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

How does the response of egocentric boundary cells depend upon the coordinate system of environmental features?

Neurons in the retrosplenial (RSC) (Alexander et al., 2020a; LaChance and Hasselmo, 2024) and postrhinal cortex (POR) respond to environmental boundaries and configurations in egocentric coordinates relative to an animals current position. Neurons in these structures and adjacent structures also respond to spatial dimensions of self- motion such as running velocity (Carstensen et al., 2021; Robinson et al., 2024). Data and modeling suggest that these responses could be essential for guiding behaviors such as barrier avoidance and goal finding (Erdem and Hasselmo, 2012; 2014). However, these findings still leave the unanswered question: What are the features and what are the coordinate systems of these features that drive these egocentric neural responses? Here we present models of the potential circuit mechanisms generating egocentric responses in RSC. These can be generated based on coding of internal representations of barriers in head-centered coordinates of distance and angle that are transformed based on current running velocity for trajectory planning and obstacle avoidance. This hypothesis is compared with an alternate potentially complementary hypothesis that neurons in the same regions might respond to retinotopic position of features at the top, bottom or edges of walls as a precursor to head-centered coordinates. Alternate hypotheses include the forward scanning of trajectories (ray tracing) to test for collision with barriers, or the comparison of optic flow on different sides of the animal. These hypotheses generate complementary modeling predictions about how changes in environmental parameters could alter the neural responses of egocentric boundary cells that are presented here.

The role of oscillations in grid cells' toroidal topology

Persistent homology applied to the activity of grid cells in the Medial Entorhinal Cortex suggests that this activity lies on a toroidal manifold. By analyzing real data and a simple model, we show that neural oscillations play a key role in the appearance of this toroidal topology. To quantitatively monitor how changes in spike trains influence the topology of the data, we first define a robust measure for the degree of toroidality of a dataset. Using this measure, we find that small perturbations ( ~ 100 ms) of spike times have little influence on both the toroidality and the hexagonality of the ratemaps. Jittering spikes by  ~ 100-500 ms, however, destroys the toroidal topology, while still having little impact on grid scores. These critical jittering time scales fall in the range of the periods of oscillations between the theta and eta bands. We thus hypothesized that these oscillatory modulations of neuronal spiking play a key role in the appearance and robustness of toroidal topology and the hexagonal spatial selectivity is not sufficient. We confirmed this hypothesis using a simple model for the activity of grid cells, consisting of an ensemble of independent rate-modulated Poisson processes. When these rates were modulated by oscillations, the network behaved similarly to the real data in exhibiting toroidal topology, even when the position of the fields were perturbed. In the absence of oscillations, this similarity was substantially lower. Furthermore, we find that the experimentally recorded spike trains indeed exhibit temporal modulations at the eta and theta bands, and that the ratio of the power in the eta band to that of the theta band, [Formula: see text], correlates with the critical jittering time at which the toroidal topology disappears.

Modeling hippocampal spatial cells in rodents navigating in 3D environments

Studies on the neural correlates of navigation in 3D environments are plagued by several issues that need to be solved. For example, experimental studies show markedly different place cell responses in rats and bats, both navigating in 3D environments. In this study, we focus on modelling the spatial cells in rodents in a 3D environment. We propose a deep autoencoder network to model the place and grid cells in a simulated agent navigating in a 3D environment. The input layer to the autoencoder network model is the HD layer, which encodes the agent's HD in terms of azimuth (θ) and pitch angles (ϕ). The output of this layer is given as input to the Path Integration (PI) layer, which computes displacement in all the preferred directions. The bottleneck layer of the autoencoder model encodes the spatial cell-like responses. Both grid cell and place cell-like responses are observed. The proposed model is verified using two experimental studies with two 3D environments. This model paves the way for a holistic approach using deep neural networks to model spatial cells in 3D navigation.

The grid-cell normative model: Unifying 'principles'

A normative model for the emergence of entorhinal grid cells in the brain's navigational system has been proposed (Sorscher et al., 2023. Neuron 111, 121-137). Using computational modeling of place-to-grid cell interactions, the authors characterized the fundamental nature of grid cells through information processing. However, the normative model does not consider certain discoveries that complement or contradict the conditions for such emergence. By briefly reviewing current evidence, we draw some implications on the interplay between place cell replay sequences and intrinsic grid cell oscillations related to the hippocampal-entorhinal navigation system that can extend the normative model.

Flexible coding of time or distance in hippocampal cells

Analysis of neuronal activity in the hippocampus of behaving animals has revealed cells acting as 'Time Cells', which exhibit selective spiking patterns at specific time intervals since a triggering event, and 'Distance Cells', which encode the traversal of specific distances. Other neurons exhibit a combination of these features, alongside place selectivity. This study aims to investigate how the task performed by animals during recording sessions influences the formation of these representations. We analyzed data from a treadmill running study conducted by Kraus et al., 2013, in which rats were trained to run at different velocities. The rats were recorded in two trial contexts: a 'fixed time' condition, where the animal ran on the treadmill for a predetermined duration before proceeding, and a 'fixed distance' condition, where the animal ran a specific distance on the treadmill. Our findings indicate that the type of experimental condition significantly influenced the encoding of hippocampal cells. Specifically, distance-encoding cells dominated in fixed-distance experiments, whereas time-encoding cells dominated in fixed-time experiments. These results underscore the flexible coding capabilities of the hippocampus, which are shaped by over-representation of salient variables associated with reward conditions.

