Models of Place and Grid Cell Firing and Theta Rhythmicity.

Abstract

Neuronal firing in the hippocampal formation (HF) of freely-moving rodents shows striking examples of spatialorganization in the form of place, directional, boundary vector and grid cells. The firing of place and grid cells shows an intriguing form of temporal organization known as “theta phase precession”. We review the mechanisms underlying theta phase precession of place cell firing, ranging from membrane potential oscillations to recurrent connectivity, and the relevant intra- and extra-cellular data. We then consider the use of these models to explain the spatial structure of grid cell firing, and review the relevant intra- and extra-cellular data. Finally, we consider the likely interaction between place cells, grid cells and boundary vector cells in estimating self-location as a compromise between path-integration and environmental information.

Historically the functional extra-cellular electrophysiology of the hippocampus of freely-moving rodents has been dominated by two findings. The local field potential (LFP) shows a large-amplitude 4-10Hz theta rhythm [1], and the majority of principal cells have firing restricted to specific environmental locations (“place cells” [2]). These two phenomena combine in the observation that place cell firing shows “theta phase precession”, whereby spikes are fired at successively earlier phases of the theta rhythm of the local field potential (“LFP”) as the animal moves through the firing field or “place field” [3], see Figure 1. The firing phase correlates better with the distance traveled through the place field than with other variables, such as the time spent in the place field: providing a phase code for the animal’s location in addition to the firing rate code [4;5]. Phase precession occurs because place cell firing is modulated at a frequency slightly higher than LFP theta frequency, and this frequency difference increases with running speed [3;6], see [7;8] for supporting evidence.

Figure 1.

Theta-phase precession of place cell firing. (a) As a rat runs on a track, a place cell in the hippocampus fires as the animal passes through a specific region (the “place field,” b). This firing rate code for location is also a temporal code (c): spikes (ticks) are fired at successively earlier phases of the theta rhythm of the local field potential (blue trace), referred to as “theta-phase precession.” The theta phase of firing correlates with the distance traveled through the place field (d) better than with other variables, such as time spent in the place field or the place cell’s instantaneous firing rate. Firing phase precesses from late to early phases of theta as the animal runs through the start, middle and end of the place field, irrespective of whether spikes are fired at a low or high rate on that run (e; mean and s.e.m. over n=34 cells). Adapted from [5]. (f) Schematic representation of the data of Harvey et al. (2009) showing local field potential theta (blue) intracellular membrane potential (MPO, black) and spikes (ticks) from a place cell as a mouse runs from left to right through the place field in a virtual arena. Note that the intracellular MPO increases in amplitude and in frequency relative to the LFP theta when the mouse is in the place field, and that the spikes are aligned to the intracellular MPO. The firing rate peaks in the center of the place field, while the membrane potential shows a more asymmetric ramp-like depolarization.

Several further significant findings have emerged including the discovery of additional spatial cell types in the wider Hippocampal Formation. First, head-direction cells which fire according to the orientation of the head relative to the environment have been found in several locations along Papez’s circuit [9] including medial entorhinal cortex (mEC) [10]. Second, grid cells have been discovered in mEC: cells which fire in a series of environmental locations which form a triangular grid across the environment [11]. Grid cells are most numerous in layer II of mEC, where their firing also shows theta-phase precession [12], but are also found in deeper layers of mEC [10] and in pre- and para-subiculum [13]. Third, boundary vector cells (BVCs), a spatial cell type theoretically predicted to provide one component of place cell firing fields [14-16] have been found in the subiculum [17] and mEC [13;18]. The firing of BVCs is determined by the location of the rat relative to environmental boundaries. Finally, technological advances have allowed intra-cellular recording [19] from place cells in head-fixed mice navigating a virtual environment, and in freely moving animals [20]. These intra-cellular data show a theta-band membrane potential oscillation (MPO) that increases in frequency above that of the LFP when the animal enters the place field, with spikes being fired at the peaks of the MPO, see Figure 1f.

