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The Journal of Neuroscience, March 21, 2007, 27(12):3211-3229; doi:10.1523/JNEUROSCI.4724-06.2007
 Previous Article | Next Article 
Behavioral/Systems/Cognitive
Scale-Invariant Memory Representations Emerge from Moiré Interference between Grid Fields That Produce Theta Oscillations: A Computational Model
Hugh T. Blair,1
Adam C. Welday,1 and
Kechen Zhang2
1University of California, Los Angeles Psychology Department, Los Angeles, California 90095, and 2Department of Biomedical Engineering, Johns Hopkins University School of Medicine, Baltimore, Maryland 21205
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Abstract
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The dorsomedial entorhinal cortex (dMEC) of the rat brain contains a remarkable population of spatially tuned neurons called grid cells (Hafting et al., 2005 ). Each grid cell fires selectively at multiple spatial locations, which are geometrically arranged to form a hexagonal lattice that tiles the surface of the rat's environment. Here, we show that grid fields can combine with one another to form moiré interference patterns, referred to as "moiré grids," that replicate the hexagonal lattice over an infinite range of spatial scales. We propose that dMEC grids are actually moiré grids formed by interference between much smaller "theta grids," which are hypothesized to be the primary source of movement-related theta rhythm in the rat brain. The formation of moiré grids from theta grids obeys two scaling laws, referred to as the length and rotational scaling rules. The length scaling rule appears to account for firing properties of grid cells in layer II of dMEC, whereas the rotational scaling rule can better explain properties of layer III grid cells. Moiré grids built from theta grids can be combined to form yet larger grids and can also be used as basis functions to construct memory representations of spatial locations (place cells) or visual images. Memory representations built from moiré grids are automatically endowed with size invariance by the scaling properties of the moiré grids. We therefore propose that moiré interference between grid fields may constitute an important principle of neural computation underlying the construction of scale-invariant memory representations.
Key words: hippocampus; place cell; entorhinal cortex; grid cells; theta rhythm; recognition memory
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Introduction
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Invariant memory representations make it possible to recognize familiar stimuli independently of their size, position, or context. For example, a resized version of a previously encountered visual image (such as a familiar object viewed from a novel distance) can easily be recognized as familiar, despite the fact that the perceived size of the stimulus has changed. Scale-invariant stimulus recognition is not merely a perceptual phenomenon but seems to reflect underlying properties of the memory representations that encode the familiar stimulus (Jolicoeur, 1987; Biederman and Cooper, 1992). In mammals, the medial temporal lobe memory system is critical for storing memories of familiar stimuli (Eichenbaum, 2004; Squire et al., 2004), so structures within the medial temporal lobe, such as the hippocampus and entorhinal cortex, may contain neural substrates for encoding invariant memory representations. Supporting this, neurons in the human hippocampus and entorhinal cortex can respond invariantly to images of specific people and objects (Fried et al., 1997; Kreiman et al., 2000; Quiroga et al., 2005).
The rodent hippocampus contains neurons called place cells, which are thought to encode memories of specific spatial locations, because each place cell fires selectively whenever the rat returns to a familiar location in space (O'Keefe and Dostrovsky, 1971; O'Keefe and Nadel, 1978; Thompson and Best, 1990; Wilson and McNaughton, 1993). Some place cells seem to encode scale-invariant memory representations of familiar locations, because they can rescale their firing fields when the spatial environment is resized (Muller and Kubie, 1987; O'Keefe and Burgess, 1996; Sharp, 1999; Huxter et al., 2003). Hence, as in humans, neurons in the rodent hippocampus demonstrate a capacity for invariant memory coding.
Hippocampal place cells receive input from neurons called grid cells in the dorsomedial entorhinal cortex (dMEC) (Fyhn et al., 2004; Hafting et al., 2005). Here, we present a computational theory proposing that the hexagonal firing fields of dMEC grid cells are moiré interference patterns, or "moiré grids," formed from smaller "theta grids" which are hypothesized to be a primary source of theta rhythm in the EEG. Simulations show that this theory can account for the dorsoventral topography of grid field sizes in dMEC (Hafting et al., 2005; Sargolini et al., 2006a) and may also explain why grid cells in layers II and III of dMEC exhibit different phase relationships with the theta EEG (Hafting et al., 2006). We show that moiré grids formed by theta grids can serve as scalable basis functions for constructing size-invariant memory representations, and this may explain how hippocampal place cells rescale their firing fields when a familiar spatial environment is resized. We also show that if a visual image is constructed from a basis set of moiré grids, then the image can automatically be represented at many different sizes within the visual field. In conclusion, we hypothesize that moiré interference between grid fields is a fundamental computational principle underlying the construction of size-invariant memory representations by the nervous system and that small grid fields that produce theta oscillations are the elementary building blocks for constructing such representations in the rodent hippocampus.
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Materials and Methods
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All simulations were performed using the Matlab programming language (MathWorks, Natick, MA). Firing rate maps of simulated grid cells and place cells were represented by a square matrix of pixels, with each pixel representing the firing rate of the simulated cell at a fixed location.
Definitions of grid field parameters.
As a rat navigates through an open-field environment, grid cells in dMEC fire at multiple vertex locations, which form a hexagonal lattice that tiles the surface of the environment (Hafting et al., 2005). Such a hexagonal lattice can be characterized by three parameters (Fig. 1): the distance between adjacent grid vertices ( ), the rotational orientation of the grid ( ), and the spatial phase of the grid (x0, y0). In the rat brain, different grid cells are tuned to different values of these parameters. The firing field of any particular grid cell maintains stable values of these parameters over time within an unchanging spatial environment (Hafting et al., 2005).
Cosine grating model of theta grids.
