Differences in Visual-Spatial Input May Underlie Different Compression Properties of Firing Fields for Grid Cell Modules in Medial Entorhinal Cortex

Firing fields of grid cells in medial entorhinal cortex show compression or expansion after manipulations of the location of environmental barriers. This compression or expansion could be selective for individual grid cell modules with particular properties of spatial scaling. We present a model for differences in the response of modules to barrier location that arise from different mechanisms for the influence of visual features on the computation of location that drives grid cell firing patterns. These differences could arise from differences in the position of visual features within the visual field. When location was computed from the movement of visual features on the ground plane (optic flow) in the ventral visual field, this resulted in grid cell spatial firing that was not sensitive to barrier location in modules modeled with small spacing between grid cell firing fields. In contrast, when location was computed from static visual features on walls of barriers, i.e. in the more dorsal visual field, this resulted in grid cell spatial firing that compressed or expanded based on the barrier locations in modules modeled with large spacing between grid cell firing fields. This indicates that different grid cell modules might have differential properties for computing location based on visual cues, or the spatial radius of sensitivity to visual cues might differ between modules.     

Modeling boundary vector cell firing given optic flow as a cue

Boundary vector cells in entorhinal cortex fire when a rat is in locations at a specific distance from walls of an environment. This firing may originate from memory of the barrier location combined with path integration, or the firing may depend upon the apparent visual input image stream. The modeling work presented here investigates the role of optic flow, the apparent change of patterns of light on the retina, as input for boundary vector cell firing. Analytical spherical flow is used by a template model to segment walls from the ground, to estimate self-motion and the distance and allocentric direction of walls, and to detect drop-offs. Distance estimates of walls in an empty circular or rectangular box have a mean error of less than or equal to two centimeters. Integrating these estimates into a visually driven boundary vector cell model leads to the firing patterns characteristic for boundary vector cells. This suggests that optic flow can influence the firing of boundary vector cells.     

Evaluation of the Oscillatory Interference Model of Grid Cell Firing through Analysis and Measured Period Variance of Some Biological Oscillators

Models of the hexagonally arrayed spatial activity pattern of grid cell firing in the literature generally fall into two main categories: continuous attractor models or oscillatory interference models. Burak and Fiete (2009, PLoS Comput Biol) recently examined noise in two continuous attractor models, but did not consider oscillatory interference models in detail. Here we analyze an oscillatory interference model to examine the effects of noise on its stability and spatial firing properties. We show analytically that the square of the drift in encoded position due to noise is proportional to time and inversely proportional to the number of oscillators. We also show there is a relatively fixed breakdown point, independent of many parameters of the model, past which noise overwhelms the spatial signal. Based on this result, we show that a pair of oscillators are expected to maintain a stable grid for approximately t = 5µ ³ /(4πσ) ² seconds where µ is the mean period of an oscillator in seconds and σ² its variance in seconds². We apply this criterion to recordings of individual persistent spiking neurons in postsubiculum (dorsal presubiculum) and layers III and V of entorhinal cortex, to subthreshold membrane potential oscillation recordings in layer II stellate cells of medial entorhinal cortex and to values from the literature regarding medial septum theta bursting cells. All oscillators examined have expected stability times far below those seen in experimental recordings of grid cells, suggesting the examined biological oscillators are unfit as a substrate for current implementations of oscillatory interference models. However, oscillatory interference models can tolerate small amounts of noise, suggesting the utility of circuit level effects which might reduce oscillator variability. Further implications for grid cell models are discussed.     

How Entorhinal Grid Cells May Learn Multiple Spatial Scales from a Dorsoventral Gradient of Cell Response Rates in a Self-organizing Map

