article: An important task of the brain is to represent the outside world. It is unclear how the brain may do this, however, as it can only rely on neural responses and has no independent access to external stimuli in order to “decode” what those responses mean. We investigate what can be learned about a space of stimuli using only the action potentials (spikes) of cells with stereotyped—but unknown—receptive fields. Using hippocampal place cells as a model system, we show that one can (1) extract global features of the environment and (2) construct an accurate representation of space, up to an overall scale factor, that can be used to track the animal' s position. Unlike previous approaches to reconstructing position from place cell activity, this information is derived without knowing place fields or any other functions relating neural responses to position. We find that simply knowing which groups of cells fire together reveals a surprising amount of structure in the underlying stimulus space; this may enable the brain to construct its own internal representations.
  Stimulus reconstruction, as implemented by the scientist, typically involves three steps: (i) characterizing the space of relevant stimuli; (ii) constructing functions relating stimuli to neuronal responses; and (iii) using these functions, together with new neuronal activity, in order to “decode” new stimuli [1]–[11]. For example, in the case of hippocampal place cells, the ‘space of stimuli’ may be the animal' s current spatial environment; for every place cell one computes a place field, i. e. , a function that assigns a firing rate to each position in space. Place fields, together with place cell activity, can then be used to infer the animal' s position [2], [4], [7]. Notably, the scientist relies on a priori assumptions about the nature of the relevant stimulus space in (i), and uses independent measurements of previously observed stimuli in order to construct the functions in (ii). While these functions (or “neural codes”) come in a variety of forms, such as receptive fields, tuning curves, spike-triggered averages, adaptive filters and conditional probability distributions [1]–[11], they all require using independent observations of prior stimuli for their construction. Presumably, the brain also uses neuronal spiking activity to reconstruct the stimulus. The brain, however, does not have access to independent stimulus measurements; neuronal activity alone must represent the external world. How does the brain do it? While much effort has been devoted to developing biologically plausible methods to implement the “decoding” of step (iii) [1], [3], [6], [10], [11], it is generally assumed that the structure of stimulus space (step (i) ) and the functions (such as receptive fields or tuning curves) of step (ii) are both present and easily available to downstream structures in the brain. Although it is possible that receptive fields are imprinted in synaptic weights, tuned throughout development and learning, this story is complicated by the observation that receptive fields in some brain areas—particularly in hippocampus—undergo rapid context-dependent remapping [2], [12]–[19]. This leads naturally to the question: can anything be inferred about a stimulus from spikes alone? We address this question in the context of hippocampal place cells. In rodents, spatial information is reflected in the activity of place cells, i. e. , pyramidal cells in areas CA1 and CA3 of dorsal hippocampus that fire in a restricted area of the spatial environment—the place field—and are mostly silent outside [20], [21]. We will use the term place field to refer both to the function and to the region in space where the firing rates are significantly above baseline. Place fields remap, and a place cell can alternate between multiple stable place fields as an animal is switched from one familiar environment to another [14], [15]. Although much work has gone into trying to understand how place fields are formed [13], [22]–[25], a different and rather unexplored question is how the output of hippocampal place cells (without access to corresponding place fields) might be used by downstream structures in order to reconstruct position and the underlying space. At first glance, it is not obvious that anything at all may be learned about a particular environment—or the animal' s position within it—using the spiking activity of place cells alone. Indeed, previous approaches to reconstructing position from place cell activity have all required knowing the corresponding place fields [2], [4], [7]. Furthermore, the values of instantaneous firing rates [2], [4] and the precise timing of spikes with respect to the theta rhythm [7] have been used, together with place fields, in order to improve position-estimation precision beyond the place field diameter. It has also been suggested that other cell types, such as head direction cells, play a vital role in deciphering position information [22]. Because place fields exhibit complex dynamics and place cells do more than just coding for place [26]–[30], it is important to identify—at least in theory—minimal aspects of neural activity that yield sufficient information for construction of an accurate representation of space. In this work we show that a great deal of information about a physical environment can be obtained using only very coarse features of population spiking activity. We define a ‘cell group’ as a collection of cells that collectively fire significantly above baseline within a broad (∼250 ms) temporal window; we do not call them ‘cell assemblies’ to avoid confusion with different timescales and degrees of sensory control implied by this term [31]–[34]. We find that the simple knowledge of which groups of hippocampal place cells fire together is enough to (1) extract global topological features of the environment, and (2) reconstruct an accurate geometric representation of physical space within which the animal' s position can be faithfully tracked. This is made possible by using standard tools from algebraic topology and graph theory; neither place fields, nor precise spike timing, nor any prior independent measurements of position are needed.
