Grid Cell Model Research Articles

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

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

Hippocampal theta oscillations are slower in humans than in rodents: implications for models of spatial navigation and memory

The theta oscillation is a neuroscience enigma. When a rat runs through an environment, large-amplitude theta oscillations (4-10 Hz) reliably appear in the hippocampus's electrical activity. The consistency of this pattern led to theta playing a central role in theories on the neural basis of mammalian spatial navigation and memory. However, in fact, hippocampal oscillations at 4-10 Hz are rare in humans and in some other species. This presents a challenge for theories proposing theta as an essential component of the mammalian brain, including models of place and grid cells. Here, I examine this issue by reviewing recent research on human hippocampal oscillations using direct brain recordings from neurosurgical patients. This work indicates that the human hippocampus does indeed exhibit rhythms that are functionally similar to theta oscillations found in rodents, but that these signals have a slower frequency of approximately 1-4 Hz. I argue that oscillatory models of navigation and memory derived from rodent data are relevant for humans, but that they should be modified to account for the slower frequency of the human theta rhythm.

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

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

A model for greed cells in 3-D environments

Individual medial entorhinal cortex (mEC) ‘grid’ cells provide a representation of space that appears to be essentially invariant across environments, modulo simple transformations, in contrast to multiple, rapidly acquired hippocampal maps; it may therefore be established gradually, during rodent development. lf this is the case, then the topology of the environment in which the development takes place should affect the way the grid final configuration appears. Until now, models of grid cells have dealt only with planar, two-dimensional topologies. We extend our single-cell adaptation model [1] to include the third dimension in the environment. We study two non-planar topologies: the sphere surface [2] the fully three-dimensional space. What grid cell firing maps would we expect to observe? In the first condition the model predicts a sequence of spherical harmonics, each of which is the optimal asymptotic solution for the self-organizing adaptation process, within a certain range of the world radius. In the second, it predicts that the grid fields should assume a configuration analogous to an hexagonal close-packing lattice. New experiments that make use of virtual reality on a revolving sphere, or in which rats are reared inside a spherical cage, will likely be a direct test for our model on the sphere, while experiments with flying bats will provide evidence on the feasibility of a genuine 3-dimensional representation of space in terms of grid units.

Impact of noise on theta-nested gamma oscillations in a spiking continuous attractor model of grid cells

Grid cells in the medial entorhinal cortex (MEC) encode location through firing fields that form grid-like maps of the environment. At the same time network activity in the MEC is dominated by oscillations in the theta (4-12 Hz) and gamma (30-100 Hz) bands. Our recent experimental data established that feedback inhibition between excitatory stellate cells and inhibitory fast spiking interneurons dominates the synaptic connectivity in layer II of the MEC, and that continuous attractor models derived from these properties are sufficient to explain both the network oscillations and grid firing fields [1]. Here we use a spiking continuous attractor network model of layer II stellate cells and fast spiking interneurons to examine the effect of intrinsic noise on the emergence of theta nested-gamma oscillations. In this model, stellate cells connect exclusively to interneurons, while interneurons contact only stellate cells. When driven with a theta (8 Hz) modulated external input, we observed network synchronization in the gamma range during the trough of the theta signal, which depended on the amount of intrinsic Gaussian noise generated independently at each neuron. Without noise, the network exhibited aberrant hyper-synchronous firing of a subpopulation of stellate cells at the beginning of each theta cycle. The stellate cell firing resulted in a strong feedback inhibitory postsynaptic currents from fast spiking interneurons that had an additional synchronizing effect. The strong inhibitory feedback abolished nested gamma oscillations. With increasing noise, the network entered an intermediate state in which we observed theta nested-gamma oscillations [1]. When noise level was set even higher, the nested gamma oscillations disappeared without increasing the amount of inhibitory coupling. We also present the impact of noise on bump attractor formation and gridness score of the grid-like receptive fields. Together, these results show that the presence of noise may have profound implications on the stability of theta-nested gamma oscillations in attractor neural circuits with feedback inhibition.