The detailed extra- and intra-cellular data concerning the spatio-temporal patterns of neuronal activity in the hippocampal formation of behaving rodents provides a model system for understanding the neural bases of behavior. Here we review the main computational models of place cell firing and theta phase precession, and how these models are being applied to explain grid cell firing. Finally we consider how place cells might calculate self-location by combining two types of input [2;21-23]: information concerning path integration, mediated by grid cells resulting from theta-band neuronal oscillations [24-27] and environmental information, mediated by boundary-vector cells [13;16;17].

Models of theta-phase precession of place cell firing

Computational work on theta-phase precession has fallen into two broad camps, those focusing on mechanisms within an individual place cell and those focusing on mechanisms arising from the interaction between large networks of neurons. Models from these two viewpoints are reviewed below, and need not be mutually exclusive. Time will tell which types of mechanism prove most productive in explaining the experimental findings and whether both types of mechanism are necessary for a comprehensive explanation.

Oscillatory mechanisms and intra-cellular data

The Dual Oscillator model [3;6, Figure 2a] posits that the basic mechanism producing precession is an interaction of two oscillations of different frequency: a baseline somatic oscillation at the LFP theta frequency, and an active dendritic oscillation whose frequency increases above baseline when the animal is in the place field, see [28]. The sum of these oscillations produces a membrane potential oscillation (MPO) wavelet in the place cell with a frequency above LFP theta and an amplitude peaking in the middle of the place field. Spikes are fired at the peaks of the MPO, and so show phase precession because the MPO has a higher frequency than LFP theta, Figure 2a. The increase in frequency of the active oscillation is assumed to be proportional to running speed, so that the phase advance of spikes reflects distance traveled through the place field, rather than time spent in the field. The two oscillations need not combine additively: the somatic input could be a shunt inhibition leading to a multiplicative model, Figure 2b (see also Figure 1 in [25]). Note that the net activity of a population of place cells will be modulated at the (slower) LFP theta frequency even though all individual cells are showing phase precession, i.e. oscillating above this frequency when in their place field [29;30].

Figure 2.

Oscillatory models of phase precession. (a) Dual Oscillator model. The somatic MPO (black) is the sum of two oscillatory inputs having different frequencies in the place field: a somatic input proportional to LFP theta (sin(2πf_bt), blue), and a dendritic input (sin(2πf_at), red) whose frequency (f_a) increases above LFP theta frequency (f_b) in the place field (proportionally to running speed, see Figure 3c). The somatic MPO frequency exceeds LFP theta and its amplitude peaks in the middle of the field. Spikes occur at the peaks of the somatic MPO (ticks; f_b=8Hz, f_a=9Hz in place field). (b) Dual Oscillator model with shunt inhibition. The somatic input can be a shunt (i.e. modulatory) input. Here somatic input ([3+sin(2πf_bt)]/2, blue) multiplies dendritic input (sin(2πf_at), red) to give the somatic MPO (black). (c-e) Somato-Dendritic Interference. The somatic MPO (black) is the sum of two theta-frequency inputs: a somatic input (sin(2πft), blue), and a dendritic input (Dsin(2πft+δ), red) with amplitude D, which increases through the place field, and phase advance δ. The MPO can show significant phase precession, but has a decreasing amplitude in the place field: MPO=Rsin(2πft+φ), where R²=1+D²+2Dcos(δ) and φ=arctan{sin(δ)/[cos(δ)+1/D]}. (c) MPO theta-phase (φ) and amplitude (R) for different values of dendritic phase advance (δ). The dendritic amplitude D increases like 2t as the animal runs through the place field from t=0 to 1.0s, f=8Hz. (d) Example somatic input (blue), dendritic input (red) and MPO (black) for δ=162°. (e) Somato-Dendritic interference with divisive inhibition: dendritic input D[1+sin(2πft)] is divided by somatic input [1.5+sin(2πft)], producing a MPO with low plateaus early in the place field and higher sharper peaks later in the place field. The increasing dendritic amplitude D delays the peak of the flat low early MPOs more than the sharper higher late MPOs, producing phase precession. (f) Depolarizing Ramp model. Somatic input is proportional to LFP theta (blue), dendritic input is a ramp-like depolarization in the place field (red). The resulting somatic MPO (black) crosses a firing threshold (green) successively earlier on each cycle of LFP theta, producing successively earlier phases (and greater numbers) of spikes.

