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J Neurosci. Author manuscript; available in PMC 2011 December 22.

Published in final edited form as:

PMCID: PMC3143841

NIHMSID: NIHMS305055

Joseph D. Monaco, Zanvyl Krieger Mind/Brain Institute, Department of Neuroscience, Johns Hopkins University, Baltimore, MD, USA;

Corresponding: J.D.M., Email: ude.uhj@ocanomj, Johns Hopkins School of Medicine, 720 Rutland Ave, Traylor 407, Baltimore, MD, 21205-2109, USA

The publisher's final edited version of this article is available free at J Neurosci

See other articles in PMC that cite the published article.

Hippocampal place fields, the local regions of activity recorded from place cells in exploring rodents, can undergo large changes in relative location during remapping. This process would appear to require some form of modulated global input. Grid-cell responses recorded from layer II of medial entorhinal cortex in rats have been observed to realign concurrently with hippocampal remapping, making them a candidate input source. However, this realignment occurs coherently across colocalized ensembles of grid cells (Fyhn et al., 2007). The hypothesized entorhinal contribution to remapping depends on whether this coherence extends to all grid cells, which is currently unknown. We study whether dividing grid cells into small numbers of independently realigning modules can both account for this localized coherence and allow for hippocampal remapping. To do this, we construct a model in which place-cell responses arise from network competition mediated by global inhibition. We show that these simulated responses approximate the sparsity and spatial specificity of hippocampal activity while fully representing a virtual environment without learning. Place field locations and the set of active place cells in one environment can be independently rearranged by changes to the underlying grid-cell inputs. We introduce new measures of remapping to assess the effectiveness of grid-cell modularity and to compare shift realignments with other geometric transformations of grid-cell responses. Complete hippocampal remapping is possible with a small number of shifting grid modules, indicating that entorhinal realignment may be able to generate place-field randomization despite substantial coherence.

The locations of rodent hippocampal place fields (O’Keefe & Dostrovsky, 1971) can be randomly rearranged from one environment to the next during a process known as remapping (Bostock et al., 1991; Wills et al., 2005; J. K. Leutgeb et al., 2005). The freedom with which place fields remap suggests a link between the local spatial representations found in hippocampus and the global representation of grid cells (Hafting et al., 2005; Fyhn et al., 2008). Grid cells in medial entorhinal cortex (MEC) project to hippocampus (Witter, 2007b) and their periodic spatial responses realign during remapping (Fyhn et al., 2007). These shifts provide an attractive candidate mechanism for remapping in which grid-cell inputs cause large displacements in place-field locations. However, the realignment of colocalized grid cells during remapping is highly coherent (Fyhn et al., 2007). This apparent uniformity must be reconciled with the random reassignment of place-field locations during remapping.

In light of experimental evidence for modularity in MEC (Witter & Moser, 2006; Walling et al., 2006), including recent observations of modularity in grid-cell geometry (Stensland et al., 2010), we study whether grid-cell modules, within which grids realign coherently, can resolve this conundrum. Previous discussions (O’Keefe & Burgess, 2005; McNaughton et al., 2006) and models (Fuhs & Touretzky, 2006; Hayman & Jeffery, 2008) have considered place-cell remapping through independent realignment of grid-cell inputs, as well as partial remapping produced by less complete grid realignments (Fuhs & Touretzky, 2006). Our particular focus is on: 1) determining the number of independently realigning modules needed to produce statistically complete place-cell remapping; 2) studying the impact of assigning grid cells to modules either randomly or on the basis of their grid spacing (spatial-frequency-based modules); and 3) comparing the efficacy of different forms of grid-cell realignment, including shifts, rotations, enlargement of grid scale (Barry et al., 2009) and changes in grid ellipticity (Barry et al., 2007; Stensland et al., 2010). The second focus is inspired by the topographic organization of grid spacing along the dorsoventral axis of MEC (Hafting et al., 2005; Kjelstrup et al., 2008) and evidence for clustering of grid scales (Barry et al., 2007). In sum, our investigations provide a theoretical interpretation of clustering and modularity within MEC.

Our results are based on a model that transforms a periodic grid representation of space into one matching the sparse activity and high spatial specificity observed in hippocampus (O’Keefe & Dostrovsky, 1971; Wilson & McNaughton, 1993; Guzowski et al., 1999). The model is meant to reproduce the first-pass activity of place cells in an unfamiliar environment (Hill, 1978; Frank et al., 2004; Karlsson & Frank, 2008) by combining fixed grid-to-place connectivity with global feedback inhibition among place cells (Buzsáki et al., 2007; Pelletier & Lacaille, 2008). This initial place-cell activity may determine the spatial representations that are ultimately learned with continued exploration (Savelli & Knierim, 2010). The simulated responses here based on randomly aligned grid inputs and uniformly distributed synaptic weights allow for flexible and independent remapping of place-field locations.

A simulated place network is defined by the grid-to-place weight matrix **W** that is created at the beginning of every simulation. For connectivity *C*, this matrix is constructed from a 1000-component reference vector that has 1000 (1 *- C*) components set to zero, and the remaining 1000*C* components randomly sampled uniformly over the range [0, 1). Each row of **W** is then set to a randomly shuffled permutation of this reference vector. Having place units with identical, but shuffled, afferent weights avoids the contribution of sampling effects to the heterogeneity of place unit activity. The focus here is to allow the grid configuration and place unit competition to drive the diversity of responses across the network. The model is integrated using the fourth-order Runge–Kutta algorithm over discrete 5-ms timesteps. Place units are initialized to zero activity.

