Grid cells are believed to play an important role in both spatial and non-spatial cognition tasks. A recent study observed the emergence of grid cells in an LSTM for path integration. The connection between biological and artificial neural networks underlying the seemingly similarity, as well as the application domain of grid cells in deep neural networks (DNNs), expect further exploration. This work demonstrated that grid cells could be replicated in either pure vision based or vision guided path integration DNNs for navigation under a proper setting of training parameters. We also show that grid-like behaviors arise in feedforward DNNs for non-spatial tasks. Our findings support that the grid coding is an effective representation for both biological and artificial networks.
Grid cells in the entorhinal cortex, together with head direction, place, speed and border cells, are major contributors to the organization of spatial representations in the brain. In this work we introduce a novel theoretical and algorithmic framework able to explain the emergence of hexagonal grid-like response patterns from head direction cells' responses. We show that this pattern is a result of minimal variance encoding of neurons. The novelty lies into the formulation of the encoding problem through the modern Frame Theory language, specifically that of equiangular Frames, providing new…
About a decade ago grid cells were discovered in the medial entorhinal cortex of rat. Their peculiar firing patterns, which correlate with periodic locations in the environment, led to early hypothesis that grid cells may provide some form of metric for space. Subsequent research has since uncovered a wealth of new insights into the characteristics of grid cells and their neural neighborhood, the parahippocampal-hippocampal region, calling for a revision and refinement of earlier grid cell models. This survey paper aims to provide a comprehensive summary of grid cell research published in the …
Grid cells in the entorhinal cortex are believed to establish their regular, spatially correlated firing patterns by path integration of the animal's motion. Mechanisms for path integration, e.g. in attractor network models, predict stochastic drift of grid responses, which is not observed experimentally. We demonstrate a biologically plausible mechanism of dynamic self-organization by which border cells, which fire at environmental boundaries, can correct such drift in grid cells. In our model, experience-dependent Hebbian plasticity during exploration allows border cells to learn connectivit…
Many recent models study the downstream projection from grid cells to place cells, while recent data has pointed out the importance of the feedback projection. We thus asked how grid cells are affected by the nature of the input from the place cells. We propose a two-layered neural network with feedforward weights connecting place-like input cells to grid cell outputs. Place-to-grid weights were learned via a generalized Hebbian rule. The architecture of this network highly resembles neural networks used to perform Principal Component Analysis (PCA). Our results indicate that if the components…
Grid cells recorded in the parahippocampal formation of freely moving rodents provide a strikingly periodic representation of self-location whose underlying mechanism has been the subject of intense interest. Our previous work(1) showed that grid cells represent the most stable subset of a larger continuum of spatially periodic cells (SPCs) which deviate from the hexagonal symmetry observed in grid cells. Recently Navratilova et al(2) suggested that our findings reflected poor isolation of the spikes from multiple grid cells, rather than the existence of actual non-grid SPCs. Here we refute th…
Grid cells play a principal role in enabling mammalian cognitive representations of ambient environments. The key property of these cells -- the regular arrangement of their firing fields -- is commonly viewed as means for establishing spatial scales or encoding specific locations. However, using grid cells' spiking outputs for deducing spatial orderliness proves to be a strenuous task, due to fairly irregular activation patterns triggered by the animal's sporadic visits to the grid fields. The following discussion addresses statistical mechanisms enabling emergent regularity of grid cell firi…
Research on network mechanisms and coding properties of grid cells assume that the firing rate of a grid cell in each of its fields is the same. Furthermore, proposed network models predict spatial regularities in the firing of inhibitory interneurons that are inconsistent with experimental data. In this paper, by analyzing the response of grid cells recorded from rats during free navigation, we first show that there are strong variations in the mean firing rate of the fields of individual grid cells and thus show that the data is inconsistent with the theoretical models that predict similar p…