Simulation of oscillatory dynamics induced by an approximation of grid cell output

Grid cells, in entorhinal cortex (EC) and related structures, signal animal location relative to hexagonal tilings of 2D space. A number of modeling papers have addressed the question of how grid firing behaviors emerge using (for example) ideas borrowed from dynamical systems (attractors) or from coupled oscillator theory. Here we use a different approach: instead of asking how grid behavior emerges, we take as a given the experimentally observed intracellular potentials of superficial medial EC neurons during grid firing. Employing a detailed neural circuit model modified from a lateral EC model, we then ask how the circuit responds when group of medial EC principal neurons exhibit such potentials, simultaneously with a simulated theta frequency input from the septal nuclei. The model predicts the emergence of robust theta-modulated gamma/beta oscillations, suggestive of oscillations observed in an in vitro medial EC experimental model (Cunningham, M.O., Pervouchine, D.D., Racca, C., Kopell, N.J., Davies, C.H., Jones, R.S.G., Traub, R.D., and Whittington, M.A. (2006). Neuronal metabolism governs cortical network response state. Proc. Natl. Acad. Sci. U S A 103: 5597-5601). Such oscillations result because feedback interneurons tightly synchronize with each other - despite the varying phases of the grid cells - and generate a robust inhibition-based rhythm. The lack of spatial specificity of the model interneurons is consistent with the lack of spatial periodicity in parvalbumin interneurons observed by Buetfering, C., Allen, K., and Monyer, H. (2014). Parvalbumin interneurons provide grid cell-driven recurrent inhibition in the medial entorhinal cortex. Nat. Neurosci. 17: 710-718. If in vivo EC gamma rhythms arise during exploration as our model predicts, there could be implications for interpreting disrupted spatial behavior and gamma oscillations in animal models of Alzheimer's disease and schizophrenia. Noting that experimental intracellular grid cell potentials closely resemble cortical Up states and Down states, during which fast oscillations also occur during Up states, we propose that the co-occurrence of slow principal cell depolarizations and fast network oscillations is a general property of the telencephalon, in both waking and sleep states.

Cortical Integration of Vestibular and Visual Cues for Navigation, Visual Processing, and Perception

Despite increasing evidence of its involvement in several key functions of the cerebral cortex, the vestibular sense rarely enters our consciousness. Indeed, the extent to which these internal signals are incorporated within cortical sensory representation and how they might be relied upon for sensory-driven decision-making, during, for example, spatial navigation, is yet to be understood. Recent novel experimental approaches in rodents have probed both the physiological and behavioral significance of vestibular signals and indicate that their widespread integration with vision improves both the cortical representation and perceptual accuracy of self-motion and orientation. Here, we summarize these recent findings with a focus on cortical circuits involved in visual perception and spatial navigation and highlight the major remaining knowledge gaps. We suggest that vestibulo-visual integration reflects a process of constant updating regarding the status of self-motion, and access to such information by the cortex is used for sensory perception and predictions that may be implemented for rapid, navigation-related decision-making.

Bimodular continuous attractor neural networks with static and moving stimuli

We investigated the dynamical behaviors of bimodular continuous attractor neural networks, each processing a modality of sensory input and interacting with each other. We found that when bumps coexist in both modules, the position of each bump is shifted towards the other input when the intermodular couplings are excitatory and is shifted away when inhibitory. When one intermodular coupling is excitatory while another is moderately inhibitory, temporally modulated population spikes can be generated. On further increase of the inhibitory coupling, momentary spikes will emerge. In the regime of bump coexistence, bump heights are primarily strengthened by excitatory intermodular couplings, but there is a lesser weakening effect due to a bump being displaced from the direct input. When bimodular networks serve as decoders of multisensory integration, we extend the Bayesian framework to show that excitatory and inhibitory couplings encode attractive and repulsive priors, respectively. At low disparity, the bump positions decode the posterior means in the Bayesian framework, whereas at high disparity, multiple steady states exist. In the regime of multiple steady states, the less stable state can be accessed if the input causing the more stable state arrives after a sufficiently long delay. When one input is moving, the bump in the corresponding module is pinned when the moving stimulus is weak, unpinned at intermediate stimulus strength, and tracks the input at strong stimulus strength, and the stimulus strengths for these transitions increase with the velocity of the moving stimulus. These results are important to understanding multisensory integration of static and dynamic stimuli.

Grid cell disruption in a mouse model of early Alzheimer's disease reflects reduced integration of self-motion cues

Converging evidence from human and rodent studies suggests that disrupted grid cell coding in the medial entorhinal cortex (MEC) underlies path integration behavioral deficits during early Alzheimer's disease (AD). However, grid cell firing relies on both self-motion cues and environmental features, and it remains unclear whether disrupted grid coding can account for specific path integration deficits reported during early AD. Here, we report in the J20 transgenic amyloid beta (Aβ) mouse model of early AD that grid cells were spatially unstable toward the center of the arena, had qualitatively different spatial components that aligned parallel to the borders of the environment, and exhibited impaired integration of distance traveled via reduced theta phase precession. Our results suggest that disrupted early AD grid coding reflects reduced integration of self-motion cues but not environmental information via geometric boundaries, providing evidence that grid cell impairments underlie path integration deficits during early AD.