To determine the active place fields for each place unit, we find contiguous areas of activity greater than 20% of the unit’s peak rate (Muller & Kubie, 1989). Then, putative fields with rates greater than 20% of the population maximum rate and field size of at least 50 sq. cm. are counted as active place fields.

Simulated grid-cell responses are constructed from interference patterns of three two-dimensional sinusoidal gratings oriented 60 degrees apart. Similar formulations for simulating grid response fields have been used in previous theoretical and computational studies of grid cell function (Solstad et al., 2006; Fuhs & Touretzky, 2006; Blair et al., 2007; Almeida et al., 2009). An exponential nonlinearity is used to shape the profile of the resulting subfields to be approximately Gaussian and to qualitatively match the shape of the firing fields of observed grid-cell responses.

The grid responses are normalized to the range [0, 1]. Each grid is characterized by three parameters, *s*, *ψ* and , which determine the grid spacing, orientation, and spatial phase (relative offset of the peak nearest the mid-point of the environment), respectively. For a given position x in the environment, grid activity for these parameters is given by

$$g(\mathbf{x})=\frac{1}{R[3]}R\left[\sum _{i=0}^{2}cos\left(\frac{4\pi}{\sqrt{3}s}\mathbf{u}({\theta}_{i}-\psi )\xb7(\mathbf{x}-\phi )\right)\right]$$

(1)

where u(*θ*) = cos(*θ*)*,* sin(*θ*) is a unit vector in the direction *θ*. The array of angles *θ _{i}* {−

We implemented a genetic algorithm to search the parameter space of fan-in connectivity *C*, inhibitory strength *J*, and threshold *λ* (Equation 3). This was necessary due to nonlinear interactions among these variables (Figure 4). The fitness function was defined as the inverse variance from the target values of spatial map properties described in the Results. Each generation consisted of 512 simulations of random grid/place-network pairs with parameters sampled from contracting hypercubes centered on the last-generation winners. A coarse-grained search (keeping the top 25% of winners) was performed starting from *C* {0.1, 0.9}, *J* {150, 2 × 10^{4}}, and λ {0, 1.1 × 10^{3}} that converged in 6 generations. Based on those results, a fine-grained search (keeping the top 10% of winners) was performed starting from *C* {0.2, 0.35}, *J* {1.8 × 10^{3}, 4.0 × 10^{3}}, and λ {0.0, 6.5} that converged in 5 generations. Final parameter values (Equation 3) are an average of neighboring winners from both searches.

We simulate a 1 sq. m. environment sampled over a 100*×*100 element array, so that each pixel represents 1 sq. cm. Spatial maps were constructed by setting x to the mid-point location of pixels in the environment and allowing the activity to converge. We found high correlations between this rasterization with fixed input and continuous-time simulations of the same environment using a naturalistic trajectory based on a random walk (not shown). We use a checkered pattern consisting of every other pixel in the 1 sq. m. area to improve the efficiency of our simulations; this does not significantly decrease accuracy because the scale of spatial activity is significantly larger than a single pixel. The first pixel is clamped for 10*τ* seconds and all subsequent pixels are clamped for 5*τ* seconds, where *τ* is the rate time-constant (Equation 3). These dwell times are sufficient for place-unit activity to converge. Responses for unsampled pixels are interpolated as the average of their sampled neighbors. This interpolation can yield aliasing artifacts at the edges of place fields so, to mitigate this, we median-filtered the rate maps with a 3*×*3 pixel kernel.

Population autocorrelograms are computed by two-dimensional Fourier domain multiplication of a population rate matrix with its complex conjugate. The correlogram is obtained by taking the real part of the inverse Fourier transform of this product. We normalize the correlograms by dividing by the peak correlation.

We tested three manipulations of grid responses, which we refer to as different types of realignment, to assess remapping. Unless otherwise specified, realignment parameters are randomly sampled as described here. Shift realignment is the spatial displacement observed by Fyhn et al. (2007) to be concurrent with complete remapping in hippocampus. For simulations of remapping with random modules, shift realignment is specified as a translation of the grid within the plane for a uniform random distance in the range 9–45 cm (or, 10–50% of the maximum possible grid spacing) along a uniform random direction sampled from 0–360 degrees. For simulations of remapping with frequency modules, distances are drawn from a range of 10–50% of the field spacing of the lowest spatial frequency grid in each module. Translations are applied equally to all grids within a module such that the location of any grid peak is shifted along the sampled direction by the sampled distance.

We also tested two other grid manipulations. First, changing grid ellipticity corresponds to a squeeze mapping in the plane, which is an equiareal transform that preserves field size by magnifying the plane along one ‘longitudinal’ axis and contracting proportionally along the transverse axis. This means that an ellipticity parameter *l* yields a primary elliptic flattening of *f* = 2*l/*(1 + *l*). The ellipticity parameter is randomly drawn uniformly over the range 0.0–0.2 (or, up to 20% magnification and contraction). The orientation of the longitudinal axis is drawn uniformly over the range [*−π/*2, *π/*2). Second, grid rescaling is a uniform magnification of the plane. The scale is uniformly drawn over the range 1.0–1.2 (or, up to 20% magnification).