The model presented in this study proposes that grid cells in dMEC are formed by moiré interference between smaller grid fields, referred to as theta grids because they are hypothesized to produce theta oscillations. We simulated theta grids by summing three cosine grating functions oriented 60° apart, which may be regarded as a simple Fourier model of the hexagonal lattice. This cosine grating model of theta grids was used for computational efficiency and is not intended to represent an accurate biological model of how theta grids are generated in the rat brain. The biological network for generating theta grids is likely to incorporate an attractor network for path integration (Fuhs and Touretzky, 2006; McNaughton et al., 2006), as explained in the Results and Discussion sections. To implement the cosine grating model used in our simulations, we use the Cartesian coordinate vector r = (x, y) to denote an arbitrary spatial position (one square pixel) within a simulated firing rate map. The firing rate of a theta grid G at each spatial location r was simulated as a sum of three cosine gratings as follows:
where the spatial phase of the grid (Fig. 1) was given by c = (x0, y0). The three cosine gratings were oriented along three vectors  1, 2, and 3, which were 60° apart from one another, and these gratings were rotated with respect to the grid field orientation parameter (Fig. 1) by angles of – 30°, + 30°, and + 90°. The three vectors had equal length |i| = , and the length of the vectors determined the vertex spacing of the theta grid, , according to the relation = 4 /31/2. Here, g was a monotonically increasing gain function given by g(x) = exp[a(x – b)] – 1 with a = 0.3 and b = –3/2. The summation of the three cosine functions had a minimum value of –3/2 and a maximum value of 3, so after passing through the gain function, the value of G(r) ranged from 0 to  3 (in arbitrary units). The purpose of the gain function was to disallow negative outputs, because neurons cannot have negative firing rates. The exact choice of gain function is unimportant for the moiré scaling effects reported here.
Spatial resolution of simulated rate maps.
Each square pixel in the firing rate maps corresponded to a square region of physical space. Simulations of moiré grids (see Figs. 4, 5) used pixels measuring 0.1625 cm on each side. Simulations of place cells and grid cells (see Figs. 6–9) used pixels measuring 0.65 cm on each side. Experimental firing rate maps of real place cells were plotted using pixels that measured 3.25 cm on each side; these experimental firing rate maps had to be resampled at a higher resolution to match the resolution of simulated grid fields before weighting coefficients for the model could be fitted to experimental data by Equation 16.
Smoothing of moiré grid fields.
Our theory assumes that theta grids change their vertex spacings as a function of running speed to keep the frequency of theta rhythm constant at all running speeds (see Eqs. 8–10). However, for simplicity, our simulations of place cell rescaling assumed that theta grids have a constant vertex spacing (as if the rat was always running at the same speed) for fixed values of k in Equations 17 and 18. A problem that arises from this simplification is that when a population of moiré grids is resized in unison by adjusting k, the high-frequency components of the moiré grids become shifted in phase relative to one another. The low-frequency components of the moiré grids remain in stable alignment after rescaling, but because the place map is simulated by summing individual pixels in Equation 15, and these pixels have a smaller spatial resolution than the high-frequency components that become realigned, the individual pixels of the moiré grids may no longer sum with one another to produce an output function that has been previously stored by weighting coefficients that were assigned with a different value of k. This problem can be regarded as an artifact of our simplifying assumption that theta grids have constant vertex spacings at all running speeds, because in the case in which theta grids change their vertex spacing as a function of running speed, the vertex spacing (and possibly spatial phase) of theta grids would vary on each traversal of the environment. Hence, moiré grids (and place cells formed from them) would not always fire at the same rate on each traversal through a given spatial location, because the high-frequency component of the grids would be aligned differently on each traversal at different running speeds. This variability would average out over multiple traversals, but it does not average out in our simulations, because we have abolished variability by assuming a constant running speed. To solve this problem, firing rate maps of moiré grid cells, Mi(r), were low-pass filtered by two iterations of smoothing with a square convolution kernel (denoted by K in Equations 11, 13, 17, and 18) measuring 2.0 cm on each side. Smoothing the moiré grids in this way is tantamount to averaging over the variability in the spacing and phase of theta grids that would normally occur over different traversals of the environment at different running speeds.
Experimental place cell recordings.
Target firing rate maps for our simulations were obtained by recording place cells from the CA1 region of the hippocampus in freely foraging rats (provided by A.C.W. and H.T.B.) using methods that have been reported previously (Moita et al., 2004). Briefly, male Long–Evans rats weighing 350–400 g were reduced to 85% of their ad libitum weight through limited daily feeding. Under deep isoflurane anesthesia, microdrives consisting of six tetrode bundles made from 0.0007'' microwire (Kanthal, Palm Coast, FL) were stereotaxically implanted into the dorsal CA1 layer of the hippocampus (coordinates, 3.3 mm posterior, ±3.0 mm lateral, and 1.7 mm ventral to bregma). After recovery, place cells were recorded while rats foraged for food pellets in open-field environments of varying shapes and sizes (see Results). Position data were sampled at 30 Hz by a video tracking system that monitored light-emitting diodes fixed to the rat's head, and single-unit spikes were recorded using multichannel data acquisition and cluster analysis software (Neuralynx, Tucson, AZ). Firing rate maps for each place cell were computed by binning spatial position data into square pixels measuring 3.25 cm on each side. Experimental procedures were approved by the University of California, Los Angeles Animal Care and Use Committee in accordance with federal regulations.
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Results
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Grid cells in dMEC exhibit vertex spacing lengths that range approximately between 30 and 100 cm (Hafting et al., 2005; Sargolini et al., 2006a), but our model predicts that dMEC grids are built from much smaller grid fields, referred to as theta grids, which we hypothesize to be responsible for producing theta oscillations. Hence, our model posits that at least two different types of grid fields are encoded in the rat brain: dMEC grids and theta grids. Before explaining how these two different grid populations are related to one another, it is necessary to review some general mathematical principles governing moiré interference between hexagonal grid fields.