Place cells in the hippocampus of higher mammals are critical for spatial navigation. Recent modeling clarifies how this may be achieved by how grid cells in the medial entorhinal cortex (MEC) input to place cells. Grid cells exhibit hexagonal grid firing patterns across space in multiple spatial scales along the MEC dorsoventral axis. Signals from grid cells of multiple scales combine adaptively to activate place cells that represent much larger spaces than grid cells. But how do grid cells learn to fire at multiple positions that form a hexagonal grid, and with spatial scales that increase along the dorsoventral axis? In vitro recordings of medial entorhinal layer II stellate cells have revealed subthreshold membrane potential oscillations (MPOs) whose temporal periods, and time constants of excitatory postsynaptic potentials (EPSPs), both increase along this axis. Slower (faster) subthreshold MPOs and slower (faster) EPSPs correlate with larger (smaller) grid spacings and field widths. A self-organizing map neural model explains how the anatomical gradient of grid spatial scales can be learned by cells that respond more slowly along the gradient to their inputs from stripe cells of multiple scales, which perform linear velocity path integration. The model cells also exhibit MPO frequencies that covary with their response rates. The gradient in intrinsic rhythmicity is thus not compelling evidence for oscillatory interference as a mechanism of grid cell firing. A response rate gradient combined with input stripe cells that have normalized receptive fields can reproduce all known spatial and temporal properties of grid cells along the MEC dorsoventral axis. This spatial gradient mechanism is homologous to a gradient mechanism for temporal learning in the lateral entorhinal cortex and its hippocampal projections. Spatial and temporal representations may hereby arise from homologous mechanisms, thereby embodying a mechanistic “neural relativity” that may clarify how episodic memories are learned.     

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 biologically inspired hierarchical goal directed navigation model

We propose an extended version of our previous goal directed navigation model based on forward planning of trajectories in a network of head direction cells, persistent spiking cells, grid cells, and place cells. In our original work the animat incrementally creates a place cell map by random exploration of a novel environment. After the exploration phase, the animat decides on its next movement direction towards a goal by probing linear look-ahead trajectories in several candidate directions while stationary and picking the one activating place cells representing the goal location. In this work we present several improvements over our previous model. We improve the range of linear look-ahead probes significantly by imposing a hierarchical structure on the place cell map consistent with the experimental findings of differences in the firing field size and spacing of grid cells recorded at different positions along the dorsal to ventral axis of entorhinal cortex. The new model represents the environment at different scales by populations of simulated hippocampal place cells with different firing field sizes. Among other advantages this model allows simultaneous constant duration linear look-ahead probes at different scales while significantly extending each probe range. The extension of the linear look-ahead probe range while keeping its duration constant also limits the degrading effects of noise accumulation in the network. We show the extended model’s performance using an animat in a large open field environment.     

Mass correlograms of multiple neuronal activity in the cat's extrastriate cortex

Electrical activity of a population of visually responsive cells located in the vicinity of a single functionally defined neuron was recorded in the area 18 of the cat's cerebral cortex with a single tungsten microelectrode. The correlograms calculated from the mass activity record showed an existence of a rhythmic neuronal firing with an average interval near to 3 ms. When the system was activated by a visual stimulus, a line at an optimal angle moving in an optimal direction, the rhythmic activity became regular, acquiring an oscillatory sinusoidal character. This rhythmic pattern cannot be easily recognized when the activity of a single neuron is recorded. It is possible that such rhythmic activity involving large numbers of neurons contributes to the recognition of the velocity and position of the visual stimulus.     

To keep on track during flight,fruitflies discount the skyward view

When small flying insects go off their intended course, they use the resulting pattern of motion on their eye, or optic flow, to guide corrective steering. A change in heading generates a unique, rotational motion pattern and a change in position generates a translational motion pattern, and each produces corrective responses in the wingbeats. Any image in the flow field can signal rotation, but owing to parallax, only the images of nearby objects can signal translation. Insects that fly near the ground might therefore respond more strongly to translational optic flow that occurs beneath them, as the nearby ground will produce strong optic flow. In these experiments, rigidly tethered fruitflies steered in response to computer-generated flow fields. When correcting for unintended rotations, flies weight the motion in their upper and lower visual fields equally. However, when correcting for unintended translations, flies weight the motion in the lower visual fields more strongly. These results are consistent with the interpretation that fruitflies stabilize by attending to visual areas likely to contain the strongest signals during natural flight conditions.     

Continuous Attractor Network Model for Conjunctive Position-by-Velocity Tuning of Grid Cells

The spatial responses of many of the cells recorded in layer II of rodent medial entorhinal cortex (MEC) show a triangular grid pattern, which appears to provide an accurate population code for animal spatial position. In layer III, V and VI of the rat MEC, grid cells are also selective to head-direction and are modulated by the speed of the animal. Several putative mechanisms of grid-like maps were proposed, including attractor network dynamics, interactions with theta oscillations or single-unit mechanisms such as firing rate adaptation. In this paper, we present a new attractor network model that accounts for the conjunctive position-by-velocity selectivity of grid cells. Our network model is able to perform robust path integration even when the recurrent connections are subject to random perturbations.     