We present two measures of remapping between any two spatial maps. First, to quantify spatial differences, we determine the set of place units that are active in both maps. Then, for both maps, we compute all pair-wise distances between peak firing locations. Spatial remapping is then calculated as 1 minus the Pearson product–moment correlation coefficient of the pairwise distances of one map with those of the other. Thus, spatial remapping is 0 for identical maps, ~1 for unrelated maps, and is sensitive to differences in the relative pair-wise structure of the map. Second, to quantify the degree and randomness of turnover in the active subset of place units, we construct the 3-element array α consisting of the proportions of place units active in neither, one, and both of the spatial maps. Then, activity turnover *T* [α] is computed by comparing the root-mean-square differences between α and two similar reference arrays,

$$T[\alpha ]=1-\text{RMSD}(\alpha ,\beta )/\text{RMSD}({\alpha}_{0},\beta )$$

(2)

where *β {s*^{2}*,* 2*s*(1 *- s*)*,* (1 *- s*)^{2}*}* is the expected activity array given random recruitment of active units, α_{0} * {s,* 0*,* 1 *- s}* is expected given no turnover, and *s* = 0.614 is the average network sparsity (Table 1). Thus, *T* [α] ~ 0 indicates no change to the active subset and *T* [α] 1 indicates a fully randomized active subset. For comparison with these two measures, we computed population vector (PV) decorrelation as 1 minus the element-by-element Pearson correlation of the original population rate matrix with the remapped population rate matrix, as has been used previously (Wills et al., 2005; J. K. Leutgeb et al., 2005, 2007).

To visualize some statistical distributions (Figure 2), we created smoothed probability density functions (PDFs). This method was adapted from Karlsson and Frank (2008). For a given distribution, we computed a fine-grained cumulative histogram (1,000 bins), extended its endpoints to reduce boundary effects, and convolved it with a Gaussian kernel for smoothing. To derive the PDF, we computed the differential of the smoothed cumulative data, cropped it to the original data range, and normalized the resulting densities to the trapezoidal integral. The s.d. and width of the smoothing kernel were 5% and 50%, respectively, of the data range.

To simulate activity-dependent plasticity of grid-to-place synaptic inputs, we implemented the weight changes Δ*W _{ij}* =

To simulate recurrent excitatory connections between place cells, we constructed a place-to-place weight matrix *P _{ij}* =

We developed all modeling and analysis code as a package of custom Python libraries. These libraries extensively utilize NumPy for its ndarray implementation of numerical arrays and array operations. Plots and graphs were created from simulation and analysis output with Mat-plotlib and saved in the vector-based Portable Document Format. Two-dimensional arrays were first converted to RGB image data using Matplotlib colormaps and then saved in the lossless Portable Network Graphics format using the Python Imaging Library.

We constructed a model of hippocampal spatial map formation in which grid inputs drive a recurrently inhibited network of nonlinear place units (Figure 1A, left). The responses of 500 place units receiving input from *N* = 1000 grids are described by a vector **r** that obeys the time-evolution equation

The competitive network model for forming spatial maps with example grid inputs and place outputs. **A.** Model schematic (left) indicating the feedforward inputs from the 1,000 simulated grids and a population of 500 recurrently inhibited place units. Competitive **...**

$$\tau \dot{\mathbf{r}}=-\mathbf{r}+{[tanh(\alpha \mathbf{Wg}(\mathbf{x})-J\langle \mathbf{r}\rangle -\lambda )]}_{+}$$

(3)

where [·]_{+} indicates rectification, *τ* = 50 ms is the rate time-constant, α = 100*/*(*NC*) normalizes the strength of the grid input, *J* sets the strength of inhibition and *λ* is the threshold (values given below). Note that place-unit output is restricted to the interval [0, 1], so we refer to the responses described here as a normalized firing rate. The first term in the argument of the tanh is the grid input**: g(x**) is a 1000-component vector describing grid responses for the current location x in the environment, and **W** is a matrix describing the connections from the grid inputs to the place units. **W** is constructed randomly (Methods) so that each place unit receives input from a fraction *C* of grid cells (value given below). The second term in the argument of the tanh, where **r** is the population average of place-unit activity, defines the global inhibition. Thus, we model this inhibition as global feedback interaction for the place-unit population rather than by including inhibitory interneurons explicitly.

Grid responses, g(x), are modeled phenomenologically as a regular hexagonal grid that tessellates the plane of the simulated environments (Figure 1B), emulating MEC grid-cell activity (Hafting et al., 2005). Though we employ an empirical method based on oscillatory interference to construct grid activity (Methods), our results here do not depend on whether grid-cell activity derives from oscillation-based or attractor-based mechanisms (Fuhs & Touretzky, 2006; Giocomo & Hasselmo, 2008; Burak & Fiete, 2009). The spatial structure of each simulated grid depends on three parameters: field spacing, spatial phase and orientation. For each grid, the spacing is drawn randomly, independently and uniformly from 30–90 cm, which represents the several-fold range of spatial scales observed for grid cells found at the dorsal pole of MEC (Hafting et al., 2005). Similarly, the spatial phase for each grid is set independently to a random point within an origin-centered circle of a diameter equal to half of the grid spacing. This corresponds to the observation that even colocalized grid cells exhibit unrelated spatial phases (Hafting et al., 2005). Grid orientation is also drawn randomly and uniformly over the range [0, *π*/3) (i.e., full angular sampling due to the six-fold rotational symmetry of the grids), but the same orientation is used for all of the grids (Hafting et al., 2005; Sargolini et al., 2006; Fyhn et al., 2008).