The Somato Dendritic Interference model (SDI; [31 Supplementary Figure 6], see also [28;32;33], posits oscillatory somatic and dendritic inputs to the place cell, both with the same frequency (LFP theta frequency) and a phase difference of 180°. In this model, a ramp-like depolarization of the dendrite in the place field causes the dendritic oscillation to increase in amplitude. In the initial version of the SDI model [32], the sum of the two inputs produces a MPO which switches abruptly from the phase of the somatic input to the phase of the dendritic input as the dendritic amplitude increases. A smooth transition in phase can be achieved if the dendritic oscillation leads the somatic one by less than 180° (e.g. via delayed conduction of the dendritic input to the soma). However there are significant problems with the SDI model. Significant amounts of phase precession require a choice of parameters which produce a significant decrease in the MPO amplitude within the place field (see Figures 2c-d), and this is incompatible with the in-vivo data, see [19] and Figure 1f. For this reason, a divisive somatic input has been proposed (i.e. shunt rather than subtractive inhibition [31]), so that the MPO amplitude will not decrease when the dendritic amplitude increases. Dividing two sinusoids of the same frequency generally changes the amplitude of the resulting MPO rather than its phase. Nonetheless, if parameters are chosen such that the MPO has broad plateaus separated by sharp troughs, the curvature of the MPO plateaus increases with amplitude. In this case, the increasing dendritic amplitude introduces a slope to the plateaus which significantly delays the occurrence of the peak of the flattest and lowest early MPOs but has successively less effect on the higher and increasingly curved later MPOs, see Figure 2e. This causes an apparent phase precession over the first few MPOs in the place field, but requires spikes to be fired precisely at the peaks of the very flat and low-amplitude MPOs early in the place field. In Losonczy et al.’s in-vitro simulation, most of the MPO phase precession occurs during the first few MPO cycles whose amplitudes are too low to reliably initiate spikes ([31] Figures 2b, 4d-e).

Figure 4.

Environmental inputs to place cells, and interactions with grid cells. (a-c) “Boundary vector cells” (BVCs) as the environmental inputs to place cells. (a) An example BVC tuned to respond to a boundary nearby to the East (above), and one tuned to respond at a longer distance (below). (b) A place cell’s firing can be modeled as the thresholded sum of its BVC inputs. Firing rate maps are shown for four environments (small square, circle, square rotated relative to distal cues, large square) for two different BVCs (above). The firing rate maps for the place cell are shown below. (c) A place cell’s firing patterns in several different shaped environments (top left) can be used to infer the BVCs driving its firing (top right). The inferred model can then predict how the place cell will fire in new environmental configurations (bottom right), usually producing a reasonable qualitative fit when the place cell’s firing is recorded in these new configurations (bottom left). Adapted from [16]. (d) Three example boundary vector cells recorded in the subiculum. Each column shows the receptive field (above), and firing rate map in a series of environments (below). Adapted from [17]. (e) Schematic proposed interaction between place cells (PC), BVCs and grid cells (GC). BVCs anchor place fields to the environment and place cells anchor grid cell firing fields to the environment (via phase-resetting of velocity controlled oscillators (VCOs) in the multiple oscillator model). In return, grid cells provide a path integrative input to place cells. Adapted from [24].

The Depolarizing Ramp model [34] posits a somatic input proportional to LFP theta and a dendritic depolarization that increases in a ramp-like fashion across the place field. This results in a very simple mechanism for phase precession, by which the increasing depolarization allows the somatic MPO to exceed a firing threshold successively earlier within each cycle of LFP theta. See Figure 2f. This mechanism could explain why some of the Losonczy et al. in-vitro data shows spikes being fired at successively earlier phases of the MPO, see Losonczy et al.’s [31] Figure 2a.