The influence of multiple firing events on the formation and stability of activity patterns in continuous attractor networks

Continuous attractor networks have been proposed to explain a variety of phenomena, including working memory and rodent entorhinal grid cells [1,2]. Typically, such networks consist of spatially-structured lattices of neurons in one or two dimensions with long-range inhibition and short-range excitation, which causes the network activity to spontaneously self-organize into one or more bumps of persistent activity. In two-dimensions, the activity patterns often take the form of triangular lattices, which bear a striking resemblance to the spatial patterns observed in recordings of entorhinal grid cells [3]. Rate-based neural field models of attractor networks predict the emergence of stable, stationary patterns [2,4], whereas simulations of integrate and fire neurons suggest a more complicated dynamical picture. In particular, synaptic timescales appear to play a crucial role [5,6] in that fast synapses tend to lead to transient, local synchronous activity, thus destabilizing activity patterns. Recent work on a model of V1 suggests that such firing events (MFEs) may be a generic, emergent feature of spiking networks [7]. In this work, we investigate whether and how MFEs may affect pattern stability in continuous attractor networks. While the stability of spiking neuron-based attractor networks have been studied previously [5,6], the models studied to date contain neither seperate excitatory and inhibitory populations nor a mixture of synaptic timescales (as would be expected in more realistic settings), and the effects of these features on pattern stability are not known. To investigate these questions, we have implemented continuous attractor networks in both one and two dimensions using spiking, integrate-and-fire neurons with conductance-based synapses and seperate excitatory and inhibitory populations. Through simulations, we systematically assess the effect of MFEs on the formation and stability of spatiotemporal activity patterns (bumps and grids). We also investigate the effects of multiple synaptic timescales, noise, and network heterogeneity on the stability of these patterns. An overarching goal of this work is to obtain insights into how real neural systems might maintain stable persistent activity states, such as those needed for accurately integrating sensorimotor information. Furthermore, as 2-D attractor networks have been incorporated into a number of recent grid cell models [2,8], and the stability of grid cell activity may have a significant effect on place fields [9], understanding the stability of the grid patterns formed by these networks is highly relevant to studies of the rat spatial navigation system.

Membrane potential dynamics of grid cells

During navigation, grid cells increase their spike rates in firing fields arranged on a strikingly regular triangular lattice, while their spike timing is often modulated by theta oscillations. Oscillatory interference models of grid cells predict theta amplitude modulations of membrane potential during firing field traversals, while competing attractor network models predict slow depolarizing ramps. Here, using in-vivo whole-cell recordings, we tested these models by directly measuring grid cell intracellular potentials in mice running along linear tracks in virtual reality. Grid cells had large and reproducible ramps of membrane potential depolarization that were the characteristic signature tightly correlated with firing fields. Grid cells also exhibited intracellular theta oscillations that influenced their spike timing. However, the properties of theta amplitude modulations were not consistent with the view that they determine firing field locations. Our results support cellular and network mechanisms in which grid fields are produced by slow ramps, as in attractor models, while theta oscillations control spike timing.

How reduction of theta rhythm by medial septum inactivation may covary with disruption of entorhinal grid cell responses due to reduced cholinergic transmission

Oscillations in the coordinated firing of brain neurons have been proposed to play important roles in perception, cognition, attention, learning, navigation, and sensory-motor control. The network theta rhythm has been associated with properties of spatial navigation, as has the firing of entorhinal grid cells and hippocampal place cells. Two recent studies reduced the theta rhythm by inactivating the medial septum (MS) and demonstrated a correlated reduction in the characteristic hexagonal spatial firing patterns of grid cells. These results, along with properties of intrinsic membrane potential oscillations (MPOs) in slice preparations of medial entorhinal cortex (MEC), have been interpreted to support oscillatory interference models of grid cell firing. The current article shows that an alternative self-organizing map (SOM) model of grid cells can explain these data about intrinsic and network oscillations without invoking oscillatory interference. In particular, the adverse effects of MS inactivation on grid cells can be understood in terms of how the concomitant reduction in cholinergic inputs may increase the conductances of leak potassium (K+) and slow and medium after-hyperpolarization (sAHP and mAHP) channels. This alternative model can also explain data that are problematic for oscillatory interference models, including how knockout of the HCN1 gene in mice, which flattens the dorsoventral gradient in MPO frequency and resonance frequency, does not affect the development of the grid cell dorsoventral gradient of spatial scales, and how hexagonal grid firing fields in bats can occur even in the absence of theta band modulation. These results demonstrate how models of grid cell self-organization can provide new insights into the relationship between brain learning and oscillatory dynamics.

Influence of Heterogeneities and Upscaling on CO2 Storage Prediction at Large Scale in Deep Saline Aquifer

What are the effects of spatial variability of permeability on CO2 migration and pressure response to CO2 injection in saline aquifer at industrial scale? Since the extension and intensity of perturbations will be constrained by the geological characteristics of the aquifer, spatial variability will be a key parameter for injectivity and capacity of storage projects. Predictions at regional or basin scale implies the study of geological model of millions grid cells. But for efficient simulations of multiphase flow, the number of grid cells need to be reduced. The purpose of this work is also to evaluate homogenization or simplification methods of permeability field that could be sufficiently accurate for CO2 storage predictions.

