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Edited by: David Holcman, École Normale Supérieure, France

Reviewed by: Kechen Zhang, Johns Hopkins University, United States; Thomas Wennekers, Plymouth University, United Kingdom

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Hippocampal cognitive map—a neuronal representation of the spatial environment—is widely discussed in the computational neuroscience literature for decades. However, more recent studies point out that hippocampus plays a major role in producing yet another cognitive framework—the memory space—that incorporates not only spatial, but also non-spatial memories. Unlike the cognitive maps, the memory spaces, broadly understood as “networks of interconnections among the representations of events,” have not yet been studied from a theoretical perspective. Here we propose a mathematical approach that allows modeling memory spaces constructively, as epiphenomena of neuronal spiking activity and thus to interlink several important notions of cognitive neurophysiology. First, we suggest that memory spaces have a topological nature—a hypothesis that allows treating both spatial and non-spatial aspects of hippocampal function on equal footing. We then model the hippocampal memory spaces in different environments and demonstrate that the resulting constructions naturally incorporate the corresponding cognitive maps and provide a wider context for interpreting spatial information. Lastly, we propose a formal description of the memory consolidation process that connects memory spaces to the Morris' cognitive schemas-heuristic representations of the acquired memories, used to explain the dynamics of learning and memory consolidation in a given environment. The proposed approach allows evaluating these constructs as the most compact representations of the memory space's structure.

In the neurophysiological literature, the functions of mammalian hippocampus are usually discussed from the following two main perspectives. One group of studies addresses the role of the hippocampus in representing the ambient space in a cognitive map (Tolman,

On the other hand, it was observed that hippocampal lesions result in severe disparity in episodic memory function, i.e., the ability to produce a specific memory episode and to place it into a context of preceding and succeeding events. In healthy animals, episodic sequences consistently interleave with one another, yielding an integrated, cohesive semantic structure (Wallenstein et al.,

A schematic illustration of memory space concept. _{1} and _{2} (red and blue ovals). The overlapping regions yield a smaller region in the intersection that represents a shared memory (top figure). Alternatively, one memory region can also contain another (the middle figure), or two memory regions can be separate from one another (bottom figure).

Traditionally, the cognitive map is viewed as a Cartesian map of animal's locations, distances to landmarks, angles between spatial cues and so forth (O'Keefe and Nadel,

The mathematical nature of memory space has not been addressed in computational neuroscience literature. However, general properties of the episodic memory frameworks suggest that such a space should also be viewed as primarily topological. Indeed, the “regions” or “locations” in

In the following, we propose a theoretical framework that incorporates both the cognitive maps and the memory spaces and allows modeling them constructively, as epiphenomena of neuronal activity. In particular, it allows relating the topological properties of the memory space to the parameters of the place cell spiking, e.g., to the rate and the spatial selectivity of firing. The proposed approach also allows connecting the concept of memory space to the Morris' cognitive schemas—abstract, heuristic representations of acquired knowledge, skills and memories, used to explain the dynamics of learning and memory consolidation (Tse et al.,

In Babichev et al. (_{k} encodes a spatial region _{k} that serves as a building block of the cognitive map. Secondly, it assumes that the large-scale structure of the cognitive map emerges from the connections between these regions, encoded in a population place cell assemblies—functionally interconnected groups that synaptically drive their respective reader-classifier (readout) neurons in the downstream networks (Harris et al., _{1}, _{2}, …, _{m}) between the regions _{1}, _{1},…_{m}.

A few schematic models were built in Dabaghian et al. (_{1}, _{2}, …, _{m}, can be formally represented by an “abstract simplex,” σ = [_{1}, _{2}, …, _{m}]. In mathematics, the term “simplex” usually designates a convex hull of (

Coactivity complex and the cell assembly complex. _{i0}, π_{i1}, …, π_{id}] of overlapping place fields, π_{i0} ∩ π_{i1} … ∩ π_{id} ≠ ∅. The bottom of the panel shows place field map, _{1}, _{2}, …, _{n}]. Over time,

Previous studies (Curto and Itskov,

A set of overlapping place fields, π_{i0} ∩ π_{i1} ∩ … ∩ π_{id} ≠ ∅ produced by the place cells _{i0}, _{i1}, …_{id} can be represented by an abstract simplex σ = [π_{i0}, π_{i1}, …, π_{id}]. The totality of all such simplexes produced for a given place field map _{i}, of _{i} ∩ π_{j} ≠ ∅, contributes a link _{i} ∩ π_{i} ∩ π_{k} ≠ ∅, contributes a facet

In the brain, the information is represented via temporal relationships between spike trains, rather than artificial geometric constructs such as place fields. However, the place cell spiking patterns can also be described in terms of a simplicial “coactivity” complex _{i} is represented by a vertex, σ_{i}, of _{i} and _{j}, contributes a link

Physiologically, not all combinations of coactive place cells are detected and processed by the downstream networks. Therefore, in order to describe only the physiologically relevant coactivities, one can construct a smaller “cell assembly complex”

Previous studies (Dabaghian et al.,

We now make a short mathematical digression to outline the key notions necessary for discussing the topology of memory spaces. In general, defining a topological space requires two constituents: a set