Gated transformations from egocentric to allocentric reference frames involving retrosplenial cortex, entorhinal cortex, and hippocampus

This paper reviews the recent experimental finding that neurons in behaving rodents show egocentric coding of the environment in a number of structures associated with the hippocampus. Many animals generating behavior on the basis of sensory input must deal with the transformation of coordinates from the egocentric position of sensory input relative to the animal, into an allocentric framework concerning the position of multiple goals and objects relative to each other in the environment. Neurons in retrosplenial cortex show egocentric coding of the position of boundaries in relation to an animal. These neuronal responses are discussed in relation to existing models of the transformation from egocentric to allocentric coordinates using gain fields and a new model proposing transformations of phase coding that differ from current models. The same type of transformations could allow hierarchical representations of complex scenes. The responses in rodents are also discussed in comparison to work on coordinate transformations in humans and non-human primates.

The chicken and egg problem of grid cells and place cells

Place cells and grid cells are major building blocks of the hippocampal cognitive map. The prominent forward model postulates that grid-cell modules are generated by a continuous attractor network; that a velocity signal evoked during locomotion moves entorhinal activity bumps; and that place-cell activity constitutes summation of entorhinal grid-cell modules. Experimental data support the first postulate, but not the latter two. Several families of solutions that depart from these postulates have been put forward. We suggest a modified model (spatial modulation continuous attractor network; SCAN), whereby place cells are generated from spatially selective nongrid cells. Locomotion causes these cells to move the hippocampal activity bump, leading to movement of the entorhinal manifolds. Such inversion accords with the shift of hippocampal thought from navigation to more abstract functions.

Subthreshold Oscillating Waves in Neural Tissue Propagate by Volume Conduction and Generate Interference

Subthreshold neural oscillations have been observed in several brain regions and can influence the timing of neural spikes. However, the spatial extent and function of these spontaneous oscillations remain unclear. To study the mechanisms underlying these oscillations, we use optogenetic stimulation to generate oscillating waves in the longitudinal hippocampal slice expressing optopatch proteins. We found that optogenetic stimulation can generate two types of neural activity: suprathreshold neural spikes and subthreshold oscillating waves. Both waves could propagate bidirectionally at similar speeds and go through a transection of the tissue. The propagating speed is independent of the oscillating frequency but increases with increasing amplitudes of the waves. The endogenous electric fields generated by oscillating waves are about 0.6 mV/mm along the dendrites and about 0.3 mV/mm along the cell layer. We also observed that these oscillating waves could interfere with each other. Optical stimulation applied simultaneously at each slice end generated a larger wave in the middle of the tissue (constructive interference) or destructive interference with laser signals in opposite phase. However, the suprathreshold neural spikes were annihilated when they collided. Finally, the waves were not affected by the NMDA blocker (APV) and still propagated in the presence of tetrodotoxin (TTX) but at a significantly lower amplitude. The role of these subthreshold waves in neural function is unknown, but the results show that at low amplitude, the subthreshold propagating waves lack a refractory period allowing a novel analog form of preprocessing of neural activity by interference independent of synaptic transmission.

Excitatory-inhibitory recurrent dynamics produce robust visual grids and stable attractors

Spatially modulated grid cells have been recently found in the rat secondary visual cortex (V2) during active navigation. However, the computational mechanism and functional significance of V2 grid cells remain unknown. To address the knowledge gap, we train a biologically inspired excitatory-inhibitory recurrent neural network to perform a two-dimensional spatial navigation task with multisensory input. We find grid-like responses in both excitatory and inhibitory RNN units, which are robust with respect to spatial cues, dimensionality of visual input, and activation function. Population responses reveal a low-dimensional, torus-like manifold and attractor. We find a link between functional grid clusters with similar receptive fields and structured excitatory-to-excitatory connections. Additionally, multistable torus-like attractors emerged with increasing sparsity in inter- and intra-subnetwork connectivity. Finally, irregular grid patterns are found in recurrent neural network (RNN) units during a visual sequence recognition task. Together, our results suggest common computational mechanisms of V2 grid cells for spatial and non-spatial tasks.

Place cells dynamically refine grid cell activities to reduce error accumulation during path integration in a continuous attractor model

Navigation is one of the most fundamental skills of animals. During spatial navigation, grid cells in the medial entorhinal cortex process speed and direction of the animal to map the environment. Hippocampal place cells, in turn, encode place using sensory signals and reduce the accumulated error of grid cells for path integration. Although both cell types are part of the path integration system, the dynamic relationship between place and grid cells and the error reduction mechanism is yet to be understood. We implemented a realistic model of grid cells based on a continuous attractor model. The grid cell model was coupled to a place cell model to address their dynamic relationship during a simulated animal's exploration of a square arena. The grid cell model processed the animal's velocity and place field information from place cells. Place cells incorporated salient visual features and proximity information with input from grid cells to define their place fields. Grid cells had similar spatial phases but a diversity of spacings and orientations. To determine the role of place cells in error reduction for path integration, the animal's position estimates were decoded from grid cell activities with and without the place field input. We found that the accumulated error was reduced as place fields emerged during the exploration. Place fields closer to the animal's current location contributed more to the error reduction than remote place fields. Place cells' fields encoding space could function as spatial anchoring signals for precise path integration by grid cells.