In this paper we explain the strikingly regular activity of the 'grid' cells in rodent dorsal medial entorhinal cortex (dMEC) and the spatially localized activity of the hippocampal place cells in CA3 and CA1 by assuming that the hippocampal region is constructed to support an internal dynamical model of the sensory information. The functioning of the different areas of the hippocampal-entorhinal loop and their interaction are derived from a set of information theoretical principles. We demonstrate through simple transformations of the stimulus representations that the double form of space rep…
Since their Nobel Prize winning discovery in 2005, grid cells have been studied extensively by neuroscientists. Their multi-scale periodic firing rates tiling the environment as the animal moves around has been shown as critical for path integration. Multiple experiments have shown that grid cells also fire for other representations such as olfactory, attention mechanisms, imagined movement, and concept organization potentially acting as a form of neural recycling and showing the possible brain mechanism for cognitive maps that Tolman envisioned in 1948. Grid cell integration into artificial n…
The grid firing patterns are thought to provide an efficient intrinsic metric capable of supporting universal spatial metric for mammalian spatial navigation in all environments. However, whether spatial representations of grid cells in the entorhinal cortex are determined by local environment cues or form globally coherent patterns remains undetermined. To explore this underlying mechanism, here we proposed a possible theoretical explanation to describe connection between the neural space and the physical environment and transition from a local anchored to a global coherent representation acc…
Grid cells in the dorsolateral band of the medial entorhinal cortex(dMEC) display strikingly regular periodic firing patterns on a lattice of positions in 2-D space. This helps animals to encode relative spatial location without reference to external cues. The dMEC is damaged in the early stages of Alzheimer's Disease, which affects navigation ability of a disease victim, reducing the synaptic density of neurons in the network. Within an established 2-dimensional continuous attractor neural network model of grid cell activity, we introduce damage parameterized by radius and by the strength of …
Grid cells in the brain respond when an animal occupies a periodic lattice of "grid fields" during spatial navigation. The grid scale varies along the dorso-ventral axis of the entorhinal cortex. We propose that the grid system minimizes the number of neurons required to encode location with a given resolution. We derive several predictions that match recent experiments: (i) grid scales follow a geometric progression, (ii) the ratio between adjacent grid scales is the square root of e for idealized neurons, and robustly lies in the range 1.4-1.7 for realistic neurons, (iii) the scale ratio var…
Grid cells in the rodent entorhinal cortex display strikingly regular firing responses to the animal's position in 2-D space, and have been hypothesized to form the neural substrate for dead-reckoning. However, in previous models suggested for grid cell activity, errors accumulate rapidly in the integration of velocity inputs. To produce grid-cell like responses, these models would require frequent resets triggered by external sensory cues, casting doubt on the dead-reckoning potential of the grid cell system. Here we focus on the accuracy of path integration in continuous attractor models of …
Grid cells in the entorhinal cortex encode the position of an animal in its environment using spatially periodic tuning curves of varying periodicity. Recent experiments established that these cells are functionally organized in discrete modules with uniform grid spacing. Here we develop a theory for efficient coding of position, which takes into account the temporal statistics of the animal's motion. The theory predicts a sharp decrease of module population sizes with grid spacing, in agreement with the trends seen in the experimental data. We identify a simple scheme for readout of the grid …
Grid cells, discovered more than a decade ago [5], are neurons in the brain of mammals that fire when the animal is located near certain specific points in its familiar terrain. Intriguingly, these points form, for a single cell, a two-dimensional triangular grid, not unlike our Figure 3. Grid cells are widely believed to be involved in path integration, that is, the maintenance of a location state through the summation of small displacements. We provide theoretical evidence for this assertion by showing that cells with grid-like tuning curves are indeed well adapted for the path integration t…
Distance by de-correlation: Computing distance with heterogeneous grid cells
