The emergence of grid cells: Intelligent design or just adaptation?

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.

Coupled noisy spiking neurons as velocity-controlled oscillators in a model of grid cell spatial firing

One of the two primary classes of models of grid cell spatial firing uses interference between oscillators at dynamically modulated frequencies. Generally, these models are presented in terms of idealized oscillators (modeled as sinusoids), which differ from biological oscillators in multiple important ways. Here we show that two more realistic, noisy neural models (Izhikevich's simple model and a biophysical model of an entorhinal cortex stellate cell) can be successfully used as oscillators in a model of this type. When additive noise is included in the models such that uncoupled or sparsely coupled cells show realistic interspike interval variance, both synaptic and gap-junction coupling can synchronize networks of cells to produce comparatively less variable network-level oscillations. We show that the frequency of these oscillatory networks can be controlled sufficiently well to produce stable grid cell spatial firing on the order of at least 2-5 min, despite the high noise level. Our results suggest that the basic principles of oscillatory interference models work with more realistic models of noisy neurons. Nevertheless, a number of simplifications were still made and future work should examine increasingly realistic models.

A manifold of spatial maps in the brain

Two neural systems are known to encode self-location in the brain: Place cells in the hippocampus encode unique locations in unique environments, whereas grid cells, border cells and head-direction cells in the parahippocampal cortex provide a universal metric for mapping positions and directions in all environments. These systems have traditionally been studied in very simple environments; however, natural environments are compartmentalized, nested and variable in time. Recent studies indicate that hippocampal and entorhinal spatial maps reflect this complexity. The maps fragment into interconnected, rapidly changing and tightly coordinated submaps. Plurality, fast dynamics and dynamic grouping are optimal for a brain system thought to exploit large pools of stored information to guide behavior on a second-by-second time frame in the animal's natural habitat.

Democracy-independence trade-off in oscillating dendrites and its implications for grid cells

Dendritic democracy and independence have been characterized for near-instantaneous processing of synaptic inputs. However, a wide class of neuronal computations requires input integration on long timescales. As a paradigmatic example, entorhinal grid fields have been thought to be generated by the democratic summation of independent dendritic oscillations performing direction-selective path integration. We analyzed how multiple dendritic oscillators embedded in the same neuron integrate inputs separately and determine somatic membrane voltage jointly. We found that the interaction of dendritic oscillations leads to phase locking, which sets an upper limit on the timescale for independent input integration. Factors that increase this timescale also decrease the influence that the dendritic oscillations exert on somatic voltage. In entorhinal stellate cells, interdendritic coupling dominates and causes these cells to act as single oscillators. Our results suggest a fundamental trade-off between local and global processing in dendritic trees integrating ongoing signals.

How informative are spatial CA3 representations established by the dentate gyrus?

In the mammalian hippocampus, the dentate gyrus (DG) is characterized by sparse and powerful unidirectional projections to CA3 pyramidal cells, the so-called mossy fibers. Mossy fiber synapses appear to duplicate, in terms of the information they convey, what CA3 cells already receive from entorhinal cortex layer II cells, which project both to the dentate gyrus and to CA3. Computational models of episodic memory have hypothesized that the function of the mossy fibers is to enforce a new, well-separated pattern of activity onto CA3 cells, to represent a new memory, prevailing over the interference produced by the traces of older memories already stored on CA3 recurrent collateral connections. Can this hypothesis apply also to spatial representations, as described by recent neurophysiological recordings in rats? To address this issue quantitatively, we estimate the amount of information DG can impart on a new CA3 pattern of spatial activity, using both mathematical analysis and computer simulations of a simplified model. We confirm that, also in the spatial case, the observed sparse connectivity and level of activity are most appropriate for driving memory storage-and not to initiate retrieval. Surprisingly, the model also indicates that even when DG codes just for space, much of the information it passes on to CA3 acquires a non-spatial and episodic character, akin to that of a random number generator. It is suggested that further hippocampal processing is required to make full spatial use of DG inputs.

