with additional spatial input required to oppose drift in the attractor state. We focus on properties of continuous attractor networks that are revealed by explicitly considering excitatory and inhibitory neurons, their connectivity and their membrane potential dynamics. Models at this level of detail can account for theta‐nested gamma oscillations as well as grid firing, predict spatial firing of interneurons as well as excitatory cells, show how gamma oscillations can be modulated independently from spatial computations, reveal critical roles for neuronal noise, and demonstrate that only a subset of excitatory cells in a network need have grid‐like firing fields. Evaluating experimental data against predictions from detailed network models will be important for establishing the mechanisms mediating grid firing.

Spike Pattern Structure Influences Synaptic Efficacy Variability under STDP and Synaptic Homeostasis. I: Spike Generating Models on Converging Motifs

In neural systems, synaptic plasticity is usually driven by spike trains. Due to the inherent noises of neurons and synapses as well as the randomness of connection details, spike trains typically exhibit variability such as spatial randomness and temporal stochasticity, resulting in variability of synaptic changes under plasticity, which we call efficacy variability. How the variability of spike trains influences the efficacy variability of synapses remains unclear. In this paper, we try to understand this influence under pair-wise additive spike-timing dependent plasticity (STDP) when the mean strength of plastic synapses into a neuron is bounded (synaptic homeostasis). Specifically, we systematically study, analytically and numerically, how four aspects of statistical features, i.e., synchronous firing, burstiness/regularity, heterogeneity of rates and heterogeneity of cross-correlations, as well as their interactions influence the efficacy variability in converging motifs (simple networks in which one neuron receives from many other neurons). Neurons (including the post-synaptic neuron) in a converging motif generate spikes according to statistical models with tunable parameters. In this way, we can explicitly control the statistics of the spike patterns, and investigate their influence onto the efficacy variability, without worrying about the feedback from synaptic changes onto the dynamics of the post-synaptic neuron. We separate efficacy variability into two parts: the drift part (DriftV) induced by the heterogeneity of change rates of different synapses, and the diffusion part (DiffV) induced by weight diffusion caused by stochasticity of spike trains. Our main findings are: (1) synchronous firing and burstiness tend to increase DiffV, (2) heterogeneity of rates induces DriftV when potentiation and depression in STDP are not balanced, and (3) heterogeneity of cross-correlations induces DriftV together with heterogeneity of rates. We anticipate our work important for understanding functional processes of neuronal networks (such as memory) and neural development.

Environmental boundaries as a mechanism for correcting and anchoring spatial maps

Ubiquitous throughout the animal kingdom, path integration-based navigation allows an animal to take a circuitous route out from a home base and using only self-motion cues, calculate a direct vector back. Despite variation in an animal's running speed and direction, medial entorhinal grid cells fire in repeating place-specific locations, pointing to the medial entorhinal circuit as a potential neural substrate for path integration-based spatial navigation. Supporting this idea, grid cells appear to provide an environment-independent metric representation of the animal's location in space and preserve their periodic firing structure even in complete darkness. However, a series of recent experiments indicate that spatially responsive medial entorhinal neurons depend on environmental cues in a more complex manner than previously proposed. While multiple types of landmarks may influence entorhinal spatial codes, environmental boundaries have emerged as salient landmarks that both correct error in entorhinal grid cells and bind internal spatial representations to the geometry of the external spatial world. The influence of boundaries on error correction and grid symmetry points to medial entorhinal border cells, which fire at a high rate only near environmental boundaries, as a potential neural substrate for landmark-driven control of spatial codes. The influence of border cells on other entorhinal cell populations, such as grid cells, could depend on plasticity, raising the possibility that experience plays a critical role in determining how external cues influence internal spatial representations.

Noise promotes independent control of gamma oscillations and grid firing within recurrent attractor networks

Neural computations underlying cognitive functions require calibration of the strength of excitatory and inhibitory synaptic connections and are associated with modulation of gamma frequency oscillations in network activity. However, principles relating gamma oscillations, synaptic strength and circuit computations are unclear. We address this in attractor network models that account for grid firing and theta-nested gamma oscillations in the medial entorhinal cortex. We show that moderate intrinsic noise massively increases the range of synaptic strengths supporting gamma oscillations and grid computation. With moderate noise, variation in excitatory or inhibitory synaptic strength tunes the amplitude and frequency of gamma activity without disrupting grid firing. This beneficial role for noise results from disruption of epileptic-like network states. Thus, moderate noise promotes independent control of multiplexed firing rate- and gamma-based computational mechanisms. Our results have implications for tuning of normal circuit function and for disorders associated with changes in gamma oscillations and synaptic strength.

DOI:http://dx.doi.org/10.7554/eLife.06444.001

When electrodes are placed on the scalp, or lowered into the brain itself, rhythmic waves of electrical activity are seen that reflect the coordinated firing of large numbers of neurons. The pattern of the waves varies between different brain regions, and according to what the animal or person is doing. During sleep and quiet wakefulness, slower brain waves predominate, whereas faster waves called gamma oscillations emerge during cognition—the act of processing knowledge.

Gamma waves can be readily detected in a region of the brain called the medial entorhinal cortex (MEC). This brain region is also known for its role in forming the spatial memories that allow an individual to remember how to navigate around an area they have previously visited. Individual MEC cells increase their firing rates whenever an individual is at specific locations. When these locations are plotted in two dimensions, they form a hexagonal grid: this ‘grid cell map’ enables the animal to keep track of its position as it navigates through its environment.

To determine how MEC neurons can simultaneously encode spatial locations and generate the gamma waves implicated in cognition, Solanka et al. have used supercomputing to simulate the activity of more than 1.5 million connections between MEC cells. Changing the strength of these connections had different effects on the ability of the MEC to produce gamma waves or spatial maps. However, adjusting the model to include random fluctuations in neuronal firing, or ‘noise’, was beneficial for both types of output. This is partly because noise prevented neuronal firing from becoming excessively synchronized, which would otherwise have caused seizures.

Although noise is generally regarded as disruptive, the results of Solanka et al. suggest that it helps the MEC to perform its two distinct roles. Specifically, the presence of noise enables relatively small changes in the strength of the connections between neurons to alter gamma waves—and thus affect cognition—without disrupting the neurons' ability to encode spatial locations. Given that noise reduces the likelihood of seizures, the results also raise the possibility that introducing noise into the brain in a controlled way could have therapeutic benefits for individuals with epilepsy.

DOI:http://dx.doi.org/10.7554/eLife.06444.002

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.