Models of grid cells and theta oscillations

Arising from M. M.Yartsev, M. P. Witter & N. Ulanovsky , 103–107 (2011)10.1038/nature10583 Grid cells recorded in the medial entorhinal cortex (MEC) of freely moving rodents show a markedly regular spatial firing pattern whose underlying mechanism has been the subject of intense interest. Yartsev et al.1 report that the firing of grid cells in crawling bats does not show theta rhythmicity “causally disproving a major class of computational models” of grid cell firing that rely on oscillatory interference2,3,4,5,6,7. However, their data may be consistent with these models, with the apparent lack of theta rhythmicity reflecting slow movement speeds and low firing rates. Thus, the conclusion of Yartsev et al. is not supported by their data.

Yartsev et al. reply

Replying to C. Barry, D. Bush, J. O’Keefe & N. Burgess , 10.1038/nature11276 (2012) Barry et al.1 propose that it is impossible to detect theta rhythmicity in bat grid cells because of their slow movement velocities and low firing rates; hence, they posit that our findings2 do not refute the oscillatory interference models of mammalian grid cells. To support this claim, they use a data set of rat grid cells of which only 58% were theta modulated, and constrained their analysis to periods of near immobility in the rat, a behavioural state in which theta is known to be absent3. Despite these biases, we argue that their own analysis showed that down-sampled rat cells were substantially more theta-modulated than real grid cells from bats, and we demonstrate further that the bat data have adequate statistical power to detect theta rhythmicity—if it was present in bat grid cells. Finally, Barry et al. focused solely on ‘first generation’ oscillatory interference models, ignoring our disproval of ‘second generation’ models. We thus uphold our original results and interpretation2.

A goal‐directed spatial navigation model using forward trajectory planning based on grid cells

A goal-directed navigation model is proposed based on forward linear look-ahead probe of trajectories in a network of head direction cells, grid cells, place cells and prefrontal cortex (PFC) cells. The model allows selection of new goal-directed trajectories. In a novel environment, the virtual rat incrementally creates a map composed of place cells and PFC cells by random exploration. After exploration, the rat retrieves memory of the goal location, picks its next movement direction by forward linear look-ahead probe of trajectories in several candidate directions while stationary in one location, and finds the one activating PFC cells with the highest reward signal. Each probe direction involves activation of a static pattern of head direction cells to drive an interference model of grid cells to update their phases in a specific direction. The updating of grid cell spiking drives place cells along the probed look-ahead trajectory similar to the forward replay during waking seen in place cell recordings. Directions are probed until the look-ahead trajectory activates the reward signal and the corresponding direction is used to guide goal-finding behavior. We report simulation results in several mazes with and without barriers. Navigation with barriers requires a PFC map topology based on the temporal vicinity of visited place cells and a reward signal diffusion process. The interaction of the forward linear look-ahead trajectory probes with the reward diffusion allows discovery of never-before experienced shortcuts towards a goal location.

AutonomousWalking over Obstacles by Means of LRF for Hexapod Robot COMET-IV

This paper presents an autonomous navigation system for a hydraulically driven hexapod robot (COMETIV) based on point cloud data acquired using a rotating Laser Range Finder (LRF). The size of the robot would prohibit its movement in a stochastic terrain environment if we only consider letting it avoid obstacles. However, the robot has a unique ability to walk over obstacles. We thus proposed the so-called Grid-based Walking Trajectory for Legged Robot (GWTLR) method. The method is developed on the basis of the geometric representation of a stochastic terrain in terms of grid cell characteristics. We also introduced the “Grid-cell model for COMET-IV” to assess the characteristics of the grid cells and to determine whether each of the cells is traversable or not. Finally, the shortest safe walking trajectory is generated using a search algorithm, A*. The performance of the proposed method is verified by the experimental results of the successful determination of a walking trajectory path and by completely walking over obstacles in various arrangements.

A Model Combining Oscillations and Attractor Dynamics for Generation of Grid Cell Firing

Different models have been able to account for different features of the data on grid cell firing properties, including the relationship of grid cells to cellular properties and network oscillations. This paper describes a model that combines elements of two major classes of models of grid cells: models using interactions of oscillations and models using attractor dynamics. This model includes a population of units with oscillatory input representing input from the medial septum. These units are termed heading angle cells because their connectivity depends upon heading angle in the environment as well as the spatial phase coded by the cell. These cells project to a population of grid cells. The sum of the heading angle input results in standing waves of circularly symmetric input to the grid cell population. Feedback from the grid cell population increases the activity of subsets of the heading angle cells, resulting in the network settling into activity patterns that resemble the patterns of firing fields in a population of grid cells. The properties of heading angle cells firing as conjunctive grid-by-head-direction cells can shift the grid cell firing according to movement velocity. The pattern of interaction of oscillations requires use of separate populations that fire on alternate cycles of the net theta rhythmic input to grid cells.