Each simulation starts with a random selection of the grid parameters (spacing, spatial phase and orientation) and of the grid-to-place connection matrix **W**. We refer to this set of parameters, which fully define the grid inputs and place network for a given simulation, as a grid/place-network pair. The model is then run to determine place-unit activity across the simulated environment, from which we determine place fields and compute several spatial map characteristics (Methods). Network sparsity is defined as the fraction of place units without active place fields. Coverage is the proportion of the environment covered by at least one place field. Representation is the average number of overlapping place fields at any point in the environment. For each active place unit, we compute the number of fields, peak firing rate, and peak firing location; and for each place field, we compute the size, peak firing rate, average firing rate across the field, and center-of-mass location.

We used a genetic search algorithm (Methods) to set the fixed parameters of the model: the connectivity *C*, inhibitory strength *J*, and threshold *λ*. We specified model performance targets on the basis of experimental data. First, several studies report that approximately 60% and 75% of place cells in hippocampal areas CA1 and CA3, respectively, are silent in any given environment (Guzowski et al., 1999; Lee et al., 2004; S. Leutgeb et al., 2004; Vazdarjanova & Guzowski, 2004); accordingly, we specified a target of 65% network sparsity. Second, to be able to test remapping, we required full representation of the environment by setting 100% coverage as a target. Third, we required a large dynamic range across place responses by specifying a target of 0.99 for the population peak rate. Fourth, to approach the spatial specificity of hippocampal activity, we specified that the model should minimize the number of place fields per active place unit. Finally, so that sparsity is not achieved at the cost of field size, we specified a target of 200 sq. cm. for the average field size. Based on the search results, simulations reported here use inhibition strength of *J* = 2250 (the value is large because the sparseness of network activity makes *r* small), threshold λ = 2 (Figure 1A, right) and connectivity *C* = 0.33. Thus, each place unit receives input from 330 grid cells, consistent with estimates on the range of MEC inputs to dentate granule and hippocampal pyramidal cells (100–1,000; Amaral et al., 1990). Example responses (Figure 1C) demonstrate that this search-optimized model produces hippocampal-like place activity.

To characterize the spatial activity generated by the model, we analyzed a sample of 32 spatial maps from different grid/place-network pairs with a total of 6,178 active units and 8,354 place fields. The results are summarized in Table 1. The model successfully produces spatially representative maps: on average, 98.8% of the environment is overlapped by at least one place field and each point is represented on average by 4.51 fields. The average network sparsity of 61.4% approximates observations of CA1 activity levels in novel environments (Wilson & McNaughton, 1993; Guzowski et al., 1999; Lee et al., 2004; Karlsson & Frank, 2008). The peak firing rate of 0.925 indicates that the model utilizes most of its dynamic range. Active place units have an average of 1.38 place fields, intermediate between reported values for dentate gyrus and CA3 (J. K. Leutgeb et al., 2007). Most non-specific activity is due to the 25% of active units with two fields; only 6.2% of active units have three or more fields (Figure 2A).

The average peak rate of 0.418 indicates that responses are not saturating the output nonlinearity (Figure 1A, right). Place-field peak and average firing rate distributions are positively skewed (Figure 2B, left) and qualitatively match CA1 firing rate distributions observed in novel environments (Karlsson & Frank, 2008). The average field size of 169 sq. cm. is smaller than typical hippocampal place fields (Muller & Kubie, 1989; Kjelstrup et al., 2008), and the size distribution is positively skewed such that only 5.2% of place fields are larger than 300 sq. cm. (Figure 2B, right). Network mechanisms beyond recurrent inhibition can be added to the model to produce larger place fields (see “Alternative place-cell models” below).

To assess multi-field responses, we classified the strongest place field of each active unit as primary and all others as secondary. The distribution of distances between the centers of mass of every secondary field and its respective primary field (2,356 secondary fields: 65.1± 15.8 cm, mean ± s.d.) may reflect the mean spacing of the grid-cell population (60 cm; one-sample *t* = 15.8, *p* < 10^{−}^{52}). To examine grid-like peridocity in place fields, we computed two-dimensional spatial autocorrelograms of the grid and place population responses (Methods) from an example grid/place-network pair (Figure 3A). The grid correlation shows a strong central peak and weaker radial arms corresponding to the periodic hexagonal structure of the grids. The orientation of the radial correlations matches the grid orientation chosen at the start of the simulation. The radial arms are smeared out because the correlogram averages over grids of different spacing. The place unit correlation has a narrower central peak and includes radial correlations similar to the grid autocorrelogram, but much weaker. This is another re-flection of the underlying grid input. The difference between the grid and place correlograms (Figure 3B) reveals much stronger suppression of the center-surround than of the radial grid correlations. This indicates that the model is better at local sparsification than at reducing the more remote redundancies of the periodic inputs.

The inhibitory strength *J* and threshold *λ* of the nonlinearity are critical for the competition that generates place fields in this model (Equation 3). To assess their impact, we simulated a 16*×*16 parameter grid using a single grid/place-network pair and holding all other parameters fixed at their usual values. To show the parameter dependence of key spatial coding properties, we computed bilinear interpolated maps of network sparsity, number of fields, field peak firing rate, and field area across this parameter grid (Figure 4). As expected, increasing *λ* or *J* generally leads to increases in sparsity and reductions in field number, rate, and area. Holding *λ* fixed and increasing *J* yields monotonic changes to spatial coding. However, for fixed *J*, sparsity is U–shaped and field number and rate are inverse-U–shaped with increasing *λ* This is somewhat paradoxical because increasing *λ* decreases excitatory drive but, up to a point, it yields more active place units with more and stronger place fields. This reflects a balance of the competitive dynamics in which the network is most efficiently driven with a certain amount of input, but past which the global inhibition suppresses activity.