Moiré interference between grid fields
Suppose that a target neuron receives convergent input from a pair of two different grid cells, each encoding a different grid field. The total input to the target cell would be greatest at spatial locations at which the vertices of the two input grids overlap. Consequently, the target cell would fire in a spatial pattern defined by the regions in which the two input grid fields intersect. It is demonstrated below that the intersection between two hexagonal grids forms a moiré interference pattern, which we shall refer to as a moiré grid, that replicates the hexagonal lattice on a larger spatial scale. Such moiré interference patterns emerge when any two periodic lattices are combined (regardless of their geometry), so the hexagonal lattice is not unique in this regard. But in the present analysis, we shall consider only the case of the hexagonal lattice, because this is the geometric pattern that is generated by grid cells in the rat brain. The size of moiré grid is determined by the parameters of the input grids in accordance with mathematical scaling laws that have been reviewed by Amidror (2000). Here, we present specific formulations of these scaling laws that are specialized for describing hexagonal moiré grids.
The length scaling rule
Consider two hexagonal basis grids that have the same angular orientation but different vertex spacing lengths (Fig. 2). If the vertex spacing length of one basis grid is denoted , then the spacing of the other grid can be denoted as +  , where is a constant representing the difference in spacing lengths between the two grids (expressed as a percentage of the first grid's spacing length). The intersection of two such grids forms a moiré grid with the same angular orientation as the basis grids but a larger vertex spacing length, S, where S is a scaling factor given by the following:
We refer to Equation 2 as the length scaling rule, because it defines the vertex spacing of the moiré grid as a function of the difference between the vertex lengths of the basis grids. The manner in which moiré grids are produced by the length scaling rule is very similar to the manner in which beat frequencies are produced by interference between sinusoids (Burgess et al., 2005; McNaughton et al., 2006). Just as two sinusoids with slightly different frequencies can sum to produce a lower beat frequency, Equation 2 states that two grid fields with slightly different vertex spacings can sum to produce a larger grid field (a moiré grid). Thus, grid fields can be regarded as two-dimensional (planar) analogues of one-dimensional (linear) sinusoids. As explained below, this kinship between grid fields and sinusoids links our moiré model of grid cells with previous "dual oscillator" models of hippocampal processing (O'Keefe and Recce, 1993; Lengyel et al., 2003; Burgess et al., 2005; O'Keefe and Burgess, 2005).
Moiré interference between grid fields is most robust when there is only a small difference between the vertex spacings of the basis grids (that is, when || is near zero); the moiré effect breaks down as || grows large (just as sinusoidal beat interference breaks down when the difference between input frequencies is too large). Figure 2c shows that the vertex spacing of the moiré grid can vary over several orders of magnitude as varies within a small range of values surrounding zero, and S approaches infinity as the vertex spacings of the basis grids approach equivalence (that is, as vanishes). The spatial phase of the moiré grid is determined by the phases of the two basis grids, but the vertex spacing of the moiré grid is unaffected by the spatial phases of the basis grids.
The rotational scaling rule
The length scaling rule alone does not provide a complete description of moiré interference between planar grid fields, because interference patterns can also be produced by rotating grid fields against one another (an operation which is not possible with sinusoids). When two hexagonal basis grids with identical vertex spacing are rotated against one another by an angle  , they intersect to form a moiré grid with vertex spacing S, where S is a scaling factor given by the following:
We refer to Equation 3 as the rotational scaling rule, because it defines the scale of the moiré grid as a function of the rotation angle between two basis grids with identical vertex spacings. Because this function is periodic at regular intervals of 60°, the moiré pattern formed when the grids are rotated against each other by an arbitrary angle is identical to the moiré pattern formed when the grids are rotated against each other by + N60° for any integer N. Thus, when analyzing the moiré interference patterns that are formed by rotating two hexagonal grids against one another by an angle , it is only necessary to consider angles on the interval 0°  < 60°. The angular orientation of the moiré grid formed by rotational scaling is rotated by 30° from the angle that is intermediate between the angles of the two basis grids:
where is the orientation of the moiré grid, and 1 and 2 are the orientations of the two basis grids that form the moiré grid. The spatial phases of the basis grids are absent from the equations above because the phases of the basis grids do not influence the vertex spacing or orientation of the moiré grid (they affect only the phase of the moiré grid). A relationship similar to the rotational scaling rule was first derived by Lord Rayleigh for parallel line gratings (Stecher, 1964). For the case of hexagonal grids, the rotational scaling principle can be derived by intuitive geometric arguments like those outlined by Oster (1969) or by methods based on Fourier spectra (Amidror, 2000).
Figure 3d shows how the scale S of the moiré grid varies as a function of the rotation angle between the basis grids in accordance with the rotational scaling rule. This relation increases monotonically without bound as approaches integer multiples of 60°. The smallest value of S occurs when = 30° + N60° for integer N, where the scaling function becomes nondifferentiable (for example, see Fig. 3d, the singular point at S = 30°). At these singular points, the moiré magnification factor is equal to S = 1/[2sin(30°/2)] = (2 + 31/2)1/2  1.93, an irrational number that we refer to as  . Because the scaling factor is irrational when S = , the moiré grid becomes an aperiodic tessellation of the plane that never exactly repeats itself (Fig. 3c).
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Figure 3. The rotational scaling rule. a, Two basis grids with identical vertex spacing and orientations which differ by angle intersect to form a moiré grid shown in black to the right of the basis grids. The vertex spacing of the moiré grid is S, where S is a scaling factor that depends on . In this example, = 6° and S = 9.55. b, Another example of a moiré grid formed by the rotational scaling rule, with = 12° and S = 4.73. c, When = 30°, the scaling factor becomes irrational (S = ), and the moiré grid becomes an aperiodic tessellation of the plane. d, The scaling factor S depends on the separation angle, , between the orientations of the basis grids (see Eq. 3); note the irrational singular point at = 30°, at which the scale of magnification is S = .
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In summary, a pair of basis grids with the same vertex spacing and different angular orientations can be combined using the rotational scaling rule to produce a moiré grid with vertex spacing ranging between and  , where is the vertex spacing of both basis grids. The spatial phase and orientation of the moiré grid are jointly determined by the phases and orientations of the two basis grids, but the vertex spacing S of the moiré grid is unaffected by the spatial phases of the basis grids, and depends only on the angle of rotation between them.
