Research ArticleNEUROSCIENCE

Connecting multiple spatial scales to decode the population activity of grid cells

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Science Advances  18 Dec 2015:
Vol. 1, no. 11, e1500816
DOI: 10.1126/science.1500816
  • Fig. 1 Decoding spatial relations from single grid cells and single grid cell modules.

    (A) Grid cells fire when the animal traverses the vertices of an internally generated hexagonal grid tiling the environment. However, counting the activity bouts of a single grid cell does not convey a measure of distance. Depending on the angle of the animal’s motion relative to the grid cell’s lattice, bumps in the spatial firing map will be encountered at spacings such as λ, Embedded Image, or Embedded Image. The periodicity of grid cell activity also makes it impossible to uniquely decode spatial location from a single grid cell. a.u., arbitrary units. (B) Nearby grid cells within a module share a common lattice orientation and scale, as depicted by the firing patterns of three idealized grid cells. The population activity repeats periodically as well; thus, it is impossible to decode the position at the population level, at least not on the basis of a single module when firing fields are set more closely than the dimensions of the environment. (C) Grid cell modules with different single-cell grid orientations could be decoded unambiguously but are not observed experimentally. As a consequence, spatial location can only be decoded by combining grid cell modules with multiple spatial scales. (D) Indeed, grid cells have lattices whose length scales λk form a discrete set, ranging from coarse to fine [figure adapted from Stensola et al. (5)]. The trajectories of a rat in a 2.2 × 2.2 m2 enclosure are shown in gray; spikes from four grid cells are shown in red.

  • Fig. 2 Reading a multiscale periodic code.

    (A) Tuning curves of four nested modules with M = 20 grid cells and evenly spaced phases. The animal’s position yields a spike vector n in each module. The likelihood P(x|n) at that scale depicts the probability of being at a certain location, given the respective spike vector. Modules with smaller spatial periods λ have more localized likelihoods, but their multiple peaks result in ambiguous position estimates. The joint likelihood given the responses of all modules, shown in gray, is highly localized and nonperiodic. The overall ML estimate is closer to the animal’s position than x0, the ML estimate of the first module. (B) All ML estimates are determined by population vectors (PVs), which are formed by assigning each position x to a phase on the unit circle, weighting the number of spikes of each cell by its preferred phase, and then summing, as shown for the first two modules. (C) These PVs can be combined for refining the position estimate, similar to how the hour and minute hands of a clock are combined to read the time of the day. In a clock, the ratio of successive scales is 12, as there are 12 hours in each half-day, and in each hour, the minute hand completes one full cycle. (D) However, the scale ratio for successive grid modules is not generally an integer [the example in (A) has a ratio of 3:2]; hence, at the next scale, a new PV refines the position estimate by using the earlier estimate xi as the center of the range of possible values for xi+1 (Eq. 3). The refined estimate x1 in (A) is close to but not identical with the ML estimate from this module. Further estimates taking into account modules 2 and 3 are recursively calculated (eq. S13). Histograms of these estimates for 213 realizations of the spike vector n are shown in colors corresponding to the different modules in (A). The relative SDs σ/λ0 highlight that the estimate at each scale successively refines the position estimate (simulation parameters: nmax = 2, κ = 2, and s = 3/2). (E) In two dimensions, the periodicity of the lattice means that the unit cell (black hexagon) can be mapped onto a torus. The position Embedded Image can be read out like a two-dimensional (2D) clock with multiple scales.

  • Fig. 3 At the population level, a periodic representation of 2D space results only for aligned grid lattices.

    (A) Four hundred neurons with randomly phase-shifted, yet aligned grid-like tuning curves yield a posterior probability distribution Embedded Image that is hexagonal (left). If the lattices are randomly oriented, the hexagonal structure disappears (right). (B) The degree of variation in the lattice orientations strongly affects the hexagonal structure in Embedded Image, as measured by its “gridness” (1). (C) Even for randomly oriented lattices, the population response can be decoded by an ideal observer; if the number of neurons is small, aligned lattices result in a lower root mean square (RMS) error. Randomly positioning the lattices, as opposed to evenly spacing them, worsens the error. Size of square box, 1 m2.

  • Fig. 4 Decoding position in two dimensions.