To test the ability of changes in grid configuration to elicit place-field remapping, we compared two spatial maps generated by a single grid/place-network pair: the first as described above, and the second in which we have realigned the grid inputs. Grid inputs were divided into equal-sized subsets or modules, such that realignment was coherent within each module but different across modules (Methods). Initially, modules were selected randomly with respect to the spatial metrics of the grids. In later simulations (Figure 7B and C) using ‘frequency modules,’ the grids were sorted by field spacing (spatial frequency) prior to being partitioned into modules. We characterized the quality of remapping using two measures: the ‘remapping strength’, which characterizes the decorrelation of the distribution of pair-wise inter-place-field distances before and after remapping, and the ‘activity turnover’, which is a measure of the randomness in the selection of active place units for the two maps (Methods). Both measures range from~0–1, where values around 0 indicate little or no remapping. Remapping strength near 1 indicates randomization of the pair-wise spatial relationships of place units active both before and after remapping. Activity turnover near 1 indicates that the subsets of place units active before and after remapping show no more overlap than expected by chance.

Statistics for sets of 64 remapping simulations under a variety of realignment conditions. The first letter (s, e, or z) of the label indicates shift, ellipticity, or rescaling, while the number indicates modularity; *srnd*, *ernd*, and *zrnd* signify fully **...**

Remapping examples based on two modules and the three grid realignment types (shifts, changes of ellipticity and uniform rescaling) are shown in Figure 5 for typical realignment parameters (cf. Fyhn et al., 2007; Barry et al., 2007, 2009). Remapping simulations here compute responses based on the initial environment A and then the realigned environment B. Restricting analysis to place units that were active in both environments, we examined changes in the location defined by the peak firing rate (Figure 5C). For shift realignments (Figure 5C, left), quite a few long-range jumps in response maxima occur. These jumps typically occur near the edges of the environment, reflecting the appearance and disappearance of activity peaks across the borders. The other transformations (Figure 5C, middle and right) show more place units that remap by small or even zero distances (median remapped distances are 3.2 and 4.5 cm for ellipticity and rescaling, respectively). The rescaling example (Figure 5C, right column) shows that units that remap short distances follow the expansion of the grid inputs about the origin. The distributions of remapped distances (Figure 5D) confirm that shift realignment more effectively remaps place fields than ellipticity and rescaling in these two-module examples.

Modular remapping examples using two random modules for three types of grid realignment. Columns correspond to shift (left), ellipticity (middle), and rescaling (right) realignments. **A.** An example grid response of 0-degree orientation, 30-cm spacing, **...**

Remapping requires not only that place fields shift by variable distances, but that they do so independently of each other. To test this, we computed the distances between the peak firing locations of pairs of place units before (Distance (A)) and after (Distance (B)) remap-ping, and show two-dimensional histograms of these distances in Figure 5E. The remapping strength measure we use is based on the decorrelation of these pair-wise data (Methods). The off-diagonal activity evident for the shift example (Figure 5E, left) reveals incoherence in the remapping, but the diagonal band indicates that the spatial map is not fully disrupted by this realignment. The high diagonal correlations evident in the ellipticity (Figure 5E, middle) and rescaling (Figure 5E, right) examples indicate that the spatial maps remained largely intact across realignment.

Although most evidence argues against incoherent grid-cell orientations, preliminary findings by Stensland et al. (2010) indicate the possibility of small changes in orientation at discrete positions along the dorsoventral axis of MEC that may be associated with changes in grid ellipticity. To investigate the effect of grid rotations and compare them to shift effects, we simulated concurrent shifts (up to the maximum possible shift of 45 cm) and rotations (up to 30 degrees) of one module against the other across a 16*×*16 grid of realignment parameters (Figure 6A). Both modular shift and rotation can elicit strong remapping independently, with 15 degrees of rotation approximately equivalent to 15 cm of shift. While remapping saturates with ~20 cm of shift, rotations under 30 degrees require shifts to fully remap (cf. divide between top two remapping quintiles, Figure 6A, left). Thus, with just two grid-cell modules, rotations without concurrent shifts do not generate full remapping, but sufficiently large shifts do not require rotations to accomplish full remapping.

Progressive modular remapping as grids are realigned from environment A to B. Using two modules, we measure remapping strength and activity turnover to track the effects of grid realignment. **A.** Remapping (left) and turnover (right) are shown for a 16 **...**

To observe continuous remapping from environment A to environment B, we performed two-module remapping simulations for an incremental series of realignments. We simulated realignments based on shift, ellipticity, and rescaling with 30 intermediate grid configurations between the two environments (Figure 6B). Remapping strength and activity turnover, computed relative to environment A, are closely correlated, indicating that these realignments similarly affect the relative spatial map structure and the active subset of place units. These remapping curves show that shift realignment can be more effective at remapping than the other transformations for randomly chosen realignment parameters.

Previous experimental remapping studies have used population vector (PV) correlations to characterize the similarity of spatial responses (Wills et al., 2005; J. K. Leutgeb et al., 2005, 2007). For the data in Figure 6B, we computed PV decorrelation relative to the spatial map of environment A (Figure 6C). PV decorrelation shows qualitatively similar remapping trends as the other measures. Across the progressive realignment, PV decorrelation is generally smaller than remapping strength for all realignment types (Figure 6C, inset). The exceptions are the strongly remapping shift realignments for which both measures have saturated at values ~1. In general, PV decorrelation provides a smoother but less sensitive measure than pair-wise remapping strength.