The optokinetic response in wild type and white zebra finches

Optic flow is a main source of information about self movement and the three-dimensional composition of the environment during locomotion. It is processed by the accessory optic system in all vertebrates. The optokinetic response is elicited by rotational optic flow, e.g. in a rotating drum lined with vertical stripes. We investigated here the effect of rotational optic flow on the optokinetic response in wild type and white zebra finches. The highest stimulus velocity eliciting an optokinetic response (upper velocity threshold) was dependent on stimulus direction and illumination level, but was not different between the colour morphs. The upper velocity threshold was higher with temporal to nasal movements in monocularly exposed birds and symmetrical with binocular exposure. Its increase with illumination level followed Fechner's law and reached a plateau at about 560 Lux. In bright daylight, white birds did not show optokinetic responses. We conclude that the altered wiring of the visual system of white birds has no influence on accessory optic system function. The unwillingness of white birds to respond with optokinetic response in bright daylight may be due to a substantial lack of inhibition within the visual system as demonstrated earlier, which may enhance the sensibility to glare.     

Modeling the impact of common noise inputs on the network activity of retinal ganglion cells

Synchronized spontaneous firing among retinal ganglion cells (RGCs), on timescales faster than visual responses, has been reported in many studies. Two candidate mechanisms of synchronized firing include direct coupling and shared noisy inputs. In neighboring parasol cells of primate retina, which exhibit rapid synchronized firing that has been studied extensively, recent experimental work indicates that direct electrical or synaptic coupling is weak, but shared synaptic input in the absence of modulated stimuli is strong. However, previous modeling efforts have not accounted for this aspect of firing in the parasol cell population. Here we develop a new model that incorporates the effects of common noise, and apply it to analyze the light responses and synchronized firing of a large, densely-sampled network of over 250 simultaneously recorded parasol cells. We use a generalized linear model in which the spike rate in each cell is determined by the linear combination of the spatio-temporally filtered visual input, the temporally filtered prior spikes of that cell, and unobserved sources representing common noise. The model accurately captures the statistical structure of the spike trains and the encoding of the visual stimulus, without the direct coupling assumption present in previous modeling work. Finally, we examined the problem of decoding the visual stimulus from the spike train given the estimated parameters. The common-noise model produces Bayesian decoding performance as accurate as that of a model with direct coupling, but with significantly more robustness to spike timing perturbations.     

Whole-field integration, not detailed analysis, is used by the crab optokinetic system to separate rotation and translation in optic flow

For optimal visual control of compensatory eye movements during locomotion it is necessary to distinguish the rotational and translational components of the optic flow field. Optokinetic eye movements can reduce the rotational component only, making the information contained in the translational flow readily available to the animal. We investigated optokinetic eye rotation in the marble rock crab, Pachygrapsus marmoratus, during translational movement, either by displacing the animal or its visual surroundings. Any eye movement in response to such stimuli is taken as an indication that the system is unable to separate the translational and the rotational components in the optic flow in a mathematically perfect way. When the crabs are translated within a pseudo-natural environment, eye movements are negligible, especially during sideways translation. When, however, crabs were placed in a gangway between two elongated rectangular sidewalls carrying dotted patterns which were translated back and forth, marked eye movements were elicited, depending on the translational velocity. To resolve this discrepancy, we tested several hypotheses about mechanisms using detailed analysis of the optic flow or whole-field integration. We found that the latter are sufficient to explain the efficient separation of translation and rotation of crabs in quasi-natural situations. Accepted: 6 May 1997     

Spectral Characteristics of Phase Sensitivity and Discharge Rate of Neurons in the Ascending Tectofugal Visual System

Drifting gratings can modulate the activity of visual neurons at the temporal frequency of the stimulus. In order to characterize the temporal frequency modulation in the cat’s ascending tectofugal visual system, we recorded the activity of single neurons in the superior colliculus, the suprageniculate nucleus, and the anterior ectosylvian cortex during visual stimulation with drifting sine-wave gratings. In response to such stimuli, neurons in each structure showed an increase in firing rate and/or oscillatory modulated firing at the temporal frequency of the stimulus (phase sensitivity). To obtain a more complete characterization of the neural responses in spatiotemporal frequency domain, we analyzed the mean firing rate and the strength of the oscillatory modulations measured by the standardized Fourier component of the response at the temporal frequency of the stimulus. We show that the spatiotemporal stimulus parameters that elicit maximal oscillations often differ from those that elicit a maximal discharge rate. Furthermore, the temporal modulation and discharge-rate spectral receptive fields often do not overlap, suggesting that the detection range for visual stimuli provided jointly by modulated and unmodulated response components is larger than the range provided by a one response component.     