School of Physical Sciences, National Institute of Science Education and Research, India（国家科学教育与研究中心物理科学学院）；Homi Bhaba National Institute, India（霍米·巴巴国立大学）；MathePhysics；NSF AI Institute for Artificial Intelligence and Fundamental Interactions (IAIFI), USA（美国国家科学基金会人工智能基础相互作用研究所（IAIFI））；Center for Doctoral Training in Mathematics of Random Systems: Analysis, Modelling and Algorithms (MRS CDT), Imperial College London & Oxford University, UK（帝国理工学院与牛津大学随机系统数学博士培训中心：分析、建模与算法中心）；Center of Mathematical Sciences and Applications (CMSA), Harvard University, USA（哈佛大学数学科学与工程应用中心）；Department of Electrical and Computer Engineering, Johns Hopkins University, USA（约翰斯·霍普金斯大学电气与计算机工程系）；Data Science and AI Institute, Johns Hopkins University, USA（约翰斯·霍普金斯大学数据科学与人工智能研究所）
University of Arizona（亚利桑那大学）；University of California, Berkeley（加州大学伯克利分校）
Place cells in the hippocampus are active when an animal visits a certain location (referred to as a place field) within an environment. Grid cells in the medial entorhinal cortex (MEC) respond at multiple locations, with firing fields that form a periodic and hexagonal tiling of the environment. The joint activity of grid and place cell populations, as a function of location, forms a neural code for space. An ensemble of codes is generated by varying grid and place cell population parameters. For each code in this ensemble, codewords are generated by stimulating a network with a discrete set …
Decades of research on the neural code underlying spatial navigation have revealed a diverse set of neural response properties. The Entorhinal Cortex (EC) of the mammalian brain contains a rich set of spatial correlates, including grid cells which encode space using tessellating patterns. However, the mechanisms and functional significance of these spatial representations remain largely mysterious. As a new way to understand these neural representations, we trained recurrent neural networks (RNNs) to perform navigation tasks in 2D arenas based on velocity inputs. Surprisingly, we find that gri…
UCLA（加州大学洛杉矶分校）；Google Research（谷歌研究院）；UT Southwestern Medical Center（德克萨斯大学西南医学中心）；UT Austin（德克萨斯大学奥斯汀分校）
The activity of the grid cell population in the medial entorhinal cortex (MEC) of the mammalian brain forms a vector representation of the self-position of the animal. Recurrent neural networks have been proposed to explain the properties of the grid cells by updating the neural activity vector based on the velocity input of the animal. In doing so, the grid cell system effectively performs path integration. In this paper, we investigate the algebraic, geometric, and topological properties of grid cells using recurrent network models. Algebraically, we study the Lie group and Lie algebra of th…
We study the statistics of spike trains of simultaneously recorded grid cells in freely behaving rats. We evaluate pairwise correlations between these cells and, using a generalized linear model (kinetic Ising model), study their functional connectivity. Even when we account for the covariations in firing rates due to overlapping fields, both the pairwise correlations and functional connections decay as a function of the shortest distance between the vertices of the spatial firing pattern of pairs of grid cells, i.e. their phase difference. The functional connectivity takes positive values bet…
Grid-cell firing fields lack local sixfold symmetry
Faculty of Biology, Ludwig-Maximilians-Universität München（慕尼黑大学动物学系）；Graduate School of Systemic Neurosciences (GSN), Ludwig-Maximilians-Universität München（慕尼黑大学系统神经科学研究生院）；Bernstein Center for Computational Neuroscience Munich（慕尼黑贝恩斯坦计算神经科学中心）
UCLA（加州大学洛杉矶分校）；Baidu Research（百度研究院）；UT Austin（德克萨斯大学奥斯汀分校）；Beijing Institute for General Artificial Intelligence (BIGAI)（北京通用人工智能研究院）
Understanding how grid cells perform path integration calculations remains a fundamental problem. In this paper, we conduct theoretical analysis of a general representation model of path integration by grid cells, where the 2D self-position is encoded as a higher dimensional vector, and the 2D self-motion is represented by a general transformation of the vector. We identify two conditions on the transformation. One is a group representation condition that is necessary for path integration. The other is an isotropic scaling condition that ensures locally conformal embedding, so that the error i…
A simple model of co-emergence of grid and place fields
University of Pennsylvania（宾夕法尼亚大学）；Tel Aviv Sourasky Medical Center（特拉维夫索拉希医疗中心）；Gray Faculty of Medical and Health Sciences, Tel Aviv University（特拉维夫大学格雷医学与健康科学学院）；Rappaport Faculty of Medicine, Technion – Israel Institute of Technology（以色列理工学院拉皮波特医学院）；Santa Fe Institute（圣塔菲研究所）
The grid cells (GCs) of the medial entorhinal cortex (MEC) and place cells (PCs) of the hippocampus are key elements of the brain network for the metric representation of space. Currently, any of the existing theoretical models can explain all aspects of GC-specific spatially selective patterns of activity. It is also not clear how these patterns are formed during the network development. On the other hand, it was previously shown that the network that can learn to extract high principal components from the activity of the place cells could provide the basis for GC-like activity patterns devel…