    (A) The hexagonal lattice has three wave vectors, Embedded Image, Embedded Image, and Embedded Image, spaced 60° apart. One can transform the hexagonal unit cell into different equally sized rectangles: Form three such rectangles so that the short edge aligns with the Embedded Image’s. Compute the population vector estimate of the position μl along the short edge of the rectangle, averaging across the long edge. For each rectangle, this yields a position estimate along the axis Embedded Image, without specifying the position in the orthogonal direction. At the height μl, draw a line parallel to the long edge in each rectangle. If the projected position estimates are exact, the three resulting lines will meet at one point, the true position of the animal. Otherwise, the three lines form an equilateral triangle, whose center is the ML solution Embedded Image. (B) The grid cell population response can yield a homing vector in egocentric coordinates to any point in the environment, such as the location of the nest (purple) or a reward (orange). Topographically rearranging the cells according to spatial phase yields a spike count map. The population vector is formed by multiplying the spike count with cosine gratings, which are aligned along the three axes of the hexagonal lattice. Each such grating is complemented by a weight function phase-shifted by 90° (not shown). The phase of the gratings determines where the homing vector points; rotating the phase of the weights shifts the vector from pointing to the nest to pointing to the reward location.

  • Fig. 5 Combining population vectors at different scales.

    (A) For each scale, there is a reference frame set by the position estimate from the previous scale. This guarantees that the correction of the position estimate lies within the hexagonal unit cell at the next scale. (B) At each scale, periodicity maps the unit cell in 2D Euclidean space onto a torus. The population vectors from the corresponding module yield a vector from the origin (circles) to the estimate of current position (star). As shown in (A), this estimate sets the origin of the coordinate system at the next scale. A linear sum of the estimates (θii) at each length scale, multiplied by weights Wi, produces a precise estimate of the homing vector. These Wi’s are functions of the ratio s = λk+1k of successive length scales. As long as the longest length scale λ0 is large enough to cover the local environment, the homing vector maps directly back onto Euclidean 2D space.

  • Fig. 6 Failures of decoding and the alignment of orientations across modules.

    Take two modules whose spatial periods are in the ratio s = λ01 = 2 and rotate the finer-scale module’s lattice orientation by an angle φ relative to the first module. (A) The expected logarithm of the posterior distribution Embedded Image when the two lattices are aligned, given that the true position is at Embedded Image. (B) Suppose that the population vector at scale λ0 yields an estimate Embedded Image. This Embedded Image centers the fundamental domain of the module with the smaller lattice scale λ1 (blue dashed lines), but now the true position must lie within the smaller fundamental domain. If it does not, as shown above, then the refinement stage of decoding will try to estimate Embedded Image and not Embedded Image. Here, ℒ11 is the fundamental domain at length scale λ1 centered at Embedded Image. For a scale ratio of s = 2, this leads to an error of λ0/2, where λ0 is the distance between firing fields at the coarsest scale. (C) If the lattices in the second module are rotated by π/6 relative to the first, the side peaks Embedded Image move toward the vertices of the fundamental domain. Note that the effective spatial period in Embedded Image also shrinks, compared to the case of aligned orientations. (D) An error at the coarser scale is “corrected” toward Embedded Image, which lies close to the vertex. (E) As the second module’s orientation varies, the expected variance of Embedded Image at a given spike count peaks at π/6. A value of Embedded Image was chosen for the posterior distribution (eq. S5). (F) This effect is most pronounced in the regime of low neuron numbers M or low spike counts nmax, which leads to the small concentration parameter Embedded Image for the posterior probability distribution. The width of the posterior distribution is inversely related to the concentration parameter Embedded Image.

  • Fig. 7 For grid cell decoding to be robust, the ratio of length scales in successive grid cell modules should be below 3/2.