Remapping may benefit from increasing the number of modules used for realignment. We tested this by running sweeps of progressive realignment simulations using from 1 to 16 modules. Realignment parameters were preserved as much as possible by drawing from a single set of 16 parameters for each sweep instead of resampling each module on every sweep. The progressive remapping curves (Figure 6D) show that shift realignment benefits substantially from having two modules and only incrementally from further modules. For ellipticity and rescaling (Figure 6D, middle and right panels), remapping improves with two modules but shows no consistent effects with further modules. The overall magnitudes are below the remapping achieved by shift realignment (Figure 6D, left panel).

To assess the effects of modularity and realignment types on remapping, we simulated sets of 64 independent remapping experiments across a number of conditions. For each realignment type, we tested 1, 2, 4, 8, and 16 modules. These sets are called sNN, eNN, and zNN for the shift, ellipticity, and rescaling (zoom) realignment types, respectively, where NN is the number of modules. In addition, we simulated conditions using 1,000 modules (i.e., every grid-cell input is realigned independently) as an upper bound on realignment incoherence. These sets are called srnd, ernd, and zrnd, respectively. Finally, as an absolute upper bound on remapping, we simulated a condition called rnd in which an entirely new set of grid inputs was sampled to define the realigned representation. Means and s.e.m. error bars are shown for all of these conditions in Figure 7A. The biggest gains in remapping occur between coherent (1 module) conditions and two-module conditions. For all realignment types, improvement in remapping levels off after the introduction of 4–8 modules. We were interested in which conditions remapped sufficiently to not be significantly different from their respective incoherent sets or the rnd set. We computed Kolmogorov–Smirnov two-sample tests between all pairs of sample sets for remapping strength and activity turnover (horizontal brackets in Figure 7A indicate non-significance at *p* > 0.05). Based on these tests, similarity here means we do not reject the null hypothesis (at 5% significance) that two samples result from the same distribution. For shift, both s16 and srnd are similar to the rnd upper bound on remapping. Both s4 and s8 were similar to rnd for remapping but not for activity turnover. For ellipticity, both e8 and e16 are similar to ernd. For rescaling, only z16 was similar to zrnd. Thus, 16 modules is generally equivalently effective at remapping as total incoherence. Shift realignment with 16 modules is the only modular condition similar to our upper bound for both remapping strength and activity turnover.

In the Introduction, we suggested that grid cells could be assigned to modules on the basis of their grid spacing, that is, spatial frequency. To study the effects of such frequency-based modules, we ran all the modular sample sets again as above but using frequency modules (Methods) instead of the random modules that we have used to this point. Frequency modules are constructed by first sorting the grid-cell population by spatial frequency and then dividing it into equal-sized subsets. The spatial frequencies themselves remain uniformly random. Trends across the modular remapping conditions are similar to and highly correlated with those for random modules (remapping: Pearson *r* = 0.998; turnover: *r* = 0.992). The shift condition shows that frequency modules are generally less effective at remapping. For example, shift realignment with 16 frequency modules yields less randomized activity turnover than 8 random modules (Figure 7B). Overall, random and frequency modules are similarly effective (Figure 7C, top), but there is a consistent turnover deficit for frequency modules that is prominent under shift realignment (Figure 7C, bottom).

The place-cell model used up to this point relies heavily on recurrent inhibition (Figure 1). Feedforward inhibition imposed as a global activity threshold has been discussed and modeled previously as an alternative mechanism (O’Keefe & Burgess, 2005; McNaughton et al., 2006; Solstad et al., 2006; Rolls et al., 2006; Fuhs & Touretzky, 2006). In contrast to our model, spatial specificity in these sum-and-threshold models relies on place cells receiving input from grid cells with correlated spatial phases, not the random spatial phases we use. Nevertheless, it is interesting to see if feedforward instead of recurrent inhibition can be used in our model. Using the same grid/place-network pair as in Figure 1, we computed linear responses with a threshold calculated to provide a sparsity of 61.6% (192/500 active place units). The place-field distributions for the recurrent (Figure 8A, left) and feedforward (Figure 8A, right) responses show that feedforward inhibition produces irregularly spaced clusters of place fields. Further, threshold-based place fields are much smaller (recurrent: 156 ± 3.98 sq. cm., mean ± s.e.m; feedforward: 116 ± 3.20 sq. cm.). Correlated clusters of place fields do not provide a useful population code for position; they simply follow the peaks in the magnitude of the grid-cell inputs (Figure 8B). The recurrent mechanism allows for a uniform representation of the environment despite variations in input strength.

