Oscillatory Interference Models Research Articles

Self-Location Based on Grid-like Representations for Artificial Agents

Self-location plays a crucial role in a framework of autonomous navigation, especially in a GNSS/radio-denied environment. At the current time, self-location for artificial agents still has to resort to the visual and laser technologies in the framework of deep neural networks, which cannot model the environments effectively, especially in some dynamic and complex scenes. Instead, researchers have attempted to transplant the navigation principle of mammals into artificial intelligence (AI) fields. As a kind of mammalian neuron, the grid cells are believed to provide a context-independent spatial metric and update the representation of self-location. By exploiting the mechanism of grid cells, we adopt the oscillatory interference model for location encoding. Furthermore, in the process of location decoding, the capacity of autonomous navigation is extended to a significantly wide range without the phase ambiguity, based on a multi-scale periodic representation mechanism supported by a step-wise phase unwrapping algorithm. Compared with the previous methods, the proposed grid-like self-location can achieve a much wider spatial range without the limitation imposed by the spatial scales of grid cells. It is also able to suppress the phase noise efficiently. The proposed method is validated by simulation results.

Neuromorphic model of hippocampus place cells using an oscillatory interference technique for hardware implementation

In this paper, we propose a simplified and robust model for place cell generation based on the oscillatory interference (OI) model concept. Aiming toward hardware implementation in bio-inspired simultaneous localization and mapping (SLAM) systems for mobile robotics, we base our model on logic operations that reduce its computational complexity. The model compensates for parameter variations in the behaviors of the population of constituent theta cells, and allows the theta cells to have square-wave oscillation profiles. The robustness of the model, with respect to mismatch in the theta cell’s base oscillation frequency and gain—as a function of modulatory inputs—is demonstrated. Place cell composed of 48 theta cells with base frequency variations with a 25% standard deviation from the mean and a gain error with 20% standard deviation from the mean only result in a 20% deformations within the place field and 0.24% outer side lobes, and an overall pattern with 0.0015 mean squared error on average. We also present how the model can be used to achieve the localization and path-tracking functionalities of SLAM. Hence, we propose a model for spatial cell formation using theta cells with behaviors that are biologically plausible and hardware implementable for real world application in neurally-inspired SLAM.

Modeling the grid cell activity on non-horizontal surfaces based on oscillatory interference modulated by gravity

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.

Grid cells are modulated by local head direction

Grid and head direction codes represent cognitive spaces for navigation and memory. Pure grid cells generate grid codes that have been assumed to be independent of head direction, whereas conjunctive cells generate grid representations that are tuned to a single head direction. Here, we demonstrate that pure grid cells also encode head direction, but through distinct mechanisms. We show that individual firing fields of pure grid cells are tuned to multiple head directions, with the preferred sets of directions differing between fields. This local directional modulation is not predicted by previous continuous attractor or oscillatory interference models of grid firing but is accounted for by models in which pure grid cells integrate inputs from co-aligned conjunctive cells with firing rates that differ between their fields. We suggest that local directional signals from grid cells may contribute to downstream computations by decorrelating different points of view from the same location.

The Art of Grid Fields: Geometry of Neuronal Time.

The discovery of grid cells in the entorhinal cortex has both elucidated our understanding of spatial representations in the brain, and germinated a large number of theoretical models regarding the mechanisms of these cells’ striking spatial firing characteristics. These models cross multiple neurobiological levels that include intrinsic membrane resonance, dendritic integration, after hyperpolarization characteristics and attractor dynamics. Despite the breadth of the models, to our knowledge, parallels can be drawn between grid fields and other temporal dynamics observed in nature, much of which was described by Art Winfree and colleagues long before the initial description of grid fields. Using theoretical and mathematical investigations of oscillators, in a wide array of mediums far from the neurobiology of grid cells, Art Winfree has provided a substantial amount of research with significant and profound similarities. These theories provide specific inferences into the biological mechanisms and extraordinary resemblances across phenomenon. Therefore, this manuscript provides a novel interpretation on the phenomenon of grid fields, from the perspective of coupled oscillators, postulating that grid fields are the spatial representation of phase resetting curves in the brain. In contrast to prior models of gird cells, the current manuscript provides a sketch by which a small network of neurons, each with oscillatory components can operate to form grid cells, perhaps providing a unique hybrid between the competing attractor neural network and oscillatory interference models. The intention of this new interpretation of the data is to encourage novel testable hypotheses.