  Although the brain may be unable to establish direct relationships (such as place fields) between neural responses and external stimuli, it can in principle compare neural responses to each other. Moreover, relationships between neural responses reflect relationships between stimuli, and hence reveal structure of the outside world. In rat hippocampus, the theta-oscillation (6–10 Hz) provides a natural timescale for organizing population activity. Cells that fire within a few theta-cycles of each other are very likely to have overlapping place fields. We define a cell group as a group of place cells that collectively fire within a two theta-cycle (250 ms) time window (Figure 1A). Note that this enables us to ignore finer spike timing effects modulated by the phase of the theta oscillation, such as phase precession [7], [31], [35]–[37]. Each place cell typically belongs to multiple cell groups (Figure 1B), and the activation of a given cell group is induced by the animal passing through the intersection of corresponding place fields. Cell groups thus yield place field intersection information (i. e. , they reveal which subsets of place fields overlap), even when the place fields themselves are unknown (Figure 1C). We first show that this intersection information can be patched together to reveal global topological features of the environment. The method for extracting global topological features does not require a metric. On the other hand, by thinking of each cell group as defining a specific location in space, we can use intersection information to construct a metric that provides relative distances between cell groups. This yields a geometric representation of the external physical space, obtained without knowing place fields. We find that this internal representation is quite faithful to the geometry of the environment. In either case, we need only make some basic assumptions about place fields. We assume that place fields exist and are stable, have similar sizes, are omni-directional, and have firing fields that are convex. These assumptions are generally satisfied for place fields of dorsal hippocampal place cells recorded from a freely foraging rat in a familiar open field environment (see Methods). We also explicitly test the importance of the assumption that place fields have similar sizes, and find that our results are in fact fairly robust to substantial variability in place-field sizes. Finally, we test our methods with multipeaked place fields, and find that our algorithms can tolerate a realistic percentage of cells having multiple firing fields, so long as the component fields are sufficiently separated and convex. What may be thought of as a ‘space of stimuli’ at one level of processing may constitute an individual stimulus at another: global features of the ‘space of positions’ become properties of individual environments that can be used to distinguish between them. Often times an animal' s physical space has “holes”—i. e. , regions in the interior of the environment where the animal is unable to go. For example, a rat may be confined to a platform with one or more holes in the middle; similarly, there may be large objects inside the environment (such as trees) providing obstructions to the animal' s path. In either case, we call the region inaccessible to the animal a hole. Holes are examples of (non-metric) topological features, because they are preserved under continuous deformations of the space. Two environments are said to be topologically equivalent (homeomorphic) if one can be continuously deformed into the other, and vice versa. Homology groups [38] (see Text S1) are topological invariants that can be used to distinguish topologically inequivalent spaces. In particular, the dimension of the first homology group H1 counts the number of holes. Higher order homology groups (H2, H3, …) count higher-dimensional “holes, ” and thus place constraints on the minimum dimensionality of the space; they are all expected to vanish for flat, two-dimensional environments. From spike trains for a population of place cells, we obtain a collection of cell groups (Figure 1A; see also Methods). The corresponding intersection information can be used to compute homology groups of the underlying environment—even though the place fields themselves are unknown. Intuitively, this works because there is a unique configuration of place fields (up to continuous deformation) consistent with a given pattern of intersections (Figure 1B and 1C). Inspired by a deep theorem in algebraic topology [38] (see Text S1), we have devised a procedure to compute homology groups from the collection of cell groups active in a given environment. This theorem has also been used in the context of sensor networks [39], and the potential utility of similar methods in the case of hippocampal place cells was independently