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The hippocampal-entorhinal circuit is considered to play an important role in the spatial cognition of animals. However, the mechanism of the information flow within the circuit and its contribution to the function of the grid-cell module are still topics of discussion. Prevailing theories suggest that grid cells are primarily influenced by self-motion inputs from the Medial Entorhinal Cortex, with place cells serving a secondary role by contributing to the visual calibration of grid cells. However, recent evidence suggests that both self-motion inputs and visual cues may collaboratively contribute to the formation of grid-like patterns. In this paper, we introduce a novel Continuous Attractor Network model based on a spatial transformation mechanism. This mechanism enables the integration of self-motion inputs and visual cues within grid-cell modules, synergistically driving the formation of grid-like patterns. From the perspective of individual neurons within the network, our model successfully replicates grid firing patterns. From the view of neural population activity within the network, the network can form and drive the activated bump, which describes the characteristic feature of grid-cell modules, namely, path integration. Through further exploration and experimentation, our model can exhibit significant performance in path integration. This study provides a new insight into understanding the mechanism of how the self-motion and visual inputs contribute to the neural activity within grid-cell modules. Furthermore, it provides theoretical support for achieving accurate path integration, which holds substantial implications for various applications requiring spatial navigation and mapping.

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The grid cells in the medial entorhinal cortex are widely recognized as a critical component of spatial cognition within the entorhinal-hippocampal neuronal circuits. To account for the hexagonal patterns, several computational models have been proposed. However, there is still considerable debate regarding the interaction between grid cells and place cells. In response, we have developed a novel grid-cell computational model based on cognitive space transformation, which established a theoretical framework of the interaction between place cells and grid cells for encoding and transforming positions between the local frame and global frame. Our model not only can generate the firing patterns of the grid cells but also reproduces the biological experiment results about the grid-cell global representation of connected environments and supports the conjecture about the underlying reason. Moreover, our model provides new insights into how grid cells and place cells integrate external and self-motion cues.

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Across species, spatial memory declines with age, possibly reflecting altered hippocampal and medial entorhinal cortex (MEC) function. However, the integrity of cellular and network-level spatial coding in aged MEC is unknown. Here, we leveraged in vivo electrophysiology to assess MEC function in young, middle-aged, and aged mice navigating virtual environments. In aged grid cells, we observed impaired stabilization of context-specific spatial firing, correlated with spatial memory deficits. Additionally, aged grid networks shifted firing patterns often but with poor alignment to context changes. Aged spatial firing was also unstable in an unchanging environment. In these same mice, we identified 458 genes differentially expressed with age in MEC, 61 of which had expression correlated with spatial firing stability. These genes were enriched among interneurons and related to synaptic transmission. Together, these findings identify coordinated transcriptomic, cellular, and network changes in MEC implicated in impaired spatial memory in aging.

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Grid firing fields have been proposed as a neural substrate for spatial localisation in general or for path integration in particular. To distinguish these possibilities, we investigate firing of grid and non-grid cells in the mouse medial entorhinal cortex during a location memory task. We find that grid firing can either be anchored to the task environment, or can encode distance travelled independently of the task reference frame. Anchoring varied between and within sessions, while spatial firing of non-grid cells was either coherent with the grid population, or was stably anchored to the task environment. We took advantage of the variability in task-anchoring to evaluate whether and when encoding of location by grid cells might contribute to behaviour. We find that when reward location is indicated by a visual cue, performance is similar regardless of whether grid cells are task-anchored or not, arguing against a role for grid representations when location cues are available. By contrast, in the absence of the visual cue, performance was enhanced when grid cells were anchored to the task environment. Our results suggest that anchoring of grid cells to task reference frames selectively enhances performance when path integration is required.

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The mechanisms underlying Parkinson's disease (PD) are complex and not fully understood, and the box-and-arrow model among other current models present significant challenges. This paper explores the potential role of the allocentric brain and especially its grid cells in several PD motor symptoms, including bradykinesia, kinesia paradoxa, freezing of gait, the bottleneck phenomenon, and their dependency on cueing. It is argued that central hubs, like the locus coeruleus and the pedunculopontine nucleus, often narrowly interpreted in the context of PD, play an equally important role in governing the allocentric brain as the basal ganglia. Consequently, the motor and secondary motor (e.g., spatially related) symptoms of PD linked with dopamine depletion may be more closely tied to erroneous computation by grid cells than to the basal ganglia alone. Because grid cells and their associated central hubs introduce both spatial and temporal information to the brain influencing velocity perception they may cause bradykinesia or hyperkinesia as well. In summary, PD motor symptoms may primarily be an allocentric disturbance resulting from virtual faulty computation by grid cells revealed by dopamine depletion in PD.