The sensori-motor model of the hippocampal place cells

The hippocampal formation contains the head direction cells, the grid cells and the place cells which work as an internal GPS for the brain. The head directions cells can sense the direction in which the animal is moving, based on which the entorhinal grid cells fire at regular intervals as the animal is at the corners of an equilateral triangle and the hippocampal place cells [1] fire when the animal is at a place in the environment. Several mathematical models [2] have been proposed to explain firing pattern of the place cells, most of which consider place cell firing to be the result of integration of either sensory or processed motor inputs (received via the grid cells). Only the oscillatory interference model mentions the role of both sensory and motor inputs in place cell firing but does not explain how this information is remembered. However, empirical observations [2-6] indicate a role for both the sensory and the grid cell inputs in place cell firing and a one to one correspondence between the grid and the place cell firing patterns. Anatomical evidence [7] indicates that an area of the medial entorhinal cortex is primarily connected to a similar area of the hippocampus along the dorso-ventral axis. All the above is not possible if the place cell firing depended on integration of inputs from several grid cells or on integration of either sensory or motor inputs alone as proposed by the current models. Moreover, the animals without binocular vision, in whom most place cell recordings have been done, ought to have different mechanism for mapping the environment than those with binocular vision. Due to lack of depth perception, the former could be using the matrix formed by grid cells to hang the objects as seen by their eyes to get an accurate estimate of position. Therefore, we propose that the place cells integrate both sensory and motor inputs to them in a Bayes' optimal manner (Fig. ​(Fig.1).1). Also each place cell is connected to only one grid cell. As the animal enters an environment, the place cell resets. While the animal moves around in the environment, the grid cell firing causes movement along the ring of intermediate cells which wraps around. As soon as the sensory inputs identify the place to be of significance and cause a place cell to fire, by Hebbian learning, synapses develop between the place cell and the intermediate neuron which is firing concurrently in the ring. These synaptic connections help the animal to retrieve the location correctly during future encounters. This model offers improvements over the prior models as it explains the empirical observations more closely. Figure 1 Sensorimotor model of the hippocampal place cells. A) Hexagonal grid cell firing gets integrated on to B) the ring of intermediate neurons. C) Together sensory and motor inputs are integrated in a Bayes' optimal manner on the place cell.

Movement dependence and layer specificity of entorhinal phase precession in two-dimensional environments.

As a rat moves, grid cells in its entorhinal cortex (EC) discharge at multiple locations of the external world, and the firing fields of each grid cell span a hexagonal lattice. For movements on linear tracks, spikes tend to occur at successively earlier phases of the theta-band filtered local field potential during the traversal of a firing field – a phenomenon termed phase precession. The complex movement patterns observed in two-dimensional (2D) open-field environments may fundamentally alter phase precession. To study this question at the behaviorally relevant single-run level, we analyzed EC spike patterns as a function of the distance traveled by the rat along each trajectory. This analysis revealed that cells across all EC layers fire spikes that phase-precess; indeed, the rate and extent of phase precession were the same, only the correlation between spike phase and path length was weaker in EC layer III. Both slope and correlation of phase precession were surprisingly similar on linear tracks and in 2D open-field environments despite strong differences in the movement statistics, including running speed. While the phase-precession slope did not correlate with the average running speed, it did depend on specific properties of the animal's path. The longer a curving path through a grid-field in a 2D environment, the shallower was the rate of phase precession, while runs that grazed a grid field tangentially led to a steeper phase-precession slope than runs through the field center. Oscillatory interference models for grid cells do not reproduce the observed phenomena.

How does the modular organization of entorhinal grid cells develop?