Generalized moiré scaling law
The length and rotational scaling rules can be combined into a single generalized scaling law that covers all cases in which two basis grids intersect to form a moiré grid, regardless of whether the basis grids differ in their vertex spacing, angular orientation, or both. In general, when two hexagonal basis grids with different vertex spacings, and + , are rotated against one another by an angle, , they intersect to form a moiré grid with vertex spacing S, where S is a scaling factor given by the following:
Variables and in Equation 5 are the same as defined above, and  = min {, 60° – } to abbreviate the minimum angle term from Equation 3. The angular orientation of the moiré grid is as follows:
where 1 and 2 are the orientations of the two basis grids as in Equation 4. Nishijima and Oster (1964) have provided a derivation of this generalized scaling rule for the case of straight-line gratings, and the results for hexagonal grids turn out to be similar. It can be verified algebraically that in the special case when the two grids have identical orientation (that is, = 1 – 2 = 0), Equation 5 reduces to the length scaling rule of Equation 2. Similarly, in the special case when the two grids have identical spacing (that is, = 0), Equation 5 reduces to the rotational scaling rule of Equation 3, and Equation 6 is equivalent to Equation 4 (taking into account that + N60° represents the same grid orientation for all integer N because of hexalateral symmetry).
Small grid fields produce theta oscillations
The moiré scaling principles outlined above have two important implications for spatial information processing by the grid cell network. First, output from grid cells in dMEC could converge on downstream target neurons to form moiré grids that have much larger vertex spacings than dMEC grid cells (because moiré grids always have larger vertex spacings than the input grids from which they are formed). Second, and perhaps more importantly, dMEC grids might themselves be moiré grids that are formed by summing together much smaller input grids. In this section, we show that small grid fields can produce the movement-related theta oscillations that are known to exist in the hippocampus and entorhinal cortex. We shall hypothesize that these theta grids are the elementary building blocks for constructing spatial memory representations.
Theta grids
Theta rhythm is a 6–8 Hz oscillation that synchronizes neural activity in the hippocampus and entorhinal cortex as a rat navigates through a spatial environment (Vanderwolf, 1969; Mitchell and Ranck, 1980). We hypothesize that this movement-related theta rhythm is produced by small hexagonal grid fields, similar to dMEC grids but with smaller vertex spacing lengths, which we shall refer to as theta grids. To understand how such miniature grid fields could cause theta oscillations during movement, consider a grid field with a small vertex spacing length of = 5.0 cm, and suppose that a rat traverses this grid field at a constant running speed of 35 cm/s. If the rat runs along a straight path that is perfectly aligned with a row of grid vertices, then it would pass through 35/5 = 7 grid vertices per second. If a burst of neural activity occurs at each vertex crossing, then the bursts would occur at a frequency of 7 Hz, which is within the frequency range of theta rhythm. More generally, the burst frequency f that would be produced as the rat runs along a row of grid vertices would be as follows:
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where V is the running speed of the rat (in cm/s), and is the vertex spacing of the grid field (in cm). Equation 7 provides an exact description of the burst frequency only when the rat runs along a straight path that is perfectly aligned with a row of grid vertices. If the rat runs along any other path through the grid, then the burst frequency would be lower than that specified by Equation 7. There are some paths along which vertices might be encountered much less frequently (for example, straight paths rotated exactly 30° from the alignment of the grid field) or even not at all (such as a curved path that avoids grid vertices by weaving in between them), and hence the burst frequency would be much slower than that specified by Equation 7. But a freely behaving rat would be unlikely to travel continuously along such a "vertex-sparse" path for an extended period of time. As long as the diameter of the grid vertices is not too small, most linear (or gently curved) paths through the grid would pass through a vertex at irregularly but similarly spaced intervals, and the mean interval distance would be very close to . Thus, Equation 7 provides a good estimate for the frequency of theta bursts that would be produced as a rat runs across a theta grid, even when the rat is not running straight along a row of vertices.
When the rat is standing still, Equation 7 implies that theta rhythm should fall silent, because f = 0/ = 0. In agreement with this, movement-related theta rhythm in the hippocampus usually falls silent when the rat stops running (Vanderwolf, 1969). However, Equation 7 would predict that the frequency of theta rhythm should be proportional to the rat's running speed with unit slope, so that when the rat runs twice as fast, the theta frequency should also double (because twice as many vertex points would be crossed in the same amount of time). Contradicting this prediction, the frequency of theta rhythm does not double when the rat's running speed doubles; instead, theta rhythm maintains a nearly constant frequency of 6–8 Hz across all running speeds. A slight increase in the theta frequency is reported as running speed increases (Whishaw and Vanderwolf, 1973; Rivas et al., 1996; Maurer et al., 2005), but the slope of this increase is much smaller than the unit slope predicted by Equation 7. Thus, experimental data contradicts our hypothesis that theta oscillations are rhythmic bursts produced as the rat traverses through small grid fields. This contradiction may be resolved by postulating that the vertex spacing of theta grids varies systematically with running speed.
Speed-dependent vertex spacing of theta grids
Theta grids can maintain a constant burst frequency by adopting a variable vertex spacing that is directly proportional to the rat's running speed. The vertex spacing of the theta grid must be larger when the rat is running quickly, and smaller when the rat is running slowly, so that the frequency of theta rhythm remains at a constant value. On first consideration, this idea might seem inconsistent with our hypothesis that dMEC grids are moiré grids constructed from theta grids. After all, grid cells in dMEC have stable vertex spacings at all running speeds, so how could they be formed from theta grids with vertex spacings that vary with the rat's running speed?