Hebbian analysis of the transformation of medial entorhinal grid-cell inputs to hippocampal place fields

The discovery of grid cells in the medial entorhinal cortex (MEC) permits the characterization of hippocampal computation in much greater detail than previously possible. The present study addresses how an integrate-and-fire unit driven by grid-cell spike trains may transform the multipeaked, spatial firing pattern of grid cells into the single-peaked activity that is typical of hippocampal place cells. Previous studies have shown that in the absence of network interactions, this transformation can succeed only if the place cell receives inputs from grids with overlapping vertices at the location of the place cell's firing field. In our simulations, the selection of these inputs was accomplished by fast Hebbian plasticity alone. The resulting nonlinear process was acutely sensitive to small input variations. Simulations differing only in the exact spike timing of grid cells produced different field locations for the same place cells. Place fields became concentrated in areas that correlated with the initial trajectory of the animal; the introduction of feedback inhibitory cells reduced this bias. These results suggest distinct roles for plasticity of the perforant path synapses and for competition via feedback inhibition in the formation of place fields in a novel environment. Furthermore, they imply that variability in MEC spiking patterns or in the rat's trajectory is sufficient for generating a distinct population code in a novel environment and suggest that recalling this code in a familiar environment involves additional inputs and/or a different mode of operation of the network.

A computational theory of episodic memory formation in the hippocampus

A quantitative computational theory of the operation of the hippocampus as an episodic memory system is described. The CA3 system operates as a single attractor or autoassociation network to enable rapid, one-trial associations between any spatial location (place in rodents or spatial view in primates) and an object or reward and to provide for completion of the whole memory during recall from any part. The theory is extended to associations between time and object or reward to implement temporal order memory, also important in episodic memory. The dentate gyrus performs pattern separation by competitive learning to produce sparse representations, producing for example neurons with place-like fields from entorhinal cortex grid cells. The dentate granule cells produce by the very small number of mossy fibre connections to CA3 a randomizing pattern separation effect important during learning but not recall that separates out the patterns represented by CA3 firing to be very different from each other, which is optimal for an unstructured episodic memory system in which each memory must be kept distinct from other memories. The direct perforant path input to CA3 is quantitatively appropriate to provide the cue for recall in CA3, but not for learning. The CA1 recodes information from CA3 to set up associatively learned backprojections to neocortex to allow subsequent retrieval of information to neocortex, providing a quantitative account of the large number of hippocampo-neocortical and neocortical-neocortical backprojections. Tests of the theory including hippocampal subregion analyses and hippocampal NMDA receptor knockouts are described and support the theory.

Knock-out of HCN1 subunit flattens dorsal-ventral frequency gradient of medial entorhinal neurons in adult mice

Layer II stellate cells at different locations along the dorsal to ventral axis of medial entorhinal cortex show differences in the frequency of intrinsic membrane potential oscillations and resonance (Giocomo et al., 2007). The frequency differences scale with differences in the size and spacing of grid-cell firing fields recorded in layer II of the medial entorhinal cortex in behaving animals. To determine the mechanism for this difference in intrinsic frequency, we analyzed oscillatory properties in adult control mice and adult mice with a global deletion of the HCN1 channel. Data from whole-cell patch recordings show that the oscillation frequency gradient along the dorsal-ventral axis previously shown in juvenile rats also appears in control adult mice, indicating that the dorsal-ventral gradient generalizes across age and species. Knock-out of the HCN1 channel flattens the dorsal-ventral gradient of the membrane potential oscillation frequency, the resonant frequency, the time constant of the "sag" potential and the amplitude of the sag potential. This supports a role of the HCN1 subunit in the mechanism of the frequency gradient in these neurons. These findings have important implications for models of grid cells and generate predictions for future in vivo work on entorhinal grid cells.

The role of competitive learning in the generation of DG fields from EC inputs

We follow up on a suggestion by Rolls and co-workers, that the effects of competitive learning should be assessed on the shape and number of spatial fields that dentate gyrus (DG) granule cells may form when receiving input from medial entorhinal cortex (mEC) grid units. We consider a simple non-dynamical model where DG units are described by a threshold-linear transfer function, and receive feedforward inputs from 1,000 mEC model grid units of various spacing, orientation and spatial phase. Feedforward weights are updated according to a Hebbian rule as the virtual rodent follows a long simulated trajectory through a single environment. Dentate activity is constrained to be very sparse. We find that indeed competitive Hebbian learning tends to result in a few active DG units with a single place field each, rounded in shape and made larger by iterative weight changes. These effects are more pronounced when produced with thousands of DG units and inputs per DG unit, which the realistic system has available, than with fewer units and inputs, in which case several DG units persists with multiple fields. The emergence of single-field units with learning is in contrast, however, to recent data indicating that most active DG units do have multiple fields. We show how multiple irregularly arranged fields can be produced by the addition of non-space selective lateral entorhinal cortex (lEC) units, which are modelled as simply providing an additional effective input specific to each DG unit. The mean number of such multiple DG fields is enhanced, in particular, when lEC and mEC inputs have overall similar variance across DG units. Finally, we show that in a restricted environment the mean size of the fields is unaltered, while their mean number is scaled down with the area of the environment.