The intra-cellular data of Harvey et al. [19] partially distinguishes between the dual oscillator and depolarizing ramp models, see Figure 1f. It shows that phase precession results from a MPO that increases in frequency and amplitude within the place field, with spikes fired at the peaks of the MPO, a result consistent with the dual oscillator mechanism. However, these data also show an asymmetrical ramp-like depolarization in the place field, which is not predicted by the dual oscillator model. By contrast, the depolarizing ramp model does predict a ramp-like depolarization which causes phase precession as spikes are fired at successively earlier phases of the MPO (which is locked to LFP theta). The crucial point of Harvey et al.’s data is that the ramp-like depolarization does not influence spike timing relative to the MPO, i.e. it is not the mechanism causing phase precession. Instead of moving relative to the MPO the spikes stay fixed with respect to it while the MPO itself precesses relative to the LFP theta. This strongly supports dual oscillator as opposed to ramp-depolarization models, and raises the question of why spike-timing reflects MPO peaks rather than an absolute firing threshold in somatic depolarization. One possibility is that the dendrites are acting as voltage controlled oscillators [3;28] and the ramp- like depolarization reflects the voltage change which is driving the frequency of the dendritic oscillator. If the dendritic oscillation produced spikelets, the timing of action potentials generation would be primarily determined by the timing of these spikelets rather the slower ramp-like variation in depolarization level, see [20] for supporting evidence and ([25] Figure 1) for related simulations.

Recurrent connections and extra-cellular data

The main alternative to the above intracellular models is the Spreading Activation model [35;36]. In this model, firing at the end of the place field is driven by feed-forward sensory inputs to the place cell at early phases of LFP theta, but activity also spreads from place cells with place fields earlier along the route to place cells with place fields further along the route. Thus, firing nearer to the start of the place field reflects input from place cells with place fields nearer to the start of the route, and occurs at later phases due to the transmission delay in the arrival of these inputs via a chain of cells with place fields earlier on the route. Changes in firing phase as the rat runs through the place field occur because the feed-forward inputs to the population of place cells change. In [35] place cell firing occurs at the peak of the synaptic input arriving within each theta cycle, consistent with the observation of spikes at the MPO peaks [37]. In [36] local oscillatory processes in the gamma and theta bands also constrain firing times.

The connections required by the spreading activation model could be formed by synaptic plasticity if the animal runs repeatedly along the route, and are consistent with a reported backward spread of place cell firing with experience [34]. However, it is less clear how the model explains phase precession in open environments in which arbitrary novel routes are taken [38-40], or the observation that blocking NMDA receptors prevents the backward spread of firing but does not affect phase precession [41].

Two aspects of the extra-cellular data on theta phase precession provide strong constraints on mechanism. First, the firing rate and firing phase dissociate: firing phase continues to precess with distance traveled through the place field irrespective of whether firing rate is rising (entering the place field) or falling (leaving the place field), and whether many or few spikes are fired on that particular run, see Figure 1e and [5]. Second, phase precession is characterized by increasing phase variability through the place field, see Figure 1d-e and [5], the opposite of that predicted by the spreading activation model.

The Depolarizing Ramp and SDI models struggle to show independence between firing phase and firing rate, since both are closely linked via the level of dendritic depolarization. To deal with this, Losonczy et al propose an additional ad hoc mechanism for the SDI model: an activity-dependent change in the time course of feed-back inhibition, so that late phase firing is inhibited in the second half of the place field. This mechanism can allow early firing phases to be maintained while dendritic drive decreases, but would not produce continued phase precession in the second half of the field. It also introduces a dependence on the neuronal activity during each run. The spreading activation model produces phase precession as a result of the physical movement of the animal past environmental cues, and is thus less consistent with observations of intrinsically generated phase precession in a running wheel [42], but note that physical motion does occur in running wheels [43], and see discussion in [44].

The divisive SDI model and the Spreading Activation model predict decreasing phase-variability through the place field. In the SDI model, firing at the start of the field results from spikes being fired at the peaks of low, flat MPOs from which spike-timing would be unreliable compared to the later more sharply peaked MPOs. The spreading activation model also predicts increased phase variance at the start of the place field (where late phase firing is driven by input from other place cells), due to dispersion in synaptic transmission from one place cell to the next, whereas firing at the end of the field reflects direct environmental input. In the Dual Oscillator model any error in the rate of increase of MPO frequency with running speed would result in increasing phase variability through the place field when multiple runs at different speeds are combined, a prediction more consistent with experimental findings.

Models of grid cell firing

Oscillatory mechanisms

Grid cells show a regularly repeating spatial firing pattern more reminiscent of an interference pattern than the uni-modal firing pattern of most place cells (Figure 3a), suggesting that the Dual Oscillator model might be extended to the grid cells in layer II of mEC [23;27]. These cells were subsequently shown to exhibit theta-phase precession, see [12] and Figure 3b.