Multiple bumps can enhance robustness to noise in continuous attractor networks

A central function of continuous attractor networks is encoding coordinates and accurately updating their values through path integration. To do so, these networks produce localized bumps of activity that move coherently in response to velocity inputs. In the brain, continuous attractors are believed to underlie grid cells and head direction cells, which maintain periodic representations of position and orientation, respectively. These representations can be achieved with any number of activity bumps, and the consequences of having more or fewer bumps are unclear. We address this knowledge gap by constructing 1D ring attractor networks with different bump numbers and characterizing their responses to three types of noise: fluctuating inputs, spiking noise, and deviations in connectivity away from ideal attractor configurations. Across all three types, networks with more bumps experience less noise-driven deviations in bump motion. This translates to more robust encodings of linear coordinates, like position, assuming that each neuron represents a fixed length no matter the bump number. Alternatively, we consider encoding a circular coordinate, like orientation, such that the network distance between adjacent bumps always maps onto 360 degrees. Under this mapping, bump number does not significantly affect the amount of error in the coordinate readout. Our simulation results are intuitively explained and quantitatively matched by a unified theory for path integration and noise in multi-bump networks. Thus, to suppress the effects of biologically relevant noise, continuous attractor networks can employ more bumps when encoding linear coordinates; this advantage disappears when encoding circular coordinates. Our findings provide motivation for multiple bumps in the mammalian grid network.

Angular and linear speed cells in the parahippocampal circuits

An essential role of the hippocampal region is to integrate information to compute and update representations. How this transpires is highly debated. Many theories hinge on the integration of self-motion signals and the existence of continuous attractor networks (CAN). CAN models hypothesise that neurons coding for navigational correlates – such as position and direction – receive inputs from cells conjunctively coding for position, direction, and self-motion. As yet, very little data exist on such conjunctive coding in the hippocampal region. Here, we report neurons coding for angular and linear velocity, uniformly distributed across the medial entorhinal cortex (MEC), the presubiculum and the parasubiculum, except for MEC layer II. Self-motion neurons often conjunctively encoded position and/or direction, yet lacked a structured organisation. These results offer insights as to how linear/angular speed – derivative in time of position/direction – may allow the updating of spatial representations, possibly uncovering a generalised algorithm to update any representation.

It remains unclear how the hippocampal region integrates position and self-motion information to update spatial representations. Here, the authors report grid and head direction cells as well as cells encoding self-motion parameters such as angular head velocity and speed, and find conjunctive representations of these different parameters.

The Role of Idiothetic Signals, Landmarks, and Conjunctive Representations in the Development of Place and Head-Direction Cells: A Self-Organizing Neural Network Model

Place and head-direction (HD) cells are fundamental to maintaining accurate representations of location and heading in the mammalian brain across sensory conditions, and are thought to underlie path integration-the ability to maintain an accurate representation of location and heading during motion in the dark. Substantial evidence suggests that both populations of spatial cells function as attractor networks, but their developmental mechanisms are poorly understood. We present simulations of a fully self-organizing attractor network model of this process using well-established neural mechanisms. We show that the differential development of the two cell types can be explained by their different idiothetic inputs, even given identical visual signals: HD cells develop when the population receives angular head velocity input, whereas place cells develop when the idiothetic input encodes planar velocity. Our model explains the functional importance of conjunctive "state-action" cells, implying that signal propagation delays and a competitive learning mechanism are crucial for successful development. Consequently, we explain how insufficiently rich environments result in pathology: place cell development requires proximal landmarks; conversely, HD cells require distal landmarks. Finally, our results suggest that both networks are instantiations of general mechanisms, and we describe their implications for the neurobiology of spatial processing.

Neural circuits and symbolic processing

The ability to use symbols is a defining feature of human intelligence. However, neuroscience has yet to explain the fundamental neural circuit mechanisms for flexibly representing and manipulating abstract concepts. This article will review the research on neural models for symbolic processing. The review first focuses on the question of how symbols could possibly be represented in neural circuits. The review then addresses how neural symbolic representations could be flexibly combined to meet a wide range of reasoning demands. Finally, the review assesses the research on program synthesis and proposes that the most flexible neural representation of symbolic processing would involve the capacity to rapidly synthesize neural operations analogous to lambda calculus to solve complex cognitive tasks.

Dynamical self-organization and efficient representation of space by grid cells

To plan trajectories and navigate, animals must maintain a mental representation of the environment and their own position within it. This "cognitive map" is thought to be supported in part by the entorhinal cortex, where grid cells are active when an animal occupies the vertices of a scaling hierarchy of periodic lattices of locations in an enclosure. Here, we review computational developments which suggest that the grid cell network is: (a) efficient, providing required spatial resolution with a minimum number of neurons, (b) self-organizing, dynamically coordinating the structure and scale of the responses, and (c) adaptive, re-organizing in response to changes in landmarks and the structure of the boundaries of spaces. We consider these ideas in light of recent discoveries of similar structures in the mental representation of abstract spaces of shapes and smells, and in other brain areas, and highlight promising directions for future research.