Optic flow-based collision-free strategies: From insects to robots

Flying insects are able to fly smartly in an unpredictable environment. It has been found that flying insects have smart neurons inside their tiny brains that are sensitive to visual motion also called optic flow. Consequently, flying insects rely mainly on visual motion during their flight maneuvers such as: takeoff or landing, terrain following, tunnel crossing, lateral and frontal obstacle avoidance, and adjusting flight speed in a cluttered environment. Optic flow can be defined as the vector field of the apparent motion of objects, surfaces, and edges in a visual scene generated by the relative motion between an observer (an eye or a camera) and the scene. Translational optic flow is particularly interesting for short-range navigation because it depends on the ratio between (i) the relative linear speed of the visual scene with respect to the observer and (ii) the distance of the observer from obstacles in the surrounding environment without any direct measurement of either speed or distance. In flying insects, roll stabilization reflex and yaw saccades attenuate any rotation at the eye level in roll and yaw respectively (i.e. to cancel any rotational optic flow) in order to ensure pure translational optic flow between two successive saccades. Our survey focuses on feedback-loops which use the translational optic flow that insects employ for collision-free navigation. Optic flow is likely, over the next decade to be one of the most important visual cues that can explain flying insects' behaviors for short-range navigation maneuvers in complex tunnels. Conversely, the biorobotic approach can therefore help to develop innovative flight control systems for flying robots with the aim of mimicking flying insects’ abilities and better understanding their flight.     

Bistable,Irregular Firing and Population Oscillations in a Modular Attractor Memory Network

Attractor neural networks are thought to underlie working memory functions in the cerebral cortex. Several such models have been proposed that successfully reproduce firing properties of neurons recorded from monkeys performing working memory tasks. However, the regular temporal structure of spike trains in these models is often incompatible with experimental data. Here, we show that the in vivo observations of bistable activity with irregular firing at the single cell level can be achieved in a large-scale network model with a modular structure in terms of several connected hypercolumns. Despite high irregularity of individual spike trains, the model shows population oscillations in the beta and gamma band in ground and active states, respectively. Irregular firing typically emerges in a high-conductance regime of balanced excitation and inhibition. Population oscillations can produce such a regime, but in previous models only a non-coding ground state was oscillatory. Due to the modular structure of our network, the oscillatory and irregular firing was maintained also in the active state without fine-tuning. Our model provides a novel mechanistic view of how irregular firing emerges in cortical populations as they go from beta to gamma oscillations during memory retrieval.     

Alternating predictive and short-term memory modes of entorhinal grid cells

Several lines of evidence indicate that the entorhinal cortex has memory functions, but such functions have not been previously found in grid cells, a cell type that provides major input to the hippocampus. We examined the firing of grid cells as rats crossed (runs) through grid cell vertices. We found that on some runs, firing tended to occur mostly inbound as the rat approached a vertex center while on other runs firing occurred mainly outbound. These results suggest that cells have a predictive mode (inbound firing) in which they represent a position ahead of the animal and a short term memory (STM) mode (outbound firing) in which they represent positions just passed through. Analysis of cell pairs showed that when vertex crossings were less than 1 second apart, the two cells tended to have the same mode. This indicates that modes are a network property. The tendency to have the same mode disappeared if crossings were separated by 2-3 seconds, suggesting that modes alternate on the time scale of seconds. There was a small but statistically significant behavioral correlate of modes: velocity was slightly less in the STM mode. Both modes were organized by theta and gamma oscillations. The results suggest that the dual requirement for hippocampal storage and recall is met by rapidly alternating modes appropriate for predicting the future and storing the recent past.     