observed in [40]. Our algorithm, described in detail in the Methods, involves constructing a simplicial complex (Figure 1B) from place field intersection information, and computing its homology groups. If the cell groups obtained from spike train data exactly reflect the correct place field intersection information, the theorem guarantees that the homology of the simplicial complex is equal to the homology of the underlying space. However, given the stochastic nature of place cell firing, there is always the possibility that we may miss cell groups corresponding to real intersections, and mistakenly detect cell groups corresponding to non-existent place field intersections. In order to verify that this procedure yields accurate results within physiologically realistic parameters, we tested it using simulated data with varying degrees of noise. Random-walk trajectories were generated in five different flat, two-dimensional environments, each of side length L (typically L∼1 m), with N = 0,1, …, 4 holes (Figure S1). In each of 300 trials, each of the five environments was covered by 70 single-peaked place fields with varying radii (0. 1–0. 15 L) and randomly-chosen centers (Figure 2A). Place cell firing was generated according to a simple model (see Methods). Differing levels of noise were introduced by removing a certain percentage of randomly-selected spikes from each spike train (for every cell) and reassigning them to occur at random times, so that they fall outside the place field (see Methods). For each trial, the first five homology groups (H0, …, H4) were computed. A trial was deemed to be ‘correct’ if and only if all homology groups matched the topology of the underlying space, and ‘incorrect’ if at least one homology group did not match. Although the correct environment could be identified using only the first homology group H1, we required the other homology groups to also match in order to ensure consistency of the overall topology (i. e. , this was not a multiple-choice framework where each trial was assigned the ‘most likely’ of the five environments; note that ‘chance level’ here is close to 0%). For low levels of noise, we found near 100% accuracy in all environments (Figure 2B). One might worry that if a cell ever spikes outside of its place field, it will activate a cell group that does not correspond to a true intersection of place fields, rendering the topology computations completely inaccurate. Remarkably, the percentage of correct trials remained very high for noise levels up to 5%; even with 10% of each cell' s spikes occurring outside the corresponding place field, more than half of all trials continued to be correct (Figure 2B). This is because the thresholding of firing rates in order to obtain cell groups (see Methods) renders the procedure quite robust to noise in the spike trains. Note that although the trajectory of the rat is itself littered with small holes (Figure 2A), the procedure only detects actual (relatively large) holes in the environment and is quite insensitive to holes that are very small compared to the sizes of place fields. As a further test, we constructed additional ‘shuffled’ data sets by pooling together spiking activity from cells in different environments. We found that each and every ‘shuffled’ trial had nonzero higher homology groups, suggesting higher-dimensional spaces. This indicates that population activity in the shuffled data sets was not generated from realistic, two-dimensional environments, and suggests that a downstream structure receiving hippocampal output could detect patterns of activity that are inconsistent with a spatial interpretation. A given cell group becomes active when the animal crosses a specific location in space, given by the intersection of the corresponding place fields. It is thus natural that, from the brain' s point of view, a location in space is itself defined by a cell group (Figure 3A and 3B). The collection of all activated cell groups thus yields a collection of points, which can be thought of as “building blocks” for an internal, discretized representation of space. A set of unrelated points, however, does not constitute a space, one must know the relationships between points (which pairs are close, and which are far away). Fortunately, there is a natural way to determine when two cell groups are “close” to each other, based on the number of place cells they have in common. We say that two cell groups are neighbors if they differ by just one place cell. By joining neighboring points with edges, one obtains a graph (Figure 3C) that is constructed purely from cell groups, without any explicit knowledge of place fields. In general, neighboring cell groups with a very high percentage of overlapping cells will represent points that are closer in space than neighbors with small overlap. We define a