Consider the case of two theta grids, G1 and G2, which have the same angular orientation but different vertex spacings, so that the vertex spacing of a moiré grid formed by these two theta grids would be determined by the length scaling rule (Eq. 2). Suppose that the vertex spacing of G1 varies with running speed so that the frequency of vertex crossings on G1 (that is, the frequency of theta rhythm) remains constant for all running speeds. We may rearrange Equation 7 to estimate the vertex spacing of G1 as follows:
where (V) is the speed-varying vertex spacing of theta grid G1, and f1 is the constant frequency of vertex crossings (theta bursts) produced by G1. Now suppose that the theta grid G1 intersects with another theta grid, G2, to form a moiré grid. For simplicity, we shall assume that G2 always has the same angular orientation as G1, so that the spacing of the moiré grid is given by the length scaling rule (differing orientation is addressed in Appendix A). We shall denote the vertex spacing of G2 as (V)[1 + (V)], where (V) is the difference between the vertex spacing of G1 and G2, expressed as a percentage of the vertex spacing of G1. This formulation is similar to that used in the length scaling rule above, except that (V) is now dependent on running speed, a modification that makes it possible to form stable moiré grids from speed-varying theta grids.
The vertex spacing of the moiré grid formed by G1 and G2 is given by S(V), where the scaling factor S is computed from the length scaling rule (Eq. 2). It can be shown algebraically by combining Equations 2 and 8 that S(V) has zero slope (and thus the vertex spacing of the moiré grid is the same for all running speeds) as long as the following is true:
where M = S(V) is a constant value denoting the fixed vertex spacing of the moiré grid, and f1 denotes the fixed frequency of vertex crossings on G1 (the theta frequency). Hence, a pair of theta grids with speed-dependent vertex spacings can combine to form a moiré grid with a speed-independent vertex spacing, so long as the relative vertex spacings of the theta grids obeys the relation specified in Equation 9. If f1 is a fixed frequency which remains constant at all running speeds, then Equation 9 implies the following:
where f2 is the frequency of vertex crossings on grid G2. Hence, if f1 is constant for all running speeds, then f2 must vary with running speed to form a moiré grid with a stable vertex spacing M. The approximation in the second step of Equation 10 is quite good, so f2 increases almost linearly with V with a very small slope that is inversely proportional to M. It is interesting to speculate that Equation 10 might explain why the frequency of theta rhythm is sometimes observed to increase very slightly with the rat's running speed (Whishaw and Vanderwolf, 1973; Rivas et al., 1996; Maurer et al., 2005). If so, then the frequency of theta rhythm recorded in dorsal dMEC (where grid cells have small M) might be expected to increase more steeply with running speed than theta rhythm recorded in ventral dMEC (where grid cells have larger M). A biologically plausible mechanism for implementing Equation 9 will be suggested later (see below, A biological mechanism for grid field rescaling). It should be noted that Equation 9 is not the only solution for keeping M constant at all running speeds while the vertex spacing of theta grids varies. As outlined in Appendix A, other solutions are possible when theta grids differ in their angular orientation instead of their vertex spacing.
Alignment of theta grids with the spatial environment
Hafting et al. (2005) have shown that dMEC grids maintain a stable spatial alignment with familiar visual landmarks. If dMEC grids are moiré grids formed from smaller theta grids, then does this imply that the theta grids must also remain in stable alignment with landmarks? No, not if theta grids change their vertex spacing with running speed to keep the frequency of theta rhythm constant. A moiré grid always has two distinct spatial frequency components: a low-frequency component (the moiré grid itself, which we equate with dMEC grids) and a high-frequency component (the underlying theta grids). The low-frequency component (the moiré grid) can continuously remain in stable alignment with the environment (as dMEC grids appear to do) while the high-frequency theta grids shift their alignment with the environment by dynamically altering their vertex spacing or spatial phase. For example, if the rat's running speed differs on two independent traversals through an environment, then the vertex spacings of theta grids would also be different on each traversal, and theta bursts must therefore occur at different locations with respect to stationary landmarks.
Clearly, the moiré grid must somehow establish and maintain a stable relationship with familiar spatial landmarks, but how can this stability be achieved if theta grids (from which moiré grids are formed) do not maintain a stable alignment with these same landmarks? There are three relationships to consider in answering this question: the alignment of theta grids with landmarks, the alignment of theta grids with moiré grids, and the alignment of moiré grids with landmarks. Importantly, it is possible for each of these three relationships to vary independently of one another. We shall propose later that theta grids are produced by a path integration network (see below, A biological mechanism for grid field rescaling). Prior path integration models have proposed that the integrator can periodically reset its position estimate with respect to visual landmarks, to correct for the accumulation of integration errors over time (Skaggs et al., 1995; Samsonovich and McNaughton, 1997). In a similar way, the theta grid path integrator might periodically reset all three relationships in the moiré network (theta grids to landmarks, theta grids to moiré grids, and moiré grids to landmarks) to correct for the accumulation of integration errors over time. But in between such "reset" events, theta grids could alter their alignment to both the environment and the moiré grids, without upsetting the alignment between the environment and the moiré grids. Therefore, in its current form, our theory does not offer specific predictions about the stability of the alignment between theta grids and the environment over time.
Sensitivity of the moiré scaling mechanism
The slopes of the moiré scaling functions are quite steep (Figs. 2c, 3d), so small fluctuations in the spacing or orientation of theta grids would cause large fluctuations in the size of moiré grids produced by the theta grids. This might be problematic in a biological network, because small levels of input noise could cause large fluctuations in the scale of the spatial representation produced by the network. Clearly, the accuracy of the spatial relationships between theta grids would have to be maintained within very strict tolerances to produce moiré grids with stable vertex spacings, and this is potentially a serious limitation of our model.
Simulations of dMEC grid fields
We shall now present simulations to demonstrate that key firing properties of dMEC grid cells can be accounted for by the hypothesis that dMEC grids are moiré grids formed from smaller theta grids. For simplicity, our simulations shall assume that theta grids have constant vertex spacings (as if the rat is always traveling at a constant running speed), because this makes it easier to visualize the moiré interference principles that we wish to demonstrate. Thus, running speed will not be included as a variable in the model equations. However, it should be understood that the principles of moiré interference that we describe in our simulations are not altered by this simplifying assumption, and in the rat brain, theta grids may have vertex spacings that vary with running speed to keep the theta frequency nearly constant (as specified by Equations 8 and 9 and in Appendix A).