Comparison of place field distributions generated by variants of the recurrent inhibition model. All place fields in a simulation are shown schematically (A,C) as a circle of equivalent diameter and plus (+) sign centered at field center-of-mass. Simulations **...**

The place fields generated by our model are atypically small and insufficiently sparse (Table 1; Figure 8A, left). We explored two mechanisms for improving these features. First, noting that experience within an environment can produce larger firing fields (Wallenstein & Hasselmo, 1997; Mehta et al., 1997, 2000; Frank et al., 2004), we introduced activity-dependent plasticity on the connections between the grid and place cells (Methods). Following these activity-dependent weight changes, the place-unit responses demonstrated higher sparsity (71.4%, 143/500 active place units) and slightly larger place field size (156 ± 3.98 sq. cm before and 166 ± 5.67 sq. cm after; K–S two-sample test, *p* < 0.05). Second, we approximated (Methods) the effects of excitatory recurrent collaterals between place cells (Samsonovich & McNaughton, 1997; Redish & Touretzky, 1998; Tsodyks, 1999; Witter, 2007a). The resulting place-field distribution (Figure 8C) shows both less coverage of the environment (85%) due to higher network sparsity (77.2%, 114/500 active place units) and substantially larger place fields (325 ± 13.1 sq. cm.) than the simulation with recurrent inhibition alone (Table 1). The lower-to-upper quartile range of place-field areas (181–429 sq. cm.) is greater than the median of the original distribution (150 sq. cm.; Figure 2B, right). Furthermore, the “novel” and “familiar” (before and after these alterations of the model) representations overlapped quite well. Applying our pair-wise measure, familiarization provoked remapping of 17.9% (the turnover measure does not apply, as it assumes similar network sparsity), which was generated primarily by small shifts: 103/113 units shifted peak firing location by *<*10 cm.

These studies show that learning and recurrent excitation can extend our original model by constructing spatial maps with the higher sparsity and larger place fields typical of hippocampal activity. The overall similarity between the place field locations of the original and extended models suggests that our results for different grid-cell transformations and numbers of modules (Figure 7) apply to familiarized as well as to novel representations.

We investigated a recurrent-inhibition model of initial place-cell activity in novel environments to assess the hypothesis of grid-cell modules as a basis of hippocampal remapping. Although simultaneously recorded grid cells in remapping experiments shift coherently (Fyhn et al., 2007), the tetrode recordings have typically been restricted to colocalized grid cells. These findings demonstrate that grid realignment is at least locally coherent. Two hypotheses were originally suggested by Fyhn et al. (2007) to reconcile the local coherence of spatial inputs to hippocampus with the randomization inherent to hippocampal remapping. First, they hypothesized independent grid-cell modules, as we have explicitly tested here. Second, they hypothesized that the spatial phases of grid cells may represent a position code for an infinitely large map of space. To elicit remapping, this position code is shifted to a new random location in the infinite map. The infinite-map hypothesis requires that grid cells of the same spatial frequency are displaced by the same amount. As a result, our simulations using frequency modules in the limit of large numbers of modules correspond to the infinite-map hypothesis. Our simulation results then show that both hypotheses can viably produce complete or nearly complete remapping. Simultaneous recordings of grid cells with significantly different spacing that nonetheless realign coherently during remapping would provide strong support for frequency-independent modules over the infinite-map hypothesis. Because we found that a small number of modules can be effective at remapping, the modularity hypothesis may require large distances between tetrodes along MEC to be proven or falsified. Such long-range recordings may be technically difficult, but the necessary experiments are being pursued (e.g., Stensland et al., 2010).

Although remapping based on grid-cell responses has been discussed (O’Keefe & Burgess, 2005; McNaughton et al., 2006) and demonstrated in modeling work (Fuhs & Touretzky, 2006; Hayman & Jeffery, 2008), the relative remapping effectiveness of various grid manipulations has not been systematically quantified. With data showing that ellipticity (Barry et al., 2007; Stensland et al., 2010) and rescaling (Barry et al., 2007, 2009) may be modes of grid-cell realignment, this sort of quantification has become necessary to understanding the relationship between the spatial activity patterns of entorhinal cortex and hippocampus. Specifically, changes in grid ellipticity are supported by observations of elliptical or compressed grids in both altered (Barry et al., 2007) and familiar (Stensland et al., 2010) environments. Preliminary observations have shown rescaling consisting of the uniform expansion of grid scale upon introduction to a novel environment (Barry et al., 2009). Theoretically, these geometric transformations could result from changes to the synaptic weights in attractor network models (McNaughton et al., 2006; Fuhs & Touretzky, 2006; Burak & Fiete, 2009) or to the frequency modulation of theta-frequency oscillators in temporal interference models (O’Keefe & Burgess, 2005; Burgess et al., 2007; Blair et al., 2008; Hasselmo, 2009) of grid cell activity. The reduced remapping capabilities of ellipticity and rescaling as putative forms of realignment could nevertheless be functional, producing for example the sort of partial or graded remapping that has been observed in CA1 (Lee et al., 2004; S. Leutgeb et al., 2004; Vazdarjanova & Guzowski, 2004).

Despite changes in grid orientation when an animal is moved to a new environment, we do not consider modularity of grid orientation to be an experimentally supported mode of realignment. If humans have grid cells, then the grid-like periodic signal from a recent functional imaging study would likely not have been apparent in the presence of any significant orientational incoherence (Doeller et al., 2010). In addition, directionality in rodent MEC is strongly coupled with the head direction system (Hargreaves et al., 2007), which is itself internally coherent during remapping conditions (Yoganarasimha et al., 2006). However, small angular rotations may accompany other geometric modifications of grid-cell responses. To examine this possibility, we tested small modular rotations (Figure 6A) and found that differential orientations can elicit strong remapping alone and complete remapping in combination with shifts.