dissimilarity index μk on neighboring cell groups as the average relative distance between the centers of adjacent regions with overlap degree k, assuming place fields of equal radius (see Methods). In principle, μk should be derivable from basic geometry, as it depends only on general and unchanging properties of physical space. We estimated μk empirically by computing the average distances between the centers of adjacent intersection regions for 30 randomly-generated sets of place fields covering the environment, and normalized the index by fixing the largest value μ1 = 1 (see Methods). We found that for, the index is well approximated by the formula, where Ncells is the number of place cells active in the environment (see Figure S2). We assume such an index can be “hard-wired” in the brain, as it has no information about any particular arrangement of place fields or any particular environment. Although we estimated μk assuming all place fields have identical size, we use exactly the same formula for μk in every reconstruction, regardless of the distribution of place field sizes we consider. The dissimilarity index can be used to assign weights to each edge in the graph. A path is a sequence of edges connecting two vertices (cell groups) in the graph; the length of a particular path is given by summing the weights along its edges. The distance between any two cell groups in the graph can then be defined as the length of a shortest path between those points (Figure 3D; see also Methods). In this manner one obtains a natural metric on cell groups. We call this graph, with cell groups as its vertices and edges between neighbors, together with the metric, the internal representation of the external space. In order to test how well the internal representation conforms to the geometry of the external space, we used simulated population spiking activity from a two-dimensional square box environment (see Methods) with differing numbers of place cells. For each number of place cells covering the environment, we randomly generated data sets for 60 trials, each trial having different place fields of radii chosen uniformly at random from the interval [0. 1 L, 0. 125 L], with randomly-chosen centers. The place field sizes were chosen to conform to the 20–25 cm range of average diameters typically observed for place cells in dorsal hippocampus for a rat exploring an open field environment of scale L∼1 m [41], [42]. For each simulated data set, we constructed an internal representation as outlined above. To assess the accuracy of the internal representations, we first computed pairwise distances between points on a fine grid spanning the L×L environment and compared them with the corresponding pairwise distances of their images in the internal representation (Figure S3). We defined the pairwise error for an individual trial (having a fixed number of place cells) as the mean error in pairwise distances when computed using the internal representation (see Methods). We found that the average pairwise error across trials had a minimum value of 0. 036 L for 90 cells (Figure 4A), or less than 1/3 the average place field radius; this indicates that relative distances between points in the internal representation are accurate to within a ball of approximately 1/9 the median place field area. To check robustness of this procedure in the case of greater place field variability, we repeated this analysis for a series of gamma-function distributions of place field radii (Figure 4B). We found that performance decreased slowly for distributions with increasing standard deviations up to ∼0. 035, and rapidly deteriorated for distributions with standard deviations greater than 0. 05 (Figure 4C). As a further test that the full geometry—and not just pairwise distances—is accurately reflected in the internal representation, we used multi-dimensional scaling (MDS) [43] to embed each graph into a two-dimensional Euclidean space, in a way that best preserves the relative distances between pairs of points (i. e. , to best preserve the metric on cell groups). Next, we “aligned” the coordinates of the embedded internal space properly so as to best match the particular coordinates used to represent the external space (Figure S4; see also Methods). Points in the external space could then be mapped into the embedded internal space by identifying corresponding cell groups (see Methods). Visually, the quality of an internal representation can be judged by mapping a coarse grid of vertical and horizontal lines from the external space into the embedded internal space, and seeing how faithfully the geometric structure is preserved. We found that the full metric geometry (including angles and relative distances) of the internal representation closely mirrored that of the external space (Figure 5A). In particular, the square shape of the box and