Simulating layer II grid cells by the length scaling rule
Grid cells in layer II of dMEC are topographically organized so that cells in dorsal regions of dMEC have small vertex spacings of 30 cm, and the spacing grows progressively larger at more ventral locations, reaching a maximum of 100 cm in the most ventral portion of dMEC (Hafting et al., 2005; Sargolini et al., 2006a). Layer II grid cells also exhibit phase precession with respect to the locally recorded theta EEG (Hafting et al., 2006), so that as the rat traverses each grid vertex, the grid cell fires at a late phase of the theta EEG cycle as a rat enters the vertex and at an early phase as the rat leaves the vertex. This is similar to theta phase precession that is observed for place cells in the hippocampus (O'Keefe and Recce, 1993; Skaggs et al., 1996; Maurer et al., 2006).
O'Keefe and Burgess (2005) have hypothesized that both of these properties of grid cells (the dorsoventral topography of their grid spacings and their theta phase precession behavior) can be accounted for by interference between sinusoids of differing frequency. Burgess et al. (2005) extended this idea into two dimensions, showing that these grid cell firing properties can be similarly accounted for by interference between hexagonal grid fields with different vertex spacings. Here, we shall demonstrate a similar result, by showing that these grid cell firing properties are inherent to moiré grids that are formed using the length scaling rule of Equation 2.
To simulate a population of layer II grid cells using the length scaling rule, a set of moiré grids was produced from "sibling pairs" of theta grids. The firing rate of the ith moiré grid cell at spatial location r was given by the following:
where Gi(r) and Gi((1 + i)r) are the ith pair of sibling theta grids, µM = 4.0 is an activation threshold, []+ denotes thresholding such that the expression inside the brackets is taken to be zero for negative values, * is the convolution operator, and K is a kernel for smoothing the moiré grid field (see Materials and Methods). The ith theta grid cell Gi(r) was simulated by the cosine grating model described in the Materials and Methods section, and its sibling Gi((1 + i)r) was derived by scaling Gi(r) about the origin of the coordinate axis. Both of the sibling theta grids, Gi(r) and Gi((1 + i)r), had the same angular orientation i, so that the orientation of the moiré grid Mi(r) was also equal to i. In simulations presented here, the vertex spacing of the first theta grid Gi(r) was always = 5.0 cm, and the spacing of the second theta grid was given by + i (with i > 0). Thus, the vertex spacing of each Mi(r) was determined solely by its i parameter.
Figure 4 (top row) shows five examples of simulated moiré grid fields Mi(r) spanning a range of vertex spacings from 40 to 80 cm along the dorsoventral axis of dMEC. To obtain this gradation of grid spacings, the value of the i parameter was incrementally changed along the interval 0.1429  i 0.0667, which produced moiré grid spacings in the range 40 Si 80 cm when = 5.0 cm (this can be verified from Equation 2). These simulations suggest that the topography of vertex spacings for layer II grid cells may arise because the i parameter is graded along the dorsoventral axis of dMEC. That is, ventrally located grid cells receive inputs from pairs of theta cells with very similar vertex spacings (i very near to zero), thereby producing large moiré grid fields. In contrast, grid cells in dorsal layer II receive input from pairs of theta grids with less similar vertex spacings (i larger but still near zero), thereby producing small moiré grid fields. This explanation is very similar to O'Keefe and Burgess' (2005) hypothesis that the topography of grid spacings in dMEC arises from a dorsoventral gradient of frequency pairings between sinusoidal oscillators, except that here, theta grid vertex spacings are substituted for sinusoidal frequencies.
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Figure 4. Simulations of layer II grid cells. Top row, Five moiré grid fields, Mi(r), produced by the length scaling rule (Eq. 2); the spacing difference (i) between theta grids is shown above each plot, and the vertex spacing (Si) of each moiré grid is shown at the top left corner of each plot (the hypothetical gradation of i along the dorsoventral axis of dMEC runs from left to right). Second row, A magnified inset of one moiré grid vertex and three different horizontal cross sections (a, b, and c) through the vertex. Third row, Magnified insets of the theta grids, with Gi(r) plotted in red, Gi((1 + i)r) plotted in green, and Mi(r) plotted in black. Fourth row, Simulated firing rates (y-axis) of the moiré grid cell (black) and theta grids (red and green) along three horizontal cross sections (columns a, b, and c). Fifth row, The phase of theta rhythm at which Mi(r) peaks on each cycle, with left-to-right traversals shown by gray arrows and right-to-left traversals shown by black arrows; the slopes of these plots show that phase precession occurs as the rat traverses the vertex in either direction (theta EEG is assumed to correspond to the green grid; see Results, Simulating layer II grid cells by the length scaling rule).
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The moiré grid vertices produced by the length scaling rule have a grainy appearance, which is caused by the high spatial frequency of the underlying theta grids from which the vertices are formed. Figure 4 (second row) shows a magnified view of a single grid vertex, to illustrate how the vertex is produced by moiré interference between the underlying pair of theta grids. Figure 4 (third row) shows a magnified view of the sibling theta grids that form the vertex, with Gi(r) plotted in red and Gi((1 + i)r) plotted in green; black regions in this plot show the points of overlap between the theta grids, which are locations where the grid cell Mi(r) would fire action potentials. The phase relationship between theta rhythm and the firing of layer II grid cells can be inferred from this plot by assuming that the peaks and troughs of the theta EEG correspond to the peaks and troughs of the theta grid plotted in green, Gi((1 + i)r). The black spots at which the grid cell fires can be seen to precess backward in phase through one cycle of the theta rhythm on each traversal of a grid vertex in any direction. To better illustrate this, Figure 4 (fourth row) shows horizontal cross sections through the grid vertex at varying distances from the vertex center (a, b, and c). In cross section, the two theta grids become a pair of detuned sinusoids (red and green), and the moiré grid vertex becomes a sequence of small bumps that together form a larger bump (black). The phase precession phenomenon emerges from interference between the sinusoids, exactly as in dual oscillator models of phase precession by place cells while rats are running on a linear track (O'Keefe and Recce, 1993; Lengyel et al., 2003; O'Keefe and Burgess, 2005). Hence, the length scaling rule can properly be regarded as an extension of these dual oscillator models into two dimensions (Burgess et al., 2005).