    (A) Comparison of the log posterior probability log[P(x|n)] for two scenarios with two modules each, but with different scale ratios s = λ01 > 1. Red shading indicates regions in which the second module increases P(x|n); blue shading represents a decrease. (B) For a grid network composed of four modules with M = 64 neurons each, the histogram shows the positions decoded from the spike count relative to the true positions, which were chosen at random 215 times. The length scale of each module was one-half of the next coarser module (s = 2), such that the modules interfered. The thin hexagon delineates the spacing between firing fields at the coarsest length scale λ0, whereas the thick inscribed hexagon is the unit cell of the lattice. The maximum of the neuron’s tuning curve was 2; the tuning curve’s shape parameter was κ = 2. (C) Spatial information in the four-module network as a function of the scale ratio s, which reaches a maximum around s = 3/2. (D) The optimal scale ratio s depends on the expected number of spikes 〈n〉 across all four modules and falls into discrete levels. (E) The RMS error for the optimal scale ratio s, relative to a 1-m2 enclosure.

  • Fig. 8 Localization beyond the spatial scale of the largest module.

    (A) For the grid scales λ = {9,7,4}, the Chinese Remainder Theorem holds: for any set of integers 0 ≤ ni ≤ λi, there is an integer m that satisfies m mod λi = ni. The λi’s in this case form a co-prime set; that is, they have no common divisors. (B) On the other hand, the grid scales λ = {9,6,4}, which form part of a geometric progression with s = λii+1 = 3/2, are not co-prime. If we treat ni as the discrete phase of the ith module’s neuronal response, then not all combinations of 0 ≤ ni ≤ λi are allowed to occur. If they do occur, error correction in the readout could map the response to the closest valid combination of phases. (C) Decoding the position (x,y) = (4.83,−2.32) from 200 realizations of the population vectors across L = 3 modules. The fundamental domain ℒ0 at the largest scale has a unit area. The ratio between the chosen spatial periods is s = 7/5, implying that the range of the code increases twentyfivefold. (D) For a smaller number of neurons and lower firing rates, modular arithmetic fails. One thousand two hundred realizations are shown. The scatter of decoded positions is not completely random but reflects ancillary peaks in the posterior probability, whose spatial positions are dictated by the scale ratio s = 7/5. The vertices of the dashed hexagon, which is exactly 52 times the size of ℒ0, represents but one subset of these ancillary peaks. (E) Discrete spatial scales imply that the log posterior probability for the model reflects the sum of sine waves with different spatial frequencies, shown here with frequencies in the ratio of 7:5. Blue and red dots indicate the maxima of the individual sine components. When these points draw close together, a secondary maximum is observed in the sum (purple line). Robust encoding requires that the true position x = 0 be much more likely than any positions corresponding to secondary maxima. (F) The probability of making catastrophic errors in the 2D plane using L = 2 modules. Such errors correspond to population vector decoding yielding a position outside of the fundamental domain ℒ0 around the true position. Samples (totaling 215) are drawn for each scale ratio s = p/q = {3/2,4/3,5/3,7/5,9/5}. Reliable decoding requires that both p and q be small.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/1/11/e1500816/DC1

    Text

    Fig. S1. Hierarchical self-similar scales enable error correction.

    Fig. S2. Noninteger scale ratios imply that the decoding algorithm must be able to rotate population vectors.

    Fig. S3. The lengths of the population vectors Formula along the hexagonal grid’s axes are correlated.

    Fig. S4. Possible distributed representations of the position vector estimate Formula of eq. S6 across a population of readout neurons.

    Fig. S5. Comparison of different decoding schemes for a single module in two dimensions.

    Fig. S6. Comparison of different decoding schemes for multiple modules in two dimensions.

    Fig. S7. If the grid scales are not organized into discrete modules, population vector decoding is no longer straightforward.

    Fig. S8. Continuum decoding requires multiple population vectors.

    References (4650)

  • Supplementary Materials

    This PDF file includes:

    • Text
    • Fig. S1. Hierarchical self-similar scales enable error correction.
    • Fig. S2. Noninteger scale ratios imply that the decoding algorithm must be able to rotate population vectors.
    • Fig. S3. The lengths of the population vectors Kl along the hexagonal grid’s axes are correlated.
    • Fig. S4. Possible distributed representations of the position vector estimate μ of eq. S6 across a population of readout neurons.
    • Fig. S5. Comparison of different decoding schemes for a single module in two dimensions.
    • Fig. S6. Comparison of different decoding schemes for multiple modules in two dimensions.
    • Fig. S7. If the grid scales are not organized into discrete modules, population vector decoding is no longer straightforward.
    • Fig. S8. Continuum decoding requires multiple population vectors.
    • References (46–50)

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