the rectilinear structure of the grid were faithfully reproduced. We quantified the accuracy of a given internal representation by computing the mismatch between the two spaces; this measure computes the average error, as a fraction of box side length L, obtained by mapping a fine grid of points from the original space into the aligned embedded internal space (see Methods). Quite similarly to the pairwise error, we found that the average mismatch decreased with increasing numbers of cells, getting as low as 3% for 120–140 cells (Figure 5B). Until now we have assumed that place fields are convex; while this is usually the case, multipeaked place fields are often observed. In open field environments of size L∼1 m, a small percentage (5–10%) of place cells have two disconnected firing fields, each of which looks like a convex, single-peaked place field [21], [44]. We will refer to these neurons as “multipeaked place cells, ” and to the connected components of the place fields simply as “fields. ” At first glance, the presence of multipeaked place cells poses a potential limitation to our study. The algebraic topology theorem no longer holds, suggesting that the algorithm we have thus far used for extracting topological features is likely to fail. In the case of the geometric reconstruction, on the other hand, we do not necessarily expect multipeaked place fields to pose a problem, so long as the component fields are individually convex. In general, the danger with multipeaked place fields is that distant regions of space may be identified as being the same. In an “across-cell” coding scheme where each neuron represents a distinct location in space, this ambiguity indeed poses serious problems [21]. When locations are represented by cell groups, however, this difficulty is easily overcome. Although the same cell may fire in two locations that are far from each other, cell groups corresponding to these distant regions will generally be very different, as other cells serve to disambiguate position. Because two cell groups are considered neighbors only in the case that they share a majority of cells in common, pairs of cell groups with only one or a few common place cells are guaranteed to represent distant positions in the internal representation. In order to test the performance of the geometric reconstruction in the case of multipeaked place cells, we simulated data as before but included small percentages (up to 11%) of multipeaked place cells while keeping the total number of firing fields covering the environment constant. In these simulations, we also required that the centers of multiple fields corresponding to the same cell be sufficiently distant; this was in order to enable disambiguation by other cells (see Methods). For 140 fields, we found the performance to be very good (Figure 6). An example reconstructed space from data containing 10% multipeaked place cells demonstrates that the algorithm naturally separates the double fields (Figure 6A and 6B; see also Figure S5). In particular, both the mismatch (Figure 6C) and pairwise error (Figure S6C) remained approximately constant ranging from 0% to 11% multipeaked place cells. For data generated from only 90 fields, however, as in Figure 5A, performance steadily decreased with increasing numbers of multipeaked place cells, as measured by both pairwise error and mismatch (Figure S6A and S6B). This is because there were not enough place fields to double-cover the environment, leading some regions within double fields to fail to be disambiguated by the presence of other place cells, and thus causing large distortions in the reconstructed space. In fact, this is precisely the problem that may cause the topology algorithm to fail. When the environment is double-covered by place fields, however, place cells with multiple fields can be detected purely from the graph of neighborhood relationships on cell groups (constructed as in Figure 3C). For each place cell, there is an induced subgraph whose vertices are the cell groups containing that cell. Each connected component of this subgraph corresponds to a distinct field (Figure S7). Having detected additional fields for multipeaked place cells, we can then assign auxiliary “place cells” to substitute the original cell labels such that each connected component corresponds to a distinct cell. We then build the simplicial complex and compute homology groups as before. Regarding the (unknown) place fields as providing an open cover of the underlying space, the added step of the topology algorithm can be thought of as making the minimal possible refinement [45] of the open cover such that the theorem again holds (see Methods). We found that with this modification, the topology algorithm maintains very good performance for the experimentally observed range of 5–10% multipeaked place cells (Figure 6D).