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We present computer simulations illustrating how the plastic integration of spatially stable inputs could contribute to the dynamic character of hippocampal spatial representations. In novel environments of slightly larger size than typical apparatus, the emergence of well-defined place fields in real place cells seems to rely on inputs from normally functioning grid cells. Theoretically, the grid-to-place transformation is possible if a place cell is able to respond selectively to a combination of suitably aligned grids. We previously identified the functional characteristics that allow a synaptic plasticity rule to accomplish this selection by synaptic competition during rat foraging behavior. Here, we show that the synaptic competition can outlast the formation of place fields, contributing to their spatial reorganization over time, when the model is run in larger environments and the topographical/modular organization of grid inputs is taken into account. Co-simulated cells that differ only by their randomly assigned grid inputs display different degrees and kinds of spatial reorganization-ranging from place-field remapping to more subtle in-field changes or lapses in firing. The model predicts a greater number of place fields and propensity for remapping in place cells recorded from more septal regions of the hippocampus and/or in larger environments, motivating future experimental standardization across studies and animal models. In sum, spontaneous remapping could arise from rapid synaptic learning involving inputs that are functionally homogeneous, spatially stable, and minimally stochastic.

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Head direction (HD) cells, which fire persistently when an animal's head is pointed in a particular direction, are widely thought to underlie an animal's sense of spatial orientation and have been identified in several limbic brain regions. Robust HD cell firing is observed throughout the thalamo-parahippocampal system, although recent studies report that parahippocampal HD cells exhibit distinct firing properties, including conjunctive aspects with other spatial parameters, which suggest they play a specialized role in spatial processing. Few studies, however, have quantified these apparent differences. Here, we performed a comparative assessment of HD cell firing characteristics across the anterior dorsal thalamus (ADN), postsubiculum (PoS), parasubiculum (PaS), medial entorhinal (MEC), and postrhinal (POR) cortices. We report that HD cells with a high degree of directional specificity were observed in all five brain regions, but ADN HD cells display greater sharpness and stability in their preferred directions, and greater anticipation of future headings compared to parahippocampal regions. Additional analysis indicated that POR HD cells were more coarsely modulated by other spatial parameters compared to PoS, PaS, and MEC. Finally, our analyses indicated that the sharpness of HD tuning decreased as a function of laminar position and conjunctive coding within the PoS, PaS, and MEC, with cells in the superficial layers along with conjunctive firing properties showing less robust directional tuning. The results are discussed in relation to theories of functional organization of HD cell tuning in thalamo-parahippocampal circuitry.

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Spatial representations in the entorhinal cortex (EC) and hippocampus (HPC) are fundamental to cognitive functions like navigation and memory. These representations, embodied in spatial field maps, dynamically remap in response to environmental changes. However, current methods, such as Pearson's correlation coefficient, struggle to capture the complexity of these remapping events, especially when fields do not overlap, or transformations are non-linear. This limitation hinders our understanding and quantification of remapping, a key aspect of spatial memory function. To address this, we propose a family of metrics based on the Earth Mover's Distance (EMD) as a versatile framework for characterizing remapping. Applied to both normalized and unnormalized distributions, the EMD provides a granular, noise-resistant, and rate-robust description of remapping. This approach enables the identification of specific cell types and the characterization of remapping in various scenarios, including disease models. Furthermore, the EMD's properties can be manipulated to identify spatially tuned cell types and to explore remapping as it relates to alternate information forms such as spatiotemporal coding. By employing approximations of the EMD, we present a feasible, lightweight approach that complements traditional methods. Our findings underscore the potential of the EMD as a powerful tool for enhancing our understanding of remapping in the brain and its implications for spatial navigation, memory studies and beyond.

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Grid cells and place cells represent the spatiotemporal continuum of an animal's past, present, and future locations. However, their spatiotemporal relationship is unclear. Here, we co-record grid and place cells in freely foraging rats. We show that average time shifts in grid cells tend to be prospective and are proportional to their spatial scale, providing a nearly instantaneous readout of a spectrum of progressively increasing time horizons ranging hundreds of milliseconds. Average time shifts of place cells are generally larger compared to grid cells and also increase with place field sizes. Moreover, time horizons display nonlinear modulation by the animal's trajectories in relation to the local boundaries and locomotion cues. Finally, long and short time horizons occur at different parts of the theta cycle, which may facilitate their readout. Together, these findings suggest that population activity of grid and place cells may represent local trajectories essential for goal-directed navigation and planning.