Grid cells: the position code, neural network models of activity, and the problem of learning

We review progress on the modeling and theoretical fronts in the quest to unravel the computational properties of the grid cell code and to explain the mechanisms underlying grid cell dynamics. The goals of the review are to outline a coherent framework for understanding the dynamics of grid cells and their representation of space; to critically present and draw contrasts between recurrent network models of grid cells based on continuous attractor dynamics and independent-neuron models based on temporal interference; and to suggest open questions for experiment and theory.

Influence of boundary removal on the spatial representations of the medial entorhinal cortex

The medial entorhinal cortex (MEC) is thought to create and update a dynamical representation of the animal's spatial location. Most suggestive of this process are grid cells, whose firing locations occur periodically in space. Prior studies in small environments were ambiguous as to whether all spatially modulated cells in MEC were variants of grid cells or whether a subset resembled classic place cells of the hippocampus. Recordings from the dorsal and ventral MEC were performed as four rats foraged in a small square box centered inside a larger one. After 6 min, without removing the rat from the enclosure, the walls of the small box were quickly removed, leaving the rat free to continue foraging in the whole area enclosed by the larger box. The rate-responses of most recorded cells (70 out of 93 cells, including 15 of 16 putative interneurons) were considered spatially modulated based on information-theoretic analysis. A number of cells that resembled classic hippocampal place cells in the small box were revealed to be grid cells in the larger box. In contrast, other cells that fired along the boundaries or corners of the small box did not show grid-cell firing in the large box, but instead fired along the corresponding locations of the large box. Remapping of the spatial response in the area corresponding to the small box after the removal of its walls was prominent in most spatially modulated cells. These results show that manipulation of local boundaries can exert a powerful influence on the spatial firing patterns of MEC cells even when the manipulations leave global cues unchanged and allow uninterrupted, self-motion-based localization. Further, they suggest the presence of landmark-related information in MEC, which might prevent cumulative drift of the spatial representation or might reset it to a previously learned configuration in a familiar environment.

A metric for space

Not all areas of neuronal systems investigation have matured to the stage where computation can be understood at the microcircuit level. In mammals, insights into cortical circuit functions have been obtained for the early stages of sensory systems, where signals can be followed through networks of increasing complexity from the receptors to the primary sensory cortices. These studies have suggested how neurons and neuronal networks extract features from the external world, but how the brain generates its own codes, in the higher-order nonsensory parts of the cortex, has remained deeply mysterious. In this terra incognita, a path was opened by the discovery of grid cells, place-modulated entorhinal neurons whose firing locations define a periodic triangular or hexagonal array covering the entirety of the animal's available environment. This array of firing is maintained in spite of ongoing changes in the animal's speed and direction, suggesting that grid cells are part of the brain's metric for representation of space. Because the crystal-like structure of the firing fields is created within the nervous system itself, grid cells may provide scientists with direct access to some of the most basic operational principles of cortical circuits.

Sources of the spatial code within the hippocampus

Neurons in the hippocampus are thought to provide information on an animal's location within its environment. Input to the hippocampus comes via afferents from the entorhinal cortex, which are separated into several major pathways serving different hippocampal regions. Recent studies show the significance of individual afferent pathways in location perception, enhancing our understanding of hippocampal function.

Grid cell mechanisms and function: contributions of entorhinal persistent spiking and phase resetting

This article presents a model of grid cell firing based on the intrinsic persistent firing shown experimentally in neurons of entorhinal cortex. In this model, the mechanism of persistent firing allows individual neurons to hold a stable baseline firing frequency. Depolarizing input from speed-modulated head direction cells transiently shifts the frequency of firing from baseline, resulting in a shift in spiking phase in proportion to the integral of velocity. The convergence of input from different persistent firing neurons causes spiking in a grid cell only when the persistent firing neurons are within similar phase ranges. This model effectively simulates the two-dimensional firing of grid cells in open field environments, as well as the properties of theta phase precession. This model provides an alternate implementation of oscillatory interference models. The persistent firing could also interact on a circuit level with rhythmic inhibition and neurons showing membrane potential oscillations to code position with spiking phase. These mechanisms could operate in parallel with computation of position from visual angle and distance of stimuli. In addition to simulating two-dimensional grid patterns, models of phase interference can account for context-dependent firing in other tasks. In network simulations of entorhinal cortex, hippocampus, and postsubiculum, the reset of phase effectively replicates context-dependent firing by entorhinal and hippocampal neurons during performance of a continuous spatial alternation task, a delayed spatial alternation task with running in a wheel during the delay period (Pastalkova et al., Science, 2008), and a hairpin maze task.