Spiking Neural Networks and Hippocampal Function: A Web-Accessible Survey of Simulations, Modeling Methods, and Underlying Theories

Computational modeling has contributed to hippocampal research in a wide variety of ways and through a large diversity of approaches, reflecting the many advanced cognitive roles of this brain region. The intensively studied neuron type circuitry of the hippocampus is a particularly conducive substrate for spiking neural models. Here we present an online knowledge base of spiking neural network simulations of hippocampal functions. First, we overview theories involving the hippocampal formation in subjects such as spatial representation, learning, and memory. Then we describe an original literature mining process to organize published reports in various key aspects, including: (i) subject area (e.g., navigation, pattern completion, epilepsy); (ii) level of modeling detail (Hodgkin-Huxley, integrate-and-fire, etc.); and (iii) theoretical framework (attractor dynamics, oscillatory interference, self-organizing maps, and others). Moreover, every peer-reviewed publication is also annotated to indicate the specific neuron types represented in the network simulation, establishing a direct link with the Hippocampome.org portal. The web interface of the knowledge base enables dynamic content browsing and advanced searches, and consistently presents evidence supporting every annotation. Moreover, users are given access to several types of statistical reports about the collection, a selection of which is summarized in this paper. This open access resource thus provides an interactive platform to survey spiking neural network models of hippocampal functions, compare available computational methods, and foster ideas for suitable new directions of research.

Optogenetic pacing of medial septum parvalbumin-positive cells disrupts temporal but not spatial firing in grid cells

Grid cells in the medial entorhinal cortex (MEC) exhibit remarkable spatial activity patterns with spikes coordinated by theta oscillations driven by the medial septal area (MSA). Spikes from grid cells progress relative to the theta phase in a phenomenon called phase precession, which is suggested as essential to create the spatial periodicity of grid cells. Here, we show that optogenetic activation of parvalbumin-positive (PV+) cells in the MSA enabled selective pacing of local field potential (LFP) oscillations in MEC. During optogenetic stimulation, the grid cells were locked to the imposed pacing frequency but kept their spatial patterns. Phase precession was abolished, and speed information was no longer reflected in the LFP oscillations but was still carried by rate coding of individual MEC neurons. Together, these results support that theta oscillations are not critical to the spatial pattern of grid cells and do not carry a crucial velocity signal.

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.

Effects of visual inputs on neural dynamics for coding of location and running speed in medial entorhinal cortex

Neuronal representations of spatial location and movement speed in the medial entorhinal cortex during the ‘active’ theta state of the brain are important for memory-guided navigation and rely on visual inputs. However, little is known about how visual inputs change neural dynamics as a function of running speed and time. By manipulating visual inputs in mice, we demonstrate that changes in spatial stability of grid cell firing correlate with changes in a proposed speed signal by local field potential theta frequency. In contrast, visual inputs do not alter the running speed-dependent gain in neuronal firing rates. Moreover, we provide evidence that sensory inputs other than visual inputs can support grid cell firing, though less accurately, in complete darkness. Finally, changes in spatial accuracy of grid cell firing on a 10 s time scale suggest that grid cell firing is a function of velocity signals integrated over past time.

Impaired speed encoding and grid cell periodicity in a mouse model of tauopathy

Dementia is associated with severe spatial memory deficits which arise from dysfunction in hippocampal and parahippocampal circuits. For spatially sensitive neurons, such as grid cells, to faithfully represent the environment these circuits require precise encoding of direction and velocity information. Here, we have probed the firing rate coding properties of neurons in medial entorhinal cortex (MEC) in a mouse model of tauopathy. We find that grid cell firing patterns are largely absent in rTg4510 mice, while head-direction tuning remains largely intact. Conversely, neural representation of running speed information was significantly disturbed, with smaller proportions of MEC cells having firing rates correlated with locomotion in rTg4510 mice. Additionally, the power of local field potential oscillations in the theta and gamma frequency bands, which in wild-type mice are tightly linked to running speed, was invariant in rTg4510 mice during locomotion. These deficits in locomotor speed encoding likely severely impact path integration systems in dementia.

Environmental deformations dynamically shift human spatial memory

Place and grid cells in the hippocampal formation are commonly thought to support a unified and coherent cognitive map of space. This mapping mechanism faces a challenge when a navigator is placed in a familiar environment that has been deformed from its original shape. Under such circumstances, many transformations could plausibly serve to map a navigator's familiar cognitive map to the deformed space. Previous empirical results indicate that the firing fields of rodent place and grid cells stretch or compress in a manner that approximately matches the environmental deformation, and human spatial memory exhibits similar distortions. These effects have been interpreted as evidence that reshaping a familiar environment elicits an analogously reshaped cognitive map. However, recent work has suggested an alternative explanation, whereby deformation-induced distortions of the grid code are attributable to a mechanism that dynamically anchors grid fields to the most recently experienced boundary, thus causing history-dependent shifts in grid phase. This interpretation raises the possibility that human spatial memory will exhibit similar history-dependent dynamics. To test this prediction, we taught participants the locations of objects in a virtual environment and then probed their memory for these locations in deformed versions of this environment. Across three experiments with variable access to visual and vestibular cues, we observed the predicted pattern, whereby the remembered locations of objects were shifted from trial to trial depending on the boundary of origin of the participant's movement trajectory. These results provide evidence for a dynamic anchoring mechanism that governs both neuronal firing and spatial memory.