Figure 3.

Multiple oscillator model of grid cell firing. (a) Example of the spatial distribution of spikes from a grid cell (red dots) as a rat forages in a 1m² box (path shown in black). Adapted from [11]. (b) Example of theta-phase precession in a grid cell from medial entorhinal cortex (layer II) as a rat runs left-to-right through two of its firing fields on a linear track. Black dots (above) show theta-phase of spikes, red dots (below) show spatial position on the track. Adapted from [12]. (c) A “velocity-controlled oscillator” (VCO) extends the dual oscillator model (see Figure 2a) to two dimensions: the active oscillation has a frequency (f_a) that varies relative to the baseline frequency (f_b) like: f_a(t)=f_b(t)+βs(t)cos(ø(t)-ø_d), where s(t) is running speed, ø(t) is running direction and ø_d is that VCO’s preferred running direction. The dependence of the active frequency on running direction and speed is shown in middle and right panels respectively (the preferred direction is left-to-right). These dependencies cause the phase of the active oscillation relative to baseline to reflect displacement along the preferred direction (see mid top panel of d). In the dual oscillator model: f_a(t)=f_b(t)+βs(t). (d) Multiple VCOs with different preferred directions combine with each other, and the somatic baseline input, to produce grid firing. Grid scale depends on the constant β as: G=2/√3β. Adapted from [25;48], see also [26].

In the Dual Oscillator model the frequency of the dendritic oscillation increases with running speed so that its phase encodes distance traveled [3;6], thus place cells could be considered “speed-controlled oscillators” [8], see Figure 3c. To generate 2D spatial firing patterns, we proposed the existence of “velocity-controlled oscillators” (VCOs), that increase in frequency according to running speed in a specific preferred direction (i.e. the dot product of velocity with preferred direction), see [24-27] and Figure 3c-d. These VCOs might be driven by synaptic input from the directionally modulated cells common in the deeper layers of mEC [10] and elsewhere along Papez’s circuit [9;45]. Their firing phase encodes displacement along their preferred direction (Figure 3d). If multiple VCOs with different preferred directions converge onto a grid cell, the grid-like spatial firing pattern can result, see Figure 3d.

Although two VCOs would be sufficient to perform path integration, and support grid cell firing (if their preferred directions differ by 60° or 120°), using three or more VCOs to drive grid cell firing confers robustness to error accumulation, and would allow simple unsupervised learning algorithms to select VCOs with preferred directions differing by multiples of 60° [24;46]. Recent analysis suggests that different VCOs should be instantiated in different cells rather than different dendrites, to avoid synchronization [47], see [25;26] for further discussion.

The multiple oscillator model makes specific predictions relating the frequency of theta-band modulation of firing rate to grid scale and running speed [25], which have been broadly verified [48]. Predictions regarding the relationship between MPOs in mEC layer II stellate cells and their depolarization level [49] and likely grid scale (inferred from their dorso-ventral location) [50] have also been broadly verified. Recent evidence suggests that the predicted VCOs exist as “theta cells” (interneurons found throughout the septo-hippocampal circuit) whose inter-burst frequency shows a cosine modulation by running direction and a linear increase with running speed (Welday et al. Soc Neurosci Abstr, 2010, 203.20).

The multiple oscillator model implies that LFP theta reflects two components: a baseline frequency and an increase in frequency with running speed that is driven by the increase in individual VCO frequencies with running speed [25]. These two components should parallel the distinction between arousal-related and movement-related theta mechanisms (see e.g. [51]), and manipulations that affect the latter component should also affect grid scale. Consistent with this suggestion, exposure to environmental novelty, which reduces theta frequency [52], causes an increase in grid scale (Barry et al., Soc Neurosci Abstr, 2009, 101.24). The observation of non-theta-modulated grid cell firing the deeper layers of mEC [10] and in bats (Yartsev et al., Soc Neurosci Abstr, 2010, 203.15), was not predicted by the multiple oscillator model. According to the model, these cells would have to inherit their grid-like firing patterns from theta modulated cells found elsewhere in the brain or during earlier exploration of the environment.