Models of Grid Cell Spatial Firing Published 2005–2011

Since the discovery of grid cells in rat entorhinal cortex, many models of their hexagonally arrayed spatial firing fields have been suggested. We review the models and organize them according to the mechanisms they use to encode position, update the positional code, read it out in the spatial grid pattern, and learn any patterned synaptic connections needed. We mention biological implementations of the models, but focus on the models on Marr’s algorithmic level, where they are not things to individually prove or disprove, but rather are a valuable collection of metaphors of the grid cell system for guiding research that are all likely true to some degree, with each simply emphasizing different aspects of the system. For the convenience of interested researchers, MATLAB implementations of the discussed grid cell models are provided at ModelDB accession 144006 or http://people.bu.edu/zilli/gridmodels.html.

Contribution of Ih to LTP, Place Cells, and Grid Cells

The brain's grid and place cells, which contribute to spatial representations of the external environment, are thought to be modulated by the hyperpolarization-activated cation current (I(h)). Giocomo et al. and Hussaini et al. now provide new insights into these cells' unique activity patterns by studying transgenic mice lacking I(h).

Grid cells without theta oscillations in the entorhinal cortex of bats

Grid cells provide a neural representation of space, by discharging when an animal traverses through the vertices of a periodic hexagonal grid spanning the environment. Although grid cells have been characterized in detail in rats, the fundamental question of what neural dynamics give rise to the grid structure remains unresolved. Two competing classes of models were proposed: network models, based on attractor dynamics, and oscillatory interference models, which propose that interference between somatic and dendritic theta-band oscillations (4-10 Hz) in single neurons transforms a temporal oscillation into a spatially periodic grid. So far, these models could not be dissociated experimentally, because rodent grid cells always co-exist with continuous theta oscillations. Here we used a novel animal model, the Egyptian fruit bat, to refute the proposed causal link between grids and theta oscillations. On the basis of our previous finding from bat hippocampus, of spatially tuned place cells in the absence of continuous theta oscillations, we hypothesized that grid cells in bat medial entorhinal cortex might also exist without theta oscillations. Indeed, we found grid cells in bat medial entorhinal cortex that shared remarkable similarities to rodent grid cells. Notably, the grids existed in the absence of continuous theta-band oscillations, and with almost no theta modulation of grid-cell spiking--both of which are essential prerequisites of the oscillatory interference models. Our results provide a direct demonstration of grid cells in a non-rodent species. Furthermore, they strongly argue against a major class of computational models of grid cells.

Phase precession and variable spatial scaling in a periodic attractor map model of medial entorhinal grid cells with realistic after‐spike dynamics

We present a model that describes the generation of the spatial (grid fields) and temporal (phase precession) properties of medial entorhinal cortical (MEC) neurons by combining network and intrinsic cellular properties. The model incorporates network architecture derived from earlier attractor map models, and is implemented in 1D for simplicity. Periodic driving of conjunctive (position × head-direction) layer-III MEC cells at theta frequency with intensity proportional to the rat's speed, moves an 'activity bump' forward in network space at a corresponding speed. The addition of prolonged excitatory currents and simple after-spike dynamics resembling those observed in MEC stellate cells (for which new data are presented) accounts for both phase precession and the change in scale of grid fields along the dorso-ventral axis of MEC. Phase precession in the model depends on both synaptic connectivity and intrinsic currents, each of which drive neural spiking either during entry into, or during exit out of a grid field. Thus, the model predicts that the slope of phase precession changes between entry into and exit out of the field. The model also exhibits independent variation in grid spatial period and grid field size, which suggests possible experimental tests of the model.

Simulation Model for Urgent Evacuation at Aircraft Accident

In this paper, the urgent evacuation behavior of passengers at aircraft accident is formulated as a simulation model which consists of two models such as a two-dimensional grid cell model and an autonomous and multiple agents' system model. The former model represents the layout of cabin interior and the allocation of passengers. On the other hand, the latter shows the behavior of evacuation of passengers from inside of a cabin to outside of an aircraft using emergency equipments such as an emergency exit and escaping slide. This autonomous agent is able to obtain the necessary information for urgent evacuation within one's field of view and decide to select the escaping route by means of obtained information concerned with escaping path to the nearest emergency exit. By the comparable study between the results of the proposed simulation and the accident analysis reports on “Garuda Indonesia Airways Accident on 1996” and “China Airline Accident on 2007 ,” it is verified that the proposed simulation system enable one to estimate the situation of urgent evacuation in the aircraft accident.

Stationary envelope synthesis (SES): A universal method for phase coding by neural oscillators

Stationary envelope synthesis (SES): A universal method for phase coding by neural oscillators