The entorhinal-hippocampal system plays a crucial role in spatial cognition and navigation. Since the discovery of grid cells in layer II of medial entorhinal cortex (MEC), several types of models have been proposed to explain their development and operation; namely, continuous attractor network models, oscillatory interference models, and self-organizing map (SOM) models. Recent experiments revealing the in vivo intracellular signatures of grid cells (Domnisoru et al., 2013; Schmidt-Heiber and Hausser, 2013), the primarily inhibitory recurrent connectivity of grid cells (Couey et al., 2013; Pastoll et al., 2013), and the topographic organization of grid cells within anatomically overlapping modules of multiple spatial scales along the dorsoventral axis of MEC (Stensola et al., 2012) provide strong constraints and challenges to existing grid cell models. This article provides a computational explanation for how MEC cells can emerge through learning with grid cell properties in modular structures. Within this SOM model, grid cells with different rates of temporal integration learn modular properties with different spatial scales. Model grid cells learn in response to inputs from multiple scales of directionally-selective stripe cells (Krupic et al., 2012; Mhatre et al., 2012) that perform path integration of the linear velocities that are experienced during navigation. Slower rates of grid cell temporal integration support learned associations with stripe cells of larger scales. The explanatory and predictive capabilities of the three types of grid cell models are comparatively analyzed in light of recent data to illustrate how the SOM model overcomes problems that other types of models have not yet handled.

Controlling phase noise in oscillatory interference models of grid cell firing.

Oscillatory interference models account for the spatial firing properties of grid cells in terms of neuronal oscillators with frequencies modulated by the animal's movement velocity. The phase of such a "velocity-controlled oscillator" (VCO) relative to a baseline (theta-band) oscillation tracks displacement along a preferred direction. Input from multiple VCOs with appropriate preferred directions causes a grid cell's grid-like firing pattern. However, accumulating phase noise causes the firing pattern to drift and become corrupted. Here we show how multiple redundant VCOs can automatically compensate for phase noise. By entraining the baseline frequency to the mean VCO frequency, VCO phases remain consistent, ensuring a coherent grid pattern and reducing its spatial drift. We show how the spatial stability of grid firing depends on the variability in VCO phases, e.g., a phase SD of 3 ms per 125 ms cycle results in stable grids for 1 min. Finally, coupling N VCOs with similar preferred directions as a ring attractor, so that their relative phases remain constant, produces grid cells with consistently offset grids, and reduces VCO phase variability of the order square root of N. The results suggest a viable functional organization of the grid cell network, and highlight the benefit of integrating displacement along multiple redundant directions for the purpose of path integration.

A Hybrid Oscillatory Interference/Continuous Attractor Network Model of Grid Cell Firing

Grid cells in the rodent medial entorhinal cortex exhibit remarkably regular spatial firing patterns that tessellate all environments visited by the animal. Two theoretical mechanisms that could generate this spatially periodic activity pattern have been proposed: oscillatory interference and continuous attractor dynamics. Although a variety of evidence has been cited in support of each, some aspects of the two mechanisms are complementary, suggesting that a combined model may best account for experimental data. The oscillatory interference model proposes that the grid pattern is formed from linear interference patterns or "periodic bands" in which velocity-controlled oscillators integrate self-motion to code displacement along preferred directions. However, it also allows the use of symmetric recurrent connectivity between grid cells to provide relative stability and continuous attractor dynamics. Here, we present simulations of this type of hybrid model, demonstrate that it generates intracellular membrane potential profiles that closely match those observed in vivo, addresses several criticisms aimed at pure oscillatory interference and continuous attractor models, and provides testable predictions for future empirical studies.