Recurrent connections

The multiple oscillator model focuses on single grid cells rather than network-level interactions. However, there must be coupling between grid cells, or their inputs. MPOs in slices are too irregular to work as posited by the multiple oscillator model. In contrast, in-vivo LFP shows a regular coherent theta rhythm - indicating widespread coordination of MPOs, and interactions between mEC cells can be seen in slices [53]. Interactions, or common inputs, are also indicated by the common orientation of grids [11;54], and the way that local populations of grid cells have grids of similar spatial scale [11] and that this scale increases dorso-ventrally in discrete steps [54]. In terms of the multiple oscillator model, coupled velocity-controlled oscillators can support sufficiently regular oscillations [55], and the model extends to coherent populations of grid cells if the velocity-controlled oscillators are arranged as ring attractors [45].

The main network-level model of grid cell firing, the Continuous Attractor model [56-58], extends preceding models of the firing of head-direction cells [22;59;60] and place cells [59;61,62]. Grid-like firing patterns across the population of cells result from symmetrical recurrent connectivity in which cells with similar response patterns experience mutual excitation while cells with dissimilar firing patterns experience mutual inhibition. Asymmetric interactions matching the animal’s speed and direction of movement causes these firing patterns to shift so that each cell exhibits a grid-like firing pattern as a function of the animal’s location. This asymmetric connectivity might be mediated by interconnections between grid cells with direction-dependent firing [10]. This can allow phase precession to occur as the rat runs through grid cell firing fields in all direction [63]. The use of spike after-depolarization to set an intrinsic rhythm of reactivation, see [36], removes the need for sensory input to initiate the sequence of firing within each theta cycle, and changes in the time-course of after-depolarization could explain dorso-ventral changes in grid scale [63].

The symmetric recurrent connectivity of Continuous Attractor models complements the Multiple Oscillator model: potentially providing additional spatial and temporal stability, either to grid cells directly [56-58], or to velocity-controlled oscillators [45]. Evidence for this aspect of attractor models includes the common orientation of different grids [11;54] and the step-wise increase in grid scale [54]. The proposed asymmetric interactions for shifting firing patterns according to the animal’s movement provides a clear alternative to the multiple oscillator model.

Interactions between the different types of spatial cell

The long-term stability of the firing patterns of place and grid cells relative to the environment indicates that firing locations are determined by environmental information as well as the mechanisms for path integration discussed above. Recording from the same place [14] or grid [54] cell during geometric manipulation of a familiar environment indicates a special influence of environmental boundaries on firing locations, see also [64]. Computational modeling of place fields [15;16] predicted the existence of input cells tuned to respond to the presence of an environmental boundary at a specific distance in a specific allocentric direction (“boundary vector cells”), see Figure 4a-c. Cells with these properties have since been found in subiculum [17] and mEC [13;18], see Figure 4d.

Place cells are thought to combine both path integrative and environmental inputs [2]. The former input is likely provided by the theta system and grid cells, while the latter input is provided by boundary vector cells and local cues such as distinctive smells or textures. The hippocampus and entorhinal cortex are mutually interconnected [65], so that return projections from place cells could provide environmental stability for the grids. See [23;24] and Figure 4f. Such an organization would be consistent with the disruption of grid cell firing by hippocampal inactivation [12]; the disruption of the grid cells, but not boundary vector cells or head direction cells, by disruption of the theta system [18;66]; and the observation of stable place cell firing prior to stable grid cell firing in early development studies [53;67]. This view predicts that the oscillatory component of place cell firing in [19] and Figure 1d reflects path integrative processes involving grid cells, while the ramp-like depolarization reflects input from boundary-vector cells or local cues and gates firing: determining the firing field and potentially affecting the numbers of spikes fired per run but not their theta-phase.

Conclusions

A range of models relate theta-phase precession in hippocampal place cell firing to membrane potential oscillations and recurrent connectivity. Some of these models also provide the basis of models of path integration and grid cell firing. However, environmental inputs are also a critical part of self location, for which boundary-vector cells may play a key role. This field has seen a recent explosion of both experimental results and computational models. Critically, computational modelling has begun to drive forward experimental advances by making testable predictions, potentially bringing this area of cognitive neuroscience into line with more establishes sciences.