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Introduction: Visual-spatial perception is a process for extracting the spatial relationship between objects in the environment. The changes in visual-spatial perception due to factors such as the activity of the sympathetic nervous system (hyperactivation) or parasympathetic nervous system (hypoactivation) can affect the internal representation of the external visual-spatial world. We formulated a quantitative model of the modulation of visual-perceptual space under action by hyperactivation or hypoactivation-inducing neuromodulating agents. We showed a Hill equation based relationship between neuromodulator agent concentration and alteration of visual-spatial perception utilizing the metric tensor to quantify the visual space. Methods: We computed the dynamics of the psilocybin (hyperactivation-inducing agent) and chlorpromazine (hypoactivation-inducing agent) in brain tissue. Then, we validated our quantitative model by analyzing the findings of different independent behavioral studies where subjects were assessed for alterations in visual-spatial perception under the action of psilocybin and under chlorpromazine. To validate the neuronal correlates, we simulated the effect of the neuromodulating agent on the computational model of the grid-cell network, and also performed diffusion MRI-based tractography to find the neural tracts between the cortical areas involved: V2 and the entorhinal cortex. Results: We applied our computational model to an experiment (where perceptual alterations were measured under psilocybin) and found that for n (Hill-coefficient) = 14.8 and k = 1.39, the theoretical prediction followed experimental observations very well (χ2 test robustly satisfied, p > 0.99). We predicted the outcome of another psilocybin-based experiment using these values (n = 14.8 and k = 1.39), whereby our prediction and experimental outcomes were well corroborated. Furthermore, we found that also under hypoactivation (chlorpromazine), the modulation of the visual-spatial perception follows our model. Moreover, we found neural tracts between the area V2 and entorhinal cortex, thus providing a possible brain network responsible for encoding visual-spatial perception. Thence, we simulated the altered grid-cell network activity, which was also found to follow the Hill equation. Conclusion: We developed a computational model of visuospatial perceptual alterations under altered neural sympathetic/parasympathetic tone. We validated our model using analysis of behavioral studies, neuroimaging assessment, and neurocomputational evaluation. Our quantitative approach may be probed as a potential behavioral screening and monitoring methodology in neuropsychology to analyze perceptual misjudgment and mishaps by highly stressed workers.

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Introduction: Spatial representations in the entorhinal cortex (EC) and hippocampus (HPC) are fundamental to cognitive functions like navigation and memory. These representations, embodied in spatial field maps, dynamically remap in response to environmental changes. However, current methods, such as Pearson's correlation coefficient, struggle to capture the complexity of these remapping events, especially when fields do not overlap, or transformations are non-linear. This limitation hinders our understanding and quantification of remapping, a key aspect of spatial memory function. Methods: We propose a family of metrics based on the Earth Mover's Distance (EMD) as a versatile framework for characterizing remapping. Results: The EMD provides a granular, noise-resistant, and rate-robust description of remapping. This approach enables the identification of specific cell types and the characterization of remapping in various scenarios, including disease models. Furthermore, the EMD's properties can be manipulated to identify spatially tuned cell types and to explore remapping as it relates to alternate information forms such as spatiotemporal coding. Discussion: We present a feasible, lightweight approach that complements traditional methods. Our findings underscore the potential of the EMD as a powerful tool for enhancing our understanding of remapping in the brain and its implications for spatial navigation, memory studies and beyond.

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Grid cells or grid-like responses have been reported in the rodent, bat and human brains during various spatial and non-spatial tasks. However, the functions of grid-like representations beyond the classical hippocampal formation remain elusive. Based on accumulating evidence from recent rodent recordings and human fMRI data, we make speculative accounts regarding the mechanisms and functional significance of the sensory cortical grid cells and further make theory-driven predictions. We argue and reason the rationale why grid responses may be universal in the brain for a wide range of perceptual and cognitive tasks that involve locomotion and mental navigation. Computational modeling may provide an alternative and complementary means to investigate the grid code or grid-like map. We hope that the new discussion will lead to experimentally testable hypotheses and drive future experimental data collection.