Theta resonance and synaptic modulation scale activity patterns in the medial entorhinal cortex stellate cells

Stellate cells (SCs) of the medial entorhinal cortex (MEC) are rich in hyperpolarization-activated cyclic nucleotide-gated (HCN) channels, which are known to effectively shape their activity patterns. The explanatory mechanisms, however, have remained elusive. One important but previously unassessed possibility is that HCN channels control the gain of synaptic inputs to these cells. Here, we test this possibility in rat brain slices, while subjecting SCs to a stochastic synaptic bombardment using the dynamic clamp. We show that in the presence of synaptic noise, HCN channels mainly exert their influence by increasing the relative signal gain in the theta frequency through the theta modulation of stochastic synaptic inputs. This subthreshold synaptic modulation then translates into a spiking resonance, which steepens with excitation in the presence of HCN channels. We present here a systematic assessment of synaptic theta modulation and trace its implications to the suprathreshold control of firing rate motifs. Such analysis was yet lacking in the SC literature. Furthermore, we assess the impact of noise statistics on this gain modulation and indicate possible mechanisms for the emergence of membrane theta oscillations and synaptic ramps, as observed in vivo. We support the data with a computational model that further unveils a competing role of inhibition, suggesting important implications for MEC computations.

Neuronal vector coding in spatial cognition

Several types of neurons involved in spatial navigation and memory encode the distance and direction (that is, the vector) between an agent and items in its environment. Such vectorial information provides a powerful basis for spatial cognition by representing the geometric relationships between the self and the external world. Here, we review the explicit encoding of vectorial information by neurons in and around the hippocampal formation, far from the sensory periphery. The parahippocampal, retrosplenial and parietal cortices, as well as the hippocampal formation and striatum, provide a plethora of examples of vector coding at the single neuron level. We provide a functional taxonomy of cells with vectorial receptive fields as reported in experiments and proposed in theoretical work. The responses of these neurons may provide the fundamental neural basis for the (bottom-up) representation of environmental layout and (top-down) memory-guided generation of visuospatial imagery and navigational planning.

Navigating Through Time: A Spatial Navigation Perspective on How the Brain May Encode Time

Interval timing, which operates on timescales of seconds to minutes, is distributed across multiple brain regions and may use distinct circuit mechanisms as compared to millisecond timing and circadian rhythms. However, its study has proven difficult, as timing on this scale is deeply entangled with other behaviors. Several circuit and cellular mechanisms could generate sequential or ramping activity patterns that carry timing information. Here we propose that a productive approach is to draw parallels between interval timing and spatial navigation, where direct analogies can be made between the variables of interest and the mathematical operations necessitated. Along with designing experiments that isolate or disambiguate timing behavior from other variables, new techniques will facilitate studies that directly address the neural mechanisms that are responsible for interval timing.

Overview of computational models of hippocampus and related structures: Introduction to the special issue

Extensive computational modeling has focused on the hippocampal formation and associated cortical structures. This overview describes some of the factors that have motivated the strong focus on these structures, including major experimental findings and their impact on computational models. This overview provides a framework for describing the topics addressed by individual articles in this special issue of the journal Hippocampus.

An uncertainty principle for neural coding: Conjugate representations of position and velocity are mapped onto firing rates and co-firing rates of neural spike trains

The hippocampal system contains neural populations that encode an animal's position and velocity as it navigates through space. Here, we show that such populations can embed two codes within their spike trains: a firing rate code (R) conveyed by within‐cell spike intervals, and a co‐firing rate code (R˙) conveyed by between‐cell spike intervals. These two codes behave as conjugates of one another, obeying an analog of the uncertainty principle from physics: information conveyed in R comes at the expense of information in R˙, and vice versa. An exception to this trade‐off occurs when spike trains encode a pair of conjugate variables, such as position and velocity, which do not compete for capacity across R and R˙. To illustrate this, we describe two biologically inspired methods for decoding R and R˙, referred to as sigma and sigma‐chi decoding, respectively. Simulations of head direction and grid cells show that if firing rates are tuned for position (but not velocity), then position is recovered by sigma decoding, whereas velocity is recovered by sigma‐chi decoding. Conversely, simulations of oscillatory interference among theta‐modulated “speed cells” show that if co‐firing rates are tuned for position (but not velocity), then position is recovered by sigma‐chi decoding, whereas velocity is recovered by sigma decoding. Between these two extremes, information about both variables can be distributed across both channels, and partially recovered by both decoders. These results suggest that populations with different spatial and temporal tuning properties—such as speed versus grid cells—might not encode different information, but rather, distribute similar information about position and velocity in different ways across R and R˙. Such conjugate coding of position and velocity may influence how hippocampal populations are interconnected to form functional circuits, and how biological neurons integrate their inputs to decode information from firing rates and spike correlations.

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.