Grid cells in the entorhinal cortex fire when animals that are exploring a certain region of space occupy the vertices of a triangular grid that spans the environment. Different neurons feature triangular grids that differ in their properties of periodicity, orientation and ellipticity. Taken together, these grids allow the animal to maintain an internal, mental representation of physical space. Experiments show that grid cells are modular, i.e. there are groups of neurons which have grids with similar periodicity, orientation and ellipticity. We use statistical physics methods to derive a rel…
What sort of grid cells do we expect to see in bats exploring a three-dimensional environment? How long will it take for them to emerge? We address these questions within our self-organization model based on firing-rate adaptation. The model indicates that the answer to the first question may be simple, and to the second one rather complex. The mathematical analysis of the simplified version of the model points at asymptotic states resembling FCC and HCP crystal structures, which are calculated to be very close to each other in terms of cost function. The simulation of the full model, however,…
Autonomous Underwater Vehicles (AUVs) are platforms used for research and exploration of marine environments. However, these types of vehicles face many challenges that hinder their widespread use in the industry. One of the main limitations is obtaining accurate position estimation, due to the lack of GPS signal underwater. This estimation is usually done with Kalman filters. However, new developments in the neuroscience field have shed light on the mechanisms by which mammals are able to obtain a reliable estimation of their current position based on external and internal motion cues. A new …
Lattices abound in nature - from the crystal structure of minerals to the honey-comb organization of ommatidia in the compound eye of insects. Such regular arrangements provide solutions for optimally dense packings, efficient resource distribution and cryptographic schemes, highlighting the importance of lattice theory in mathematics and physics, biology and economics, and computer science and coding theory. Do lattices also play a role in how the brain represents information? To answer this question, we focus on higher-dimensional stimulus domains, with particular emphasis on neural represen…
The Principle of Isomorphism: A Theory of Population Activity in Grid Cells and Beyond
School of Systems Science, Beijing Normal University（北京师范大学系统科学学院）；Institute of Natural Sciences, Shanghai Jiao Tong University（上海交通大学自然科学研究院）；School of Psychology, Shanghai Jiao Tong University（上海交通大学心理学院）；Chinese Institute for Brain Research, Beijing（北京脑科学与类脑研究中心）；School of Physics and Astronomy, Shanghai Jiao Tong University（上海交通大学物理与天文学院）
Fudan University（复旦大学）；Beijing Normal University（北京师范大学）
The entorhinal-hippocampal circuit plays a critical role in higher brain functions, especially spatial cognition. Grid cells in the medial entorhinal cortex (MEC) periodically fire with different grid spacing and orientation, which makes a contribution that place cells in the hippocampus can uniquely encode locations in an environment. But how sparse firing granule cells in the dentate gyrus are formed from grid cells in the MEC remains to be determined. Recently, the fruit fly olfactory circuit provides a variant algorithm (called locality-sensitive hashing) to solve this problem. To investig…
A Hopfield network is an auto-associative, distributive model of neural memory storage and retrieval. A form of error-correcting code, the Hopfield network can learn a set of patterns as stable points of the network dynamic, and retrieve them from noisy inputs -- thus Hopfield networks are their own decoders. Unlike in coding theory, where the information rate of a good code (in the Shannon sense) is finite but the cost of decoding does not play a role in the rate, the information rate of Hopfield networks trained with state-of-the-art learning algorithms is of the order ${\log(n)}/{n}$, a qua…
Institute of Cognitive Neuroscience, UCL（伦敦大学学院认知神经科学研究所）；Centre for Artificial Intelligence, UCL（伦敦大学学院人工智能中心）；Wellcome Centre for Integrative Neuroimaging, University of Oxford（牛津大学惠康整合神经影像中心）；Sainsbury Wellcome Centre, UCL（伦敦大学学院桑斯伯里惠康中心）
Knowing how the effects of directed actions generalise to new situations (e.g. moving North, South, East and West, or turning left, right, etc.) is key to rapid generalisation across new situations. Markovian tasks can be characterised by a state space and a transition matrix and recent work has proposed that neural grid codes provide an efficient representation of the state space, as eigenvectors of a transition matrix reflecting diffusion across states, that allows efficient prediction of future state distributions. Here we extend the eigenbasis prediction model, utilising tools from Fourier…