  We have shown that, in the case of hippocampal place cell activity, global topological features of a two-dimensional environment as well as an accurate geometric reconstruction of physical space—including the animal' s position within it—can be inferred from spikes alone. In either case, one need only assume that place fields exist and have a stereotypical form; knowledge of actual place fields or any other prior independent measurements of position is not needed. This provides a general framework for building up stimulus spaces (or ‘cognitive maps’ [46]) using only neural activity from a relatively homogeneous population of neurons, such as dorsal hippocampal place cells. Even after obtaining a geometric representation of space, global topological features (if needed) must still be computed. Although we may be able to “see” topological features of the stimulus space by looking at a two-dimensional embedding of the internal representation, this does not mean no further computation is necessary; it merely reflects the fact that our visual system is able to do the computation. Moreover, global features of a ‘space of stimuli’ at one level of processing may become properties of an individual (composite) stimulus at another. Interestingly, although the computation of topological features also has cell groups as its starting point, it does not require constructing a geometric representation of space, and hence bypasses the need for a metric. At first glance, our internal representation is perhaps reminiscent of the ‘cognitive graph’ in [47], [48], as it is also a graph constructed to represent a physical environment. The ‘cognitive graph, ’ however, was envisioned as an actual neural (sub-) network realized in the hippocampus, with an individual place cell for every vertex and a synapse for every edge. The metric was encoded in synaptic weights between place cells, and determined via an LTP learning rule. This implies that geometric distortion would result from a biased sampling of the environment by the animal' s trajectory. Although each vertex in the ‘cognitive graph’ was intended to represent the center of the corresponding place field, it is always the case that many place cells are simultaneously active at any given location in space, suggesting that this graph is not suitable to represent specific positions given population place cell activity. Moreover, the existence of multipeaked place cells presents a seemingly insurmountable challenge in this and any paradigm where place cell firing is presumed to signal proximity to a single place [21]. In contrast, our internal representation graph has a vertex for every group of reliably co-firing neurons, and is closer in spirit to Hebb' s cell assemblies [34] than to a literal neural network representation of space. Since distances between cell groups result from the combinatorics of their overlaps, the fact that multiple cells co-fire in response to a given stimulus is not a nuisance but a necessary condition for inferring the structure of the underlying space. An essential feature is that, unlike in the case of Kohonen maps [49], geometric relationships between external stimuli are revealed even when neurons in the network have no a priori topographic structure (as in hippocampus). Moreover, the resulting metric on cell groups is insensitive to a biased sampling of the environment or to the particular values of synaptic weights within the hippocampus, and does not depend on nearby stimuli occurring in temporal proximity to each other, as would be required to infer stimulus space structure using spike train metrics [50]. Instead, we have used only a very coarse aspect of population spiking activity—the set of all cell groups—while fully exploiting the fact that co-firing cells have overlapping receptive fields. These results suggest that it may be possible for maps of the environment to be constructed in downstream brain areas purely from cell groups. If this is the case, we would expect that geometric distortions in the animal' s spatial perception would arise as a consequence of uneven place field coverage of an environment: the animal should overestimate distances in a region of higher place field density, and underestimate distances in regions with significantly lower place field density. This prediction, if confirmed by experiment, would provide evidence that only cell groups are used in constructing internal representations of space. If, on the other hand, such perceptual distortions are not observed, we can be almost certain that some other aspect of neural spiking activity contributes. Interestingly, because in our simulations we chose place field centers uniformly at random from within the environment, we consistently had a lower density of place field coverage near the boundaries. This, in fact, led to greater geometric distortions near the boundaries of our spatial reconstructions than in the interior. To compensate for this, one might expect there to be a greater number of place fields near the boundary of an environment. Such an effect has, in fact, been reported experimentally [51], [52], and may be regarded as a postdiction. We have considered environments that are flat and two-dimensional; however, it is easy to generalize our procedures to stimulus spaces that are higher-dimensional and/or curved. Recent experiments suggest that three-dimensional hippocampal place fields