Entorhinal velocity signals reflect environmental geometry

The entorhinal cortex contains neurons that represent self-location, including grid cells that fire in periodic locations and velocity signals that encode running speed and head direction. Although the size and shape of the environment influence grid patterns, whether entorhinal velocity signals are equally influenced or provide a universal metric for self-motion across environments remains unknown. Here we report that speed cells rescale after changes to the size and shape of the environment. Moreover, head direction cells reorganize in an experience-dependent manner to align with the axis of environmental change. A knockout mouse model allows dissociation of the coordination between cell types, with grid and speed cells, but not head direction cells, responding in concert to environmental change. These results point to malleability in the coding features of multiple entorhinal cell types and have implications for which cell types contribute to the velocity signal used by computational models of grid cells.

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.

Modulating grid cell scale and intrinsic frequencies via slow high-threshold conductances: A simplified model

Grid cells in the medial entorhinal cortex (MEC) have known spatial periodic firing fields which provide a metric for the representation of self-location and path planning. The hexagonal tessellation pattern of grid cells scales up progressively along the MEC's layer II dorsal-to-ventral axis. This scaling gradient has been hypothesized to originate either from inter-population synaptic dynamics as postulated by attractor networks, or from projected theta frequency waves to different axis levels, as in oscillatory models. Alternatively, cellular dynamics and specifically slow high-threshold conductances have been proposed to have an impact on the grid cell scale. To test the hypothesis that intrinsic hyperpolarization-activated cation currents account for both the scaled gradient and the oscillatory frequencies observed along the dorsal-to-ventral axis, we have modeled and analyzed data from a population of grid cells simulated with spiking neurons interacting through low-dimensional attractor dynamics. We observed that the intrinsic neuronal membrane properties of simulated cells were sufficient to induce an increase in grid scale and potentiate differences in the membrane potential oscillatory frequency. Overall, our results suggest that the after-spike dynamics of cation currents may play a major role in determining the grid cells' scale and that oscillatory frequencies are a consequence of intrinsic cellular properties that are specific to different levels of the dorsal-to-ventral axis in the MEC layer II.

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.

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.

A Computational Model of Visual Recognition Memory via Grid Cells

Models of face, object, and scene recognition traditionally focus on massively parallel processing of low-level features, with higher-order representations emerging at later processing stages. However, visual perception is tightly coupled to eye movements, which are necessarily sequential. Recently, neurons in entorhinal cortex have been reported with grid cell-like firing in response to eye movements, i.e., in visual space. Following the presumed role of grid cells in vector navigation, we propose a model of recognition memory for familiar faces, objects, and scenes, in which grid cells encode translation vectors between salient stimulus features. A sequence of saccadic eye-movement vectors, moving from one salient feature to the expected location of the next, potentially confirms an initial hypothesis (accumulating evidence toward a threshold) about stimulus identity, based on the relative feature layout (i.e., going beyond recognition of individual features). The model provides an explicit neural mechanism for the long-held view that directed saccades support hypothesis-driven, constructive perception and recognition; is compatible with holistic face processing; and constitutes the first quantitative proposal for a role of grid cells in visual recognition. The variance of grid cell activity along saccade trajectories exhibits 6-fold symmetry across 360 degrees akin to recently reported fMRI data. The model suggests that disconnecting grid cells from occipitotemporal inputs may yield prosopagnosia-like symptoms. The mechanism is robust with regard to partial visual occlusion, can accommodate size and position invariance, and suggests a functional explanation for medial temporal lobe involvement in visual memory for relational information and memory-guided attention.

Altered neural odometry in the vertical dimension

The brain’s spatial map is supported by place cells, encoding current location, and grid cells, which report horizontal distance traveled by producing evenly sized and spaced foci of activity (firing fields) that tile the environment surface. We investigated whether the metric properties of the cells’ activity are the same in vertical space as in horizontal. On a vertical wall, grid-cell firing fields were enlarged and more widely spaced, while place-cell firing fields were unchanged in size/shape but less prevalent. Sensitivity of single-cell and population field potential activity to running speed was reduced. Together, these results suggest that spatial encoding properties are determined by an interaction between the body-plane alignment and the gravity axis.

Entorhinal grid cells integrate sensory and self-motion inputs to provide a spatial metric of a characteristic scale. One function of this metric may be to help localize the firing fields of hippocampal place cells during formation and use of the hippocampal spatial representation (“cognitive map”). Of theoretical importance is the question of how this metric, and the resulting map, is configured in 3D space. We find here that when the body plane is vertical as rats climb a wall, grid cells produce stable, almost-circular grid-cell firing fields. This contrasts with previous findings when the body was aligned horizontally during vertical exploration, suggesting a role for the body plane in orienting the plane of the grid cell map. However, in the present experiment, the fields on the wall were fewer and larger, suggesting an altered or absent odometric (distance-measuring) process. Several physiological indices of running speed in the entorhinal cortex showed reduced gain, which may explain the enlarged grid pattern. Hippocampal place fields were found to be sparser but unchanged in size/shape. Together, these observations suggest that the orientation and scale of the grid cell map, at least on a surface, are determined by an interaction between egocentric information (the body plane) and allocentric information (the gravity axis). This may be mediated by the different sensory or locomotor information available on a vertical surface and means that the resulting map has different properties on a vertical plane than a horizontal plane (i.e., is anisotropic).