DC-shifts in amplitude in-field generated by an oscillatory interference model of grid cell firing

Oscillatory interference models propose a mechanism by which the spatial firing pattern of grid cells can arise from the interaction of multiple oscillators that shift in relative phase. These models produce aspects of the physiological data such as the phase precession dynamics observed in grid cells. However, existing oscillatory interference models did not predict the in-field DC shifts in the membrane potential of grid cells that have been observed during intracellular recordings in navigating animals. Here, we demonstrate that DC shifts can be generated in an oscillatory interference model when half-wave rectified oscillatory inputs are summed by a leaky integrate-and-fire neuron with a long membrane decay constant (100 ms). The non-linear mean of the half-wave rectified input signal is reproduced in the grid cell's membrane potential trace producing the DC shift within field. For shorter values of the decay constant integration is more effective if the input signal, comprising input from 6 head direction selective populations, is temporally spread during in-field epochs; this requires that the head direction selective populations act as velocity controlled oscillators with baseline oscillations that are phase offset from one another. The resulting simulated membrane potential matches several properties of the empirical intracellular recordings, including: in-field DC-shifts, theta-band oscillations, phase precession of both membrane potential oscillations and grid cell spiking activity relative to network theta and a stronger correlation between DC-shift amplitude and firing-rate than between theta-band oscillation amplitude and firing-rate. This work serves to demonstrate that oscillatory interference models can account for the DC shifts in the membrane potential observed during intracellular recordings of grid cells without the need to appeal to attractor dynamics.

Cellular mechanisms of spatial navigation in the medial entorhinal cortex

Neurons in the medial entorhinal cortex exhibit a grid-like spatial pattern of spike rates that has been proposed to represent a neural code for path integration. To understand how grid cell firing arises from the combination of intrinsic conductances and synaptic input in medial entorhinal stellate cells, we performed patch-clamp recordings in mice navigating in a virtual-reality environment. We found that the membrane potential signature of stellate cells during firing field crossings consisted of a slow depolarization driving spike output. This was best predicted by network models in which neurons receive sustained depolarizing synaptic input during a field crossing, such as continuous attractor network models of grid cell firing. Another key feature of the data, phase precession of intracellular theta oscillations and spiking with respect to extracellular theta oscillations, was best captured by an oscillatory interference model. Thus, these findings provide crucial new information for a quantitative understanding of the cellular basis of spatial navigation in the entorhinal cortex.

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.

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.

Modeling the influence of optic flow on grid cell firing in the absence of other cues1

Information from the vestibular, sensorimotor, or visual systems can affect the firing of grid cells recorded in entorhinal cortex of rats. Optic flow provides information about the rat’s linear and rotational velocity and, thus, could influence the firing pattern of grid cells. To investigate this possible link, we model parts of the rat’s visual system and analyze their capability in estimating linear and rotational velocity. In our model a rat is simulated to move along trajectories recorded from rat’s foraging on a circular ground platform. Thus, we preserve the intrinsic statistics of real rats’ movements. Visual image motion is analytically computed for a spherical camera model and superimposed with noise in order to model the optic flow that would be available to the rat. This optic flow is fed into a template model to estimate the rat’s linear and rotational velocities, which in turn are fed into an oscillatory interference model of grid cell firing. Grid scores are reported while altering the flow noise, tilt angle of the optical axis with respect to the ground, the number of flow templates, and the frequency used in the oscillatory interference model. Activity patterns are compatible with those of grid cells, suggesting that optic flow can contribute to their firing.

The Neural Encoding of Space in Parahippocampal Cortices

The Neural Encoding of Space in Parahippocampal Cortices

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.

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).

Periodicity without rhythmicity

Grid cells confer a spatial impression of an animal's environment on the brain. Their firing patterns in a cave-dwelling bat reopen old questions about how they do this, and pose some compelling new ones. See Letter p.103 Animals maintain a neural representation of space through the coordinated activity of grid, place and head-direction cells. Grid cells fire as the animal passes across the vertices of a periodic hexagonal grid depicting space. How these cells create the grid structure is still under debate, although recent work strongly proposes a model involving an oscillatory interference-driven transformation of temporal oscillations into the spatial grid. Yartsev et al. refute this model by characterizing a proper network of grid cells in an animal model, the Egyptian fruit bat, which naturally lacks oscillations required for the oscillatory interference model to produce grid structure. Besides directly characterizing grid cells in a non-rodent species, this study also argues against a major computational model of grid-cell production.