Remapping may involve additional computational components such as pattern separation provided by dentate gyrus (Acsady & Kali, 2007; S. Leutgeb & Leutgeb, 2007) and the integration of changes in external sensory information represented by activity carried by the lateral entorhinal projection to the hippocampus (Knierim et al., 2006; Witter, 2007b; Lisman, 2007). Grid lattices are only part of a broader functional diversity of non-spatial and spatial activity in entorhinal cortex (Hargreaves et al., 2005; Sargolini et al., 2006; Savelli et al., 2008; Solstad et al., 2008). Although MEC and hippocampus respond concurrently during remapping, we do not address whether changes in grid-cell response result from direct environmental input or other mechanisms such as hippocampal feedback (O’Keefe & Burgess, 2005; McNaughton et al., 2006; Burgess et al., 2007; Hasselmo, 2008; Burak & Fiete, 2009). The model here approaches one particular spatial mode of hippocampal processing of its cortical inputs.

A number of grid-to-place models were posited following the discovery of grid cells (Rolls et al., 2006; McNaughton et al., 2006; Solstad et al., 2006; Franzius et al., 2007). Most rely on a combination of activity thresholds, grid inputs with correlated spatial phases, and associative or competitive learning rules. Associative learning with heterosynaptic depression generates place fields in both temporal and rate coding models (Molter & Yamaguchi, 2008; Savelli & Knierim, 2010). Fuhs and Touretzky (2006) demonstrated that place fields could be randomly displaced by progressively resetting the spatial phases of its grid inputs. Hayman and Jeffery (2008) showed that learning a dendritically organized spatial-phase partition of grids could provide the context-dependence necessary for partial and complete remapping. Almeida et al. (2009) proposed a model that, similarly to our model, uses randomly aligned grid inputs and does not require learning to produce place-like activity. However, their model depends on an adaptive activity threshold and a skewed weight distribution that prevents the recruitment of independent active subsets of place units in new environments. We use feedback inhibition rather than an adaptive threshold or synaptic modification to produce spatially selective outputs from the global spatial representation of grid-cell inputs. We showed that simple threshold mechanisms, representing feedforward inhibition, tend to follow overall input strength in novel environments and do not produce informative spatial representations.

Inhibition plays a key role in our place-cell network model. Hippocampal interneurons are diverse (e.g., Sik et al., 1997) and constitute up to 20% of hippocampal cells, about one-third of which directly innervate pyramidal cells (Buzsáki et al., 2007). The dentate gyrus and subregion CA3 both have extensive recurrent and feedforward inhibitory microcircuits that are crucial to hippocampal computation (Sik et al., 1997; Acsady & Kali, 2007) and are supported by diverse mechanisms of synaptic plasticity (Pelletier & Lacaille, 2008). Inhibitory network dynamics are integral to many hippocampal functions, such as the competitive transformation of redundant inputs into more informative outputs (J. K. Leutgeb et al., 2007; Karlsson & Frank, 2008). We hypothesize that this general computation could produce place fields on the first pass through unfamiliar environments (Hill, 1978; Frank et al., 2004). Several studies have shown that interneurons in CA3 mediate disynaptic-latency feedback inhibition across long distances within the pyramidal cell layer (Glickfeld et al., 2009; Bazelot et al., 2010), which is consistent with the global inhibition modeled here. A detailed compartmental pyramid–interneuron model of fast feedback loops in CA3 demonstrated higher spike and burst rates with stronger inhibitory gains (Zeldenrust & Wadman, 2009). Our network-level model similarly uses strong feedback inhibition to enhance competition.

The activity patterns that we have modelled could form the basis of a ‘seed’ representation that becomes refined with longer-timescale familiarization (Gerstner & Abbott, 1997; Lee et al., 2004; S. Leutgeb et al., 2004; Karlsson & Frank, 2008). Indeed, Savelli and Knierim (2010) used their spatial learning model to show that the details of initial activity may critically determine the spatial representations that are learned as a novel environment becomes more familiar. Our model predicts that non-specific firing in the initial activity of place cells results from correlated grid inputs. Learning mechanisms may act to enhance spatial specificity while also broadening and shaping place fields consistent with previous modeling and experimental findings (Wallenstein & Hasselmo, 1997; Mehta et al., 1997, 2000). We showed that post-hoc application of associative synaptic modification (Figure 8C) can both enhance sparsity and broaden the place fields of our model. These effects of familiarization may also contribute to remapping.

In conclusion, the effectiveness of a small number of shifting grid modules indicates that entorhinal realignment could be significantly coherent and still contribute substantially to hippocampal remapping. Although we did not test all combinations of realignment types, combinations of various grid transformations can provide enhanced remapping with minimal grid-cell modularity. While shift-based realignment is more effective than ellipticity or scale changes, fully orthogonal remapping may not be necessary for spatial memory encoding in rodent hippocampus. These transformations could contribute to partial remapping or other graded response changes critical to hippocampal function.

This work was supported by NIH grant MH58754 and an NIH Director’s Pioneer Award, part of the NIH Roadmap for Medical Research, through grant 5-DP1-OD114-02 to L.F.A. and NIH grant P01 NS-038310 to James J. Knierim who supported J.D.M. during the writing of the manuscript.

Thanks to Eric R. Kandel, Pablo Jercog, Francesco Savelli, Caswell Barry, Grace Hwang, and the anonymous reviewers for helpful comments and discussion.

Commercial Interest: No

Conflict of Interest: None.

Joseph D. Monaco, Zanvyl Krieger Mind/Brain Institute, Department of Neuroscience, Johns Hopkins University, Baltimore, MD, USA.

L. F. Abbott, Department of Neuroscience, Department of Physiology and Cellular Biophysics, Columbia University College of Physicians and Surgeons, New York, NY, USA.

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