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Traditional facial recognition methods depend on a large number of training samples due to the massive turning of synaptic weights for low-level feature extractions. In prior work, a brain-inspired model of visual recognition memory suggested that grid cells encode translation saccadic eye movement vectors between salient stimulus features. With a small training set for each recognition type, the relative positions among the selected features for each image were represented using grid and feature label cells in Hebbian learning. However, this model is suitable only for the recognition of familiar faces, objects, and scenes. The model's performance for a given face with unfamiliar facial expressions was unsatisfactory. In this study, an improved computational model via grid cells for facial recognition was proposed. Here, the initial hypothesis about stimulus identity was obtained using the histograms of oriented gradients (HOG) algorithm. The HOG descriptors effectively captured the sample edge or gradient structure features. Thus, most test samples were successfully recognized within three saccades. Moreover, the probability of a false hypothesis and the average fixations for successful recognition were reduced. Compared with other neural network models, such as convolutional neural networks and deep belief networks, the proposed method shows the best performance with only one training sample for each face. Moreover, it is robust against image occlusion and size variance or scaling. Our results may give insight for efficient recognition with small training samples based on neural networks.

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Grid cells are crucial in path integration and representation of the external world. The spikes of grid cells spatially form clusters called grid fields, which encode important information about allocentric positions. To decode the information, studying the spatial structures of grid fields is a key task for both experimenters and theorists. Experiments reveal that grid fields form hexagonal lattice during planar navigation, and are anisotropic beyond planar navigation. During volumetric navigation, they lose global order but possess local order. How grid cells form different field structures behind these different navigation modes remains an open theoretical question. However, to date, few models connect to the latest discoveries and explain the formation of various grid field structures. To fill in this gap, we propose an interpretive plane-dependent model of three-dimensional (3D) grid cells for representing both two-dimensional (2D) and 3D space. The model first evaluates motion with respect to planes, such as the planes animals stand on and the tangent planes of the motion manifold. Projection of the motion onto the planes leads to anisotropy, and error in the perception of planes degrades grid field regularity. A training-free recurrent neural network (RNN) then maps the processed motion information to grid fields. We verify that our model can generate regular and anisotropic grid fields, as well as grid fields with merely local order; our model is also compatible with mode switching. Furthermore, simulations predict that the degradation of grid field regularity is inversely proportional to the interval between two consecutive perceptions of planes. In conclusion, our model is one of the few pioneers that address grid field structures in a general case. Compared to the other pioneer models, our theory argues that the anisotropy and loss of global order result from the uncertain perception of planes rather than insufficient training.

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Grid cells in rodent medial entorhinal cortex are thought to play a key role for spatial navigation. When the animal is freely moving in an open arena the firing fields of each grid cell tend to form a highly regular, hexagonal lattice spanning the environment. However, firing rates vary from field to field and change under contextual modifications, whereas the field locations shift at most by a small amount under such "rate remapping." The observed differences in firing rate could reflect overall activity changes or changes in the detailed spike-train statistics. As these two alternatives imply distinct neural coding schemes, we investigated whether temporal firing patterns vary from field to field and whether they change under rate remapping. Focusing on short time scales, we found that the proportion of bursts compared to all discharge events is similar in all firing fields of a given grid cell and does not change under rate remapping. For each cell, mean firing rates with bursts are proportional to mean firing rates without bursts. However, this ratio varies across cells. Additionally, we looked at how rate remapping relates to entorhinal theta-frequency oscillations. Theta-phase coding was preserved despite firing-rate changes from rate remapping but we did not observe differences between the first and second half of the theta cycle, as had been reported for CA1. Our results indicate that both, the heterogeneity between firing fields and rate remapping, are not due to altered firing patterns on short time scales but reflect location-specific changes at the firing-rate level.

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Place cells and grid cells are important neurons involved in spatial navigation in the mammalian brain. Grid cells are believed to play an important role in forming a cognitive map of the environment. Experimental observations in recent years showed that the grid pattern is not invariant but is influenced by the shape of the spatial environment. However, the cause of this deformation remains elusive. Here, we focused on the functional interactions between place cells and grid cells, utilizing the information of location relationships between the firing fields of place cells to optimize the previous grid cell feedforward generation model and expand its application to more complex environmental scenarios. Not only was the regular equilateral triangle periodic firing field structure of the grid cells reproduced, but the expected results were consistent with the experiment for the environment with various complex boundary shapes and environmental deformation. Even in the field of three-dimensional spatial grid patterns, forward-looking predictions have been made. This provides a possible model explanation for how the coupling of grid cells and place cells adapt to the diversity of the external environment to deepen our understanding of the neural basis for constructing cognitive maps.