Figure 4 (bottom row) plots the phase of theta rhythm at which the grid cell fires (y-axis) for each horizontal position along the grid vertex (x-axis). Time progresses either from left to right or right to left along the x-axis, depending on which direction the rat is moving (rightward vs leftward, respectively). The phase plot is negatively sloped with respect to position when the rat travels rightward (gray triangles), and positively sloped when the rat travels leftward (black triangles), so grid cell spiking always precesses backward through the phase of theta rhythm. It can also be seen in the two-dimensional theta grid plot (Fig. 4, row 3) that backward phase precession occurs regardless of the angle at which the cross section is taken through the grid vertex. It must be assumed that the local theta EEG in layer II measures vertex crossings on the larger theta grid (green) and not the smaller one (red), because if the theta EEG were produced by vertex crossings on the smaller theta grid, then grid cell spikes would shift in the wrong direction through the theta phase (forward instead of backward), in conflict with experimental data. To account for this, it may be assumed that all of the grid cells within a local region of layer II receive shared input from a common "master" theta grid, so that their membrane potentials oscillate synchronously to produce a field potential that is observable as theta rhythm in the EEG. In addition to input from the master grid, each grid cell may also receive input from a secondary theta grid with a smaller vertex spacing than the master grid, which would interfere with the master grid to form a moiré grid. Unlike the master grid, which is shared among all grid cells, the secondary theta grid would be different for each dMEC cell. Thus, the secondary grids would not oscillate in synchrony and would not be detectable by EEG. The secondary theta grid signal might arrive through unique synaptic inputs to each grid cell or it might somehow be encoded by intrinsic oscillatory membrane properties that are unique to each grid cell, as in previous implementations of dual oscillator phase precession models (Lengyel et al., 2003).
Simulating layer III grid cells by the rotational scaling rule
Grid cells in layer III of dMEC seem to differ from grid cells in layer II in two major respects. First, the dorsoventral gradation of grid spacings is poorly organized, so that neighboring layer III grid cells can have very different vertex spacings even though they are located at the same dorsoventral coordinate position (Sargolini et al., 2006a). Second, layer III grid cells often do not show theta phase precession during traversal of a grid vertex (Hafting et al., 2006). These firing properties suggest that layer III grid cells are not formed by the length scaling rule. It is demonstrated below that these firing properties layer III grid cells might instead be explained by the rotational scaling rule.
To simulate a population of dMEC grids using the rotational scaling rule, moiré grids were again produced from sibling pairs of theta grids, except that this time, the sibling theta grids differed in their angular orientations rather than their vertex spacings. It was mathematically convenient to obtain each sibling pair of theta grids by rotating a common "parent" theta grid by the same angle  i in opposite directions (thus, the orientation of the parent grid was intermediate between the orientations of the siblings). Using this formulation, the orientation i of the moiré grid always remained constant when the angle between the theta grids was adjusted by changing i. By Equation 4, the orientation of the parent grid must equal i – 30° to produce a moiré grid with orientation i. Hence, the orientations of the two siblings were – 30° + i and – 30° – i, and the rotation angle between the two theta grids that formed the moiré grid was as follows:
Using this method for obtaining theta grids, the firing rate of the ith moiré grid cell at location r was given by the following:
where Gi(R(–i)r) and Gi(R(+i)r) are the ith pair of sibling theta grids, and all other terms are exactly as in Equation 11 above. The siblings Gi(R(–i)r) and Gi(R(+i)r) were derived by rotating a common parent grid Gi(r) by equal angles in opposite directions using the following rotation matrix:
Both theta grids had the same vertex spacing, which was always equal to = 5.0 cm. The parent grid Gi(r) was simulated using the cosine grating model (see Materials and Methods), and the orientation of the ith parent grid was defined as i – 30°, so that the orientation of the ith moiré grid Mi(r) would be equal to i in accordance with Equation 4.
Figure 5 (top row) shows five examples of simulated moiré grid fields Mi(r) spanning the range of vertex spacings from 40 to 80 cm. To obtain this gradation of grid spacings, the value of the i parameter was incrementally changed along the interval 3.58° i 1.79°. It can be verified from the rotational scaling rule (Eq. 3) that this range of separation angles between theta grids produces moiré grid spacings in the range 40 Si 80 cm when = 5.0 cm. Hence, unlike in layer II, in which grid cell spacings were controlled by the i parameter, the vertex spacing of layer III grid cells was controlled by the i parameter. If the i parameter is graded along the dorsoventral axis of dMEC, but the i parameter is not, then this could explain why layer II grid spacings are topographically organized along the dorsoventral axis, whereas layer III grid spacings are not as rigidly organized along the dorsoventral axis (Sargolini et al., 2006a).
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Figure 5. Simulations of layer III grid cells. Top row, Five moiré grid fields, Mi(r), produced by the rotational scaling rule (Eq. 3); the separation angle (i) between theta grids is shown above each plot, and the vertex spacing (Si) of each moiré grid is shown at the top left corner of each plot. Second row, A magnified inset of one moiré grid vertex, with three different horizontal cross sections (a, b, and c) through the vertex. Third row, Magnified insets of the theta grids, with Gi(R(+i)r) plotted in red, Gi(R(–i)r) plotted in green, and Mi(r) plotted in black. Fourth row, Simulated firing rates (y-axis) of the moiré grid cell (black) and theta grids (red and green) along three horizontal cross sections (columns a, b, and c). Fifth row, The phase of theta rhythm at which Mi(r) peaks on each theta cycle for left-to-right traversals (gray arrows) and right-to-left traversals (black arrows). Lateral phase shifting can be seen by comparing across columns a, b, and c. In column b, the rat passes slightly below the center of the vertex, so Mi(r) fires at a slight phase shift from the middle of the theta cycle; in column c, the rat passes farther below the center of the vertex, so Mi(r) fires at a large phase shift from the middle of the theta cycle; in column a, the rat passes above the center of the vertex (farther from the center than in b but not as far as in c), so Mi(r) fires at a moderate phase shift from the middle of the theta cycle (note that in a, the direction of the phase shift is opposite from b and c, because the rat is passing above the vertex center rather than below it). Theta EEG is assumed to correspond to the green grid as in Figure 4.