Spiking Neurons in a Hierarchical Self-Organizing Map Model Can Learn to Develop Spatial and Temporal Properties of Entorhinal Grid Cells and Hippocampal Place Cells

Medial entorhinal grid cells and hippocampal place cells provide neural correlates of spatial representation in the brain. A place cell typically fires whenever an animal is present in one or more spatial regions, or places, of an environment. A grid cell typically fires in multiple spatial regions that form a regular hexagonal grid structure extending throughout the environment. Different grid and place cells prefer spatially offset regions, with their firing fields increasing in size along the dorsoventral axes of the medial entorhinal cortex and hippocampus. The spacing between neighboring fields for a grid cell also increases along the dorsoventral axis. This article presents a neural model whose spiking neurons operate in a hierarchy of self-organizing maps, each obeying the same laws. This spiking GridPlaceMap model simulates how grid cells and place cells may develop. It responds to realistic rat navigational trajectories by learning 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 these cells and their development in juvenile rats. The place cells represent much larger spaces than the grid cells, which enable them to support navigational behaviors. Both self-organizing maps amplify and learn to categorize the most frequent and energetic co-occurrences of their inputs. The current results build upon a previous rate-based model of grid and place cell learning, and thus illustrate a general method for converting rate-based adaptive neural models, without the loss of any of their analog properties, into models whose cells obey spiking dynamics. New properties of the spiking GridPlaceMap model include the appearance of theta band modulation. The spiking model also opens a path for implementation in brain-emulating nanochips comprised of networks of noisy spiking neurons with multiple-level adaptive weights for controlling autonomous adaptive robots capable of spatial navigation.     

Correlations and Functional Connections in a Population of Grid Cells

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 maximum entropy kinetic pairwise 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. They take positive values between cells with nearby phases and approach zero or negative values for larger phase differences. We find similar results also when, in addition to correlations due to overlapping fields, we account for correlations due to theta oscillations and head directional inputs. The inferred connections between neurons in the same module and those from different modules can be both negative and positive, with a mean close to zero, but with the strongest inferred connections found between cells of the same module. Taken together, our results suggest that grid cells in the same module do indeed form a local network of interconnected neurons with a functional connectivity that supports a role for attractor dynamics in the generation of grid pattern.     

Patterns in the discharge of simple and complex visual cortical cells

The activity of visual cortical neurons (area 17) was recorded in anaesthetized cats in response to sinusoidal drifting gratings. The statistical structure of the discharge of simple and complex cells has been studied as a function of the various parameters of a drifting grating: spatial frequency, orientation, drifting velocity and contrast. For simple cells it has been found that the interspike interval distributions in response to drifting gratings of various spatial frequencies differ only by a time scale factor. They can be reduced to a unique distribution by a linear time transformation. Variations in the spatial frequency of the grating induce variations in the mean firing rate of the cell but leave unchanged the statistical structure of the discharge. On the contrary, the statistical structure of the simple cell activity changes when the contrast or the velocity of the stimulus is varied. For complex cells it has been found that the invariance property described above for simple cells is not valid. Complex cells present in their activity in response to visual stimuli two different firing patterns: spikes organized in clusters and spikes that do not show this organization ('isolated spikes'). The clustered component is the only component of the complex cell discharge that is tuned for spatial frequency and orientation, while the isolated spike component is correlated with the contrast of the stimulus.     

Accurate Path Integration in Continuous Attractor Network Models of Grid Cells

Grid cells in the rat 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, errors accumulate rapidly when velocity inputs are integrated in existing models of grid cell activity. To produce grid-cell-like responses, these models would require frequent resets triggered by external sensory cues. Such inadequacies, shared by various models, cast doubt on the dead-reckoning potential of the grid cell system. Here we focus on the question of accurate path integration, specifically in continuous attractor models of grid cell activity. We show, in contrast to previous models, that continuous attractor models can generate regular triangular grid responses, based on inputs that encode only the rat''s velocity and heading direction. We consider the role of the network boundary in the integration performance of the network and show that both periodic and aperiodic networks are capable of accurate path integration, despite important differences in their attractor manifolds. We quantify the rate at which errors in the velocity integration accumulate as a function of network size and intrinsic noise within the network. With a plausible range of parameters and the inclusion of spike variability, our model networks can accurately integrate velocity inputs over a maximum of ∼10–100 meters and ∼1–10 minutes. These findings form a proof-of-concept that continuous attractor dynamics may underlie velocity integration in the dorsolateral medial entorhinal cortex. The simulations also generate pertinent upper bounds on the accuracy of integration that may be achieved by continuous attractor dynamics in the grid cell network. We suggest experiments to test the continuous attractor model and differentiate it from models in which single cells establish their responses independently of each other.