Saccade Velocity Driven Oscillatory Network Model of Grid Cells

Grid cells and place cells are believed to be cellular substrates for the spatial navigation functions of hippocampus as experimental animals physically navigated in 2D and 3D spaces. However, a recent saccade study on head fixated monkey has also reported grid-like representations on saccadic trajectory while the animal scanned the images on a computer screen. We present two computational models that explain the formation of grid patterns on saccadic trajectory formed on the novel Images. The first model named Saccade Velocity Driven Oscillatory Network -Direct PCA (SVDON—DPCA) explains how grid patterns can be generated on saccadic space using Principal Component Analysis (PCA) like learning rule. The model adopts a hierarchical architecture. We extend this to a network model viz. Saccade Velocity Driven Oscillatory Network—Network PCA (SVDON-NPCA) where the direct PCA stage is replaced by a neural network that can implement PCA using a neurally plausible algorithm. This gives the leverage to study the formation of grid cells at a network level. Saccade trajectory for both models is generated based on an attention model which attends to the salient location by computing the saliency maps of the images. Both models capture the spatial characteristics of grid cells such as grid scale variation on the dorso-ventral axis of Medial Entorhinal cortex. Adding one more layer of LAHN over the SVDON-NPCA model predicts the Place cells in saccadic space, which are yet to be discovered experimentally. To the best of our knowledge, this is the first attempt to model grid cells and place cells from saccade trajectory.

Spatial synchronization codes from coupled rate-phase neurons

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

Environmental deformations dynamically shift the grid cell spatial metric

In familiar environments, the firing fields of entorhinal grid cells form regular triangular lattices. However, when the geometric shape of the environment is deformed, these time-averaged grid patterns are distorted in a grid scale-dependent and local manner. We hypothesized that this distortion in part reflects dynamic anchoring of the grid code to displaced boundaries, possibly through border cell-grid cell interactions. To test this hypothesis, we first reanalyzed two existing rodent grid rescaling datasets to identify previously unrecognized boundary-tethered shifts in grid phase that contribute to the appearance of rescaling. We then demonstrated in a computational model that boundary-tethered phase shifts, as well as scale-dependent and local distortions of the time-averaged grid pattern, could emerge from border-grid interactions without altering inherent grid scale. Together, these results demonstrate that environmental deformations induce history-dependent shifts in grid phase, and implicate border-grid interactions as a potential mechanism underlying these dynamics.

Grid-like hexadirectional modulation of human entorhinal theta oscillations

The entorhinal cortex contains a network of grid cells that play a fundamental part in the brain's spatial system, supporting tasks such as path integration and spatial memory. In rodents, grid cells are thought to rely on network theta oscillations, but such signals are not evident in all species, challenging our understanding of the physiological basis of the grid network. We analyzed intracranial recordings from neurosurgical patients during virtual navigation to identify oscillatory characteristics of the human entorhinal grid network. The power of entorhinal theta oscillations showed six-fold modulation according to the virtual heading during navigation, which is a hypothesized signature of grid representations. Furthermore, modulation strength correlated with spatial memory performance. These results demonstrate the connection between theta oscillations and the human entorhinal grid network and show that features of grid-like neuronal representations can be identified from population electrophysiological recordings.

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.

Self-motion processing in visual and entorhinal cortices: inputs, integration, and implications for position coding

The sensory signals generated by self-motion are complex and multimodal, but the ability to integrate these signals into a unified self-motion percept to guide navigation is essential for animal survival. Here, we summarize classic and recent work on self-motion coding in the visual and entorhinal cortices of the rodent brain. We compare motion processing in rodent and primate visual cortices, highlighting the strengths of classic primate work in establishing causal links between neural activity and perception, and discuss the integration of motor and visual signals in rodent visual cortex. We then turn to the medial entorhinal cortex (MEC), where calculations using self-motion to update position estimates are thought to occur. We focus on several key sources of self-motion information to MEC: the medial septum, which provides locomotor speed information; visual cortex, whose input has been increasingly recognized as essential to both position and speed-tuned MEC cells; and the head direction system, which is a major source of directional information for self-motion estimates. These inputs create a large and diverse group of self-motion codes in MEC, and great interest remains in how these self-motion codes might be integrated by MEC grid cells to estimate position. However, which signals are used in these calculations and the mechanisms by which they are integrated remain controversial. We end by proposing future experiments that could further our understanding of the interactions between MEC cells that code for self-motion and position and clarify the relationship between the activity of these cells and spatial perception.

A neural-level model of spatial memory and imagery

We present a model of how neural representations of egocentric spatial experiences in parietal cortex interface with viewpoint-independent representations in medial temporal areas, via retrosplenial cortex, to enable many key aspects of spatial cognition. This account shows how previously reported neural responses (place, head-direction and grid cells, allocentric boundary- and object-vector cells, gain-field neurons) can map onto higher cognitive function in a modular way, and predicts new cell types (egocentric and head-direction-modulated boundary- and object-vector cells). The model predicts how these neural populations should interact across multiple brain regions to support spatial memory, scene construction, novelty-detection, 'trace cells', and mental navigation. Simulated behavior and firing rate maps are compared to experimental data, for example showing how object-vector cells allow items to be remembered within a contextual representation based on environmental boundaries, and how grid cells could update the viewpoint in imagery during planning and short-cutting by driving sequential place cell activity.