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Figure 5 (second row) shows a magnified view of a single grid vertex, and Figure 5 (third row) shows a magnified view of the sibling theta grids that form the vertex, with Gi(R(–i)r) plotted in red and Gi(R(+i)r) plotted in green; black regions show the points of overlap between the theta grids, which are locations at which the grid cell Mi(r) would fire action potentials. The phase relationship between theta rhythm and the firing of layer III grid cells can be inferred from this plot by assuming that the peaks and troughs of the theta EEG correspond to the peaks and troughs of Gi(R(+i)r), the grid plotted in green. Notice that the black spots, at which the grid cell fires, remain at the same phase within the theta rhythm on each traversal of a grid vertex in any direction (although there is a lateral shift of the black spots, addressed further below). To better illustrate this, Figure 5 (fourth row) shows horizontal cross sections through the grid vertex at varying distances from the vertex center (a, b, and c). In cross section, the two theta grids become a pair of sinusoids with the same frequency (red and green), and the moiré grid vertex becomes a sequence of small bumps that together form a larger bump (black). Figure 5 (bottom row) plots the phase of theta rhythm at which the grid cell fires (y-axis) at each position within the grid vertex (x-axis) as the rat traverses the vertex in either direction. The phase plot is always flat for both left-to-right traversals (gray triangles) and right-to-left traversals (black triangles), showing that the grid cell fires at the same theta phase throughout each traversal in either direction; thus, phase precession does not occur.
Lateral phase shifting: a novel prediction of the model
Although phase precession does not occur for grid cells formed by the rotational scaling rule, the phase at which the grid cell fires does depend on where the rat passes through the vertex. Figure 5 shows that when the rat passes near the center of the vertex (row b), the grid cell fires at a constant phase near the middle of the theta cycle. When the rat passes through the edges of the vertex, the grid cell fires early in the theta cycle at one edge and late in theta cycle at the other edge (rows a and c). Consequently, the grid cell exhibits lateral phase shifting from the left to the right edges of the grid vertex, rather than from the front to the back of the vertex as in classical phase precession.
This lateral phase-shifting effect is a prediction of the model that can be tested by recording grid cells from dMEC in an open-field environment. If some dMEC grid cells are formed by the rotational scaling rule, then these cells should exhibit lateral phase shifting of their spikes with respect to the local theta EEG. Hence, whenever the rat passes through a grid vertex on one side (e.g., the left edge), the grid cell's spikes should occur at a particular phase within the theta cycle (e.g., early). If the rat passes through a vertex of the same grid cell on the opposite side of its center (e.g., the right edge), then the grid cell's spikes will occur at the opposite phase within the theta cycle (e.g., late). If this prediction were to be verified, it would provide very strong evidence that some dMEC grids are indeed formed by the rotational scaling rule and thereby support our hypothesis that theta rhythm is produced by grid fields with small vertex spacings.
Simultaneous forward and lateral phase shifting
Some dMEC grids might be moiré grids which are formed by the generalized scaling rule of Equation 5, so that they are constructed from theta grids that differ both in their orientation and their vertex spacing (rather than just one or the other, as in Figs. 4 and 5). In such cases, grid cells would simultaneously exhibit standard phase precession (as in Fig. 4) as well as lateral phase shifting (as in Fig. 5). Hence, grid cells in dMEC might exhibit both forward and lateral phase shifting at the same time, rather than just one or the other, depending on how they are formed from theta grids.
Building place cells from grid cells
Place cells in the hippocampus receive inputs from grid cells in dMEC (Fyhn et al., 2004), so place cells could form their location-specific firing fields by summing inputs from dMEC grid cells (Fuhs and Touretzky, 2006; McNaughton et al., 2006, Solstad et al., 2006). To simulate location-specific firing in place cells receiving convergent input from a population of moiré grid cells, we used a two-stage feedforward model (Fig. 6a). In the first stage, sibling pairs of theta grids were combined to form moiré grid cells, Mi(r). In the second stage, outputs from Mi(r) were linearly summed to create a place cell firing field according to the following equation:
where r = (x, y) denotes an arbitrary Cartesian position within a simulated firing rate map, P(r) is the output of the model, which corresponds to the mean firing rate of a simulated place cell at location r, Mi(r) is the mean firing rate at location r of the ith moiré grid cell, N is the number of grid cells projecting onto the simulated place cell, and w1, w2, · · ·, wN are the weighting coefficients of synaptic inputs onto the place cell from each grid moiré cell. For place cell simulations presented here, the activation threshold µP was set equal to 25% of the maximum value of   = 1 wiMi(r) over all locations r.
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Figure 6. Building place cells from grid cells. a, Diagram of the two-stage model of place cells. b, Path plots (left) and firing rate maps (right) for three real place cells that were recorded in rectangular, circular, and square environments. c, Simulated firing rate maps produced by the two-stage model (see Eq. 15) with different numbers of grid cells in the basis set (N = 100, 50, 10, or 5).
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Input to the feedforward model was provided by a basis set consisting of moiré grid cells, Mi(r). Moiré grid cells can be generated using either the length scaling rule to simulate layer II grid cells (Eq. 11) or the rotational scaling rule to simulate layer III grid cells (Eq. 13). For simplicity, place cell simulations presented here used a basis set of moiré grids formed only by the length scaling rule (Eq. 11). Parameters of the simulated grid cell population |
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Abstract
Introduction