may be observable in flying bats [53]. The topology algorithm can be used in exactly the same way to detect holes or obstacles in three-dimensional environments from cell groups. The geometric reconstruction algorithm could also be used in exactly the same manner, the only difference being that a different dissimilarity index μk, computed for three-dimensional space, would need to be used. Furthermore, we believe our approach could be generalized to stimulus spaces reflected in other brain areas. The geometric methods could prove useful in discovering new structure in stimulus spaces reflected in neocortical areas, such as primary and higher order sensory cortices. Moreover, the brain appears to be particularly adept at identifying topological properties of complex objects. For example, connected components and holes in a visual object or scene are often among the most salient features. A topological approach, such as the one we have used here, could yield insight into understanding how global features of a visual object are extracted from the activity of cells with spatially localized receptive fields. Our notion of stimulus reconstruction is a significant departure from traditional “decoding” paradigms, as it does not require directly relating neuronal activity to external stimuli (as in the computation of receptive fields), or to activity in any other area of the nervous system. Moreover, while the computation of receptive fields begins with a priori assumptions about the nature of the stimulus space being represented, we recover the structure of the stimulus space itself from the structure of the induced patterns of neuronal activity. The identity of a particular stimulus, then, emerges from a combination of modality (the location and type of activated neurons) and the relationship of its corresponding cell group to all others in an internally represented space. Any necessary assumptions about receptive fields may be regarded as a kind of “universal grammar” [54] that renders stimulus space reconstruction possible. Recently it has been suggested that sequential replay, as observed in hippocampus and neocortex [55]–[58], may be a mechanism for consolidating sequences of cells that under spatial navigation conditions fire within larger time windows. The sequences reflect groups of cells that co-fire within the same theta cycle during behavior, however, and it is unclear to what extent the precise ordering matters [59], [60]. It is therefore plausible that cell groups may be communicated to cortex during replay events—on a compressed timescale—enabling identification via coincidence detection. This may allow for building representations of space and computing topology in cortex, as these computations require knowledge of the full collection of cell groups. In summary, we have shown that a surprising amount of information about the structure of stimulus space can be obtained from the combinatorics of cell groups, extracted from noisy population spiking data with a coarse time window. Although we were able to demonstrate the presence of this information constructively, whether and how the brain uses this information remains to be seen. Our results suggest, nevertheless, that combinatorial relationships between groups of cells that fire together could reflect stimulus space structure inside the brain, and may perhaps lead to a general principle of how the brain constructs representations of the outside world. (1) Place fields are omni-directional, as is typical in an open field environment, but not on a linear track [21], [61]. (2) Place fields have been previously formed and are stable. (3) The collection of place fields corresponding to observed cells covers the entire traversed environment. (4) The holes/obstacles are larger than the diameters of place fields. (5) Each (connected) component field of a single or multipeaked place field is convex. (6) Background activity is low compared to the firing inside the place fields. (7) Place fields are roughly circular and have similar sizes, as is typical in dorsal hippocampus [41], [42]. Although individual electrophysiological recordings can only simultaneously monitor a limited number of cells, it is almost certain that the hippocampus possesses enough place cells for any given environment such that the corresponding place fields cover the entire explored space many times over [44]. The convexity of component fields means that a straight line segment connecting any two points in the field will itself be entirely contained within the field. This is consistent with the observation that individual fields tend to have circular or elliptical shape [21]. Although these are reasonable assumptions about dorsal hippocampal place cells, they pose significant constraints on the quality and quantity of cells in the recording. One may assume, however, that downstream structures in the brain receiving hippocampal output do have access to this kind of data. We test our approaches for constructing internal representations and computing homology groups on simulated data that satisfy these criteria. Each environment is an L×L box, with or without holes in the interior. All length units are with respect to the side length L of the box (typically L∼1 m). In order for the topology and metric algorithms to work, we of course need the animal to fully explore the environment. In particular, the trajectory must be dense