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The brain is able to calculate the distance and direction to the desired position based on grid cells. Extensive neurophysiological studies of rodent navigation have postulated the grid cells function as a metric for space, and have inspired many computational studies to develop innovative navigation approaches. Furthermore, grid cells may provide a general encoding scheme for high-order nonspatial information. Built upon existing neuroscience and machine learning work, this paper provides theoretical clarity on that the grid cell population codes can be taken as a metric for space. The metric is generated by a shift-invariant positive definite kernel via kernel distance method and embeds isometrically in a Euclidean space, and the inner product of the grid cell population code exponentially converges to the kernel. We also provide a method to learn the distribution of grid cell population efficiently. Grid cells, as a scalable position encoding method, can encode the spatial relationships of places and enable grid cells to outperform place cells in navigation. Further, we extend the grid cell to images encoding and find that grid cells embed images into a mental map, where geometric relationships are conceptual relationships of images. The theoretical model and analysis would contribute to establishing the grid cell code as a generic coding scheme for both spatial and conceptual spaces, and is promising for a multitude of problems across spatial cognition, machine learning and semantic cognition.

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This article describes a neural model of the anatomy, neurophysiology, and functions of intrinsic and extrinsic theta rhythms in the brains of multiple species. Topics include how theta rhythms were discovered; how theta rhythms organize brain information processing into temporal series of spatial patterns; how distinct theta rhythms occur within area CA1 of the hippocampus and between the septum and area CA3 of the hippocampus; what functions theta rhythms carry out in different brain regions, notably CA1-supported functions like learning, recognition, and memory that involve visual, cognitive, and emotional processes; how spatial navigation, adaptively timed learning, and category learning interact with hippocampal theta rhythms; how parallel cortical streams through the lateral entorhinal cortex (LEC) and the medial entorhinal cortex (MEC) represent the end-points of the What cortical stream for perception and cognition and the Where cortical stream for spatial representation and action; how the neuromodulator acetylcholine interacts with the septo-hippocampal theta rhythm and modulates category learning; what functions are carried out by other brain rhythms, such as gamma and beta oscillations; and how gamma and beta oscillations interact with theta rhythms. Multiple experimental facts about theta rhythms are unified and functionally explained by this theoretical synthesis.

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Across many studies, ventromedial prefrontal cortex (vmPFC) activity has been found to correlate with subjective value during value-based decision-making. Recently, however, vmPFC has also been shown to reflect a hexagonal gridlike code during navigation through physical and conceptual space, and such gridlike codes have been proposed to enable value-based choices between novel options. Here, we first show that, in theory, a hexagonal gridlike code can in some cases mimic vmPFC activity previously attributed to subjective value, raising the possibility that the subjective value correlates previously observed in vmPFC may have actually been a misconstrued gridlike signal. We then compare the two accounts empirically, using fMRI data from a large number of subjects performing an intertemporal choice task. We find clear and unambiguous evidence that subjective value is a better description of vmPFC activity in this task than a hexagonal gridlike code. In fact, we find no significant evidence at all for a hexagonal gridlike code in vmPFC activity during intertemporal choice. This result limits the generality of gridlike modulation as description of vmPFC activity. We suggest that vmPFC may flexibly switch representational schemes so as to encode the most relevant information for the current task.

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Internal representation of the space is a fundamental and crucial function of the animal's brain. Grid cells in the medial entorhinal cortex are thought to provide an environment-invariant metric system for the navigation of the animal. Most experimental and theoretical studies have focused on the horizontal planar codes of grid cell, while how this metric coordinate system is configured in the actual three-dimensional space remains unclear. Evidence has implied the spatial cognition may not be fully volumetric. We proposed an oscillatory interference model with a novel gravity and body plane modulation to simulate grid cell activity in complex space for rodents. The animal can perceive the rotation of its body plane along the local surface by sensing the gravity, causing the modulation to the dendritic oscillations. The results not only reproduce the firing patterns of the grid cell recorded from known experiments, but also predict the grid codes in novel environments. It further demonstrates that the gravity signal is indispensable for the animal's navigation, and supports the hypothesis that the periodic firing of the grid cell is intrinsically not a volumetric code in three-dimensional space. This will provide new insights to understand the spatial representation of the actual world in the brain.

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