enough to sample the majority of cell groups. For the geometric reconstructions, this is merely a matter of resolution, as sampling fewer cell groups will lead to less precise geometric information. For the topology computations, we need to ensure that the set of all cell groups reveals the full low-order intersection information in order for the low-order homology groups to be accurate. For accurate computation of the nth homology group Hn, we need up to (n+1) -fold intersections to be detectable via cell groups. If we were only interested in the first homology group H1 (this is enough to detect holes/obstacles and distinguish between environments) we need only guarantee that pairwise intersections are accurately reflected—i. e. , the trajectory must pass through each pairwise intersection of place fields at least once. However, because we compute homology groups up to H5, in order to check consistency of the data with the interpretation as a two-dimensional environment, we have used denser trajectories in our simulations. This would not be necessary if we were only interested in H0 and H1. Note that a high-order cell group of n cells, signifying an n-fold intersection, implies all lower-order intersections. We define a cell group as a group of place cells that collectively fire within a two theta-cycle (250 ms) time window. To determine the full set of cell groups that become activated as the animal traverses the environment, we first bin population spike trains into 2-theta-cycle time bins. A certain subset of cells fires in each time bin, and we use these subsets to determine the cell groups. Because there is some probability that a given place cell will fire outside its place field, we impose a threshold on firing rates in order to determine the group of cells that fired significantly above baseline for each bin. Each resulting cell group can then be assumed to correspond to a particular intersection of place fields. We first divided population activity into a set of population vectors, i. e. , vectors in RNcells with firing rates for each cell in a given time bin. In order not to miss any cell groups due to the arbitrary choice of where bins start and end, the binning time windows were then shifted to have a total of five different starting positions (eight for topology), equally spaced within two theta-cycles, so that each spike contributed to five population vectors. All population vectors were pooled and thresholded as follows. For each cell, the firing rate in a particular population vector was considered significant if it was at least 6 times the average firing rate for that cell. Each population vector thus yielded a cell group, consisting of all cells firing significantly above baseline in a particular time bin. The thresholding is what renders the topology and reconstruction of space procedures fairly robust to noise in the spike trains. Here we describe how to compute the homology groups of a given environment from the collection of all cell groups that are active in the environment. Here we describe the construction of an internal representation of the environment from the collection of all cell groups that are active in that environment. This can be summarized in two steps: (i) construction of a graph, containing a vertex for every cell group and an edge between neighboring cell groups, and (ii) construction of a distance matrix (or metric) containing distances between any two cell groups. In order to verify that the internal representation is faithful to the geometry of the external space, we computed the average error on pairwise distances between points in the external space as estimated using the metric for the internal representation. To further validate that the full geometry is accurately reflected in the internal representation, we used multidimensional scaling (MDS) to embed the graph in two-dimensional Euclidean space in a way that best preserves the metric on cell groups. This enables comparison of the full geometries by visual inspection and by computation of the mismatch (see below).
summary: We construct our understanding of the world solely from neuronal activity generated in our brains. How do we do this? Many studies have investigated how neural activity is related to outside stimuli, and maps of these relationships (often called receptive fields) are routinely computed from data collected in neuroscience experiments. Yet how the brain can understand the meaning of this activity, without the dictionary provided by these maps, remains a mystery. We tackle this fundamental question in the context of hippocampal place cells—i. e. , neurons in rodent hippocampus whose activity is strongly correlated to the animal' s position in space. We find that the structure of stimulus space can be revealed by exploiting relationships between groups of cofiring neurons in response to different stimuli. We provide a ‘proof of principle’ by demonstrating constructively how the topology of space and the animal' s position in an environment can be derived purely from the action potentials fired by hippocampal place cells. In this way, the brain may be able to build up structured representations of stimulus spaces that are then used to represent external stimuli.
keywords: neuroscience/animal cognition
  neuroscience/sensory systems
  neuroscience/theoretical neuroscience
title: Cell Groups Reveal Structure of Stimulus Space