Modeling a “memory space” requires modifying this approach in two major aspects. First, since a memory space emerges from the spiking activity of a finite number of neurons, it must be modeled as

To build a model of a memory space, we start by noticing that simplicial complexes themselves may be viewed as topological spaces, because the relationships between simplexes in a simplicial complex Σ naturally define a set of topological proximity neighborhoods. Indeed, a neighborhood of a simplex σ is formed by a collection of simplexes that include σ (Figure

Importantly, the construction of Alexandrov spaces applies to “abstract” simplicial complexes, whose simplexes may represent collections of elements of arbitrary nature and hence possess a great contextual flexibility. In our model, individual simplexes represent combinations of coactive place cells, believed to encode memory episodes. We may therefore view the pool of coactive neuronal combinations as a topological space from two perspectives. On the one hand, one can consider a formal “space of coactivities”

We would like to note here, that since the simplexes are not structureless objects (e.g., one combination of coactive cells represented by simplex σ_{1} may overlap with another combination, represented by a simplex σ_{2}, yielding a third combination/simplex σ_{3}), they represent extended regions, rather than structureless points. As a result, the memory space

The discrete memories that comprise a memory space may be triggered by constellations of cues and/or actions, that drive the activity of a particular population of cell assemblies (Buzsaki et al.,

Intuitively, one would expect that a continuous physical trajectory should be represented by a “continuous succession” of activity regimes of the place cells that represents a continuous sequence of memory episodes. Indeed, the topological structure of the memory space provides a concrete meaning for this intuition. It can be shown that the environment

Topological properties of memory spaces can be studied from two perspectives: from the perspective of algebraic topology that captures the large-scale structure of

The algebraic-topological properties of the coactivity complexes were studied in Dabaghian et al. (_{min}. These results apply directly to the memory spaces, since the Betti numbers of a memory space _{min}, for the same set of spiking parameters (in terminology of Dabaghian et al. (

Importantly, the learning times and other global characteristics of _{min} depends mostly on the mean place field sizes and the mean peak firing rates, but it does not depend strongly on the spatial layout of the place fields or on the limited spiking variations. The question arises, how sensitive is the “fabric” of the memory space to the parameters of neuronal activity?

To address this question, we simulated ten different place field maps _{i}, _{i} cannot, in general, be continuously deformed into the memory space _{j} in the same environment. From the mathematical perspective, this outcome is not surprising: since memory spaces are topologically inequivalent to the environment (a continuous mapping

Place field maps in three simulated environments.

Further analyses point out that even if the place field map is geometrically the same but the firing rates change by less than 5%, the cell assembly networks built according to the methods outlined in Babichev et al. (

Similarity between memory spaces and place field remapping. _{i} and _{j} are independently scattered (global remapping); right panel illustrates the case in which the place field positions are fixed, but the place cells' firing rates and place field sizes are altered by 5% (rate remapping). In the latter case, most links are preserved, implying that the one-dimensional “skeleton” of the coactivity complex (Munkres,

These results can be physiologically interpreted in the context of the so-called place field remapping phenomena, which we briefly outline as follows. As mentioned in the Introduction, if the changes in the environment are gradual, then the relative order of the place fields in space remains the same and place cells exhibit only small changes in the frequency of spiking (Colgin et al., _{i} and _{j} (physiologically, one can view a place field map _{j} as a result of a remapping from a map _{i}) are large, whereas rate remapping produces much smaller variations in the structure of the memory space (Figure

Over time, the memory frameworks undergo complex changes: detailed spatial memories initially acquired by the hippocampus become coarser-grained as they consolidate into long-term memories stored in the cortex (Rosenbaum et al.,

As mentioned in the section 2, topological neighborhoods define proximity and remoteness between spatial locations. However, certain neighborhoods may carry only limited topological information. For example, if a neighborhood _{i} in a space _{k} and is involved in the same relationships with other neighborhoods as _{k}, then it only adds granularity to the topology of _{i} and producing a “reduced” space

Reduction of finite topological spaces.

To the extent to which the consolidated memory frameworks retain the structure of the memory space

Reduction of the Alexandrov spaces into their cores and the corresponding Morris' schemas.

Importantly, the reduced memory spaces

The smallest memory space obtained at the last step of the reduction process

Similar compact, schematic representations of the memory structures are frequently discussed in neurophysiological literature. For example, in Tse et al. (

Under such hypothesis, the model allows computing specific Morris' schemas from their respective memory spaces, using the physiological parameters of neuronal activity and the corresponding cell assembly network architecture. Specifically, one can identify the number of elements in a given schema, their projected locations in the environment and their shapes. For the memory spaces constructed for different place field maps of the environments shown on Figure

According to the cognitive map concept, spatial cognition is based on internalized representation of space encoded by the hippocampal network (Tolman,

Simplicial coactivity complexes, e.g., the ones discussed in the Examples 2 and 3 of section 2, are used to represent spatial information by a population of readout neurons responding to nearly simultaneous activity of the presynaptic place cells (Babichev et al., _{1} is “smaller” than σ_{2}, if σ_{2} contains σ_{1} (i.e., σ_{1} < σ_{2} if σ_{2} ∩ σ_{1} = σ_{1}). However, all topological schemas discussed in Babichev et al. (

Given the same physiological parameters (e.g., the same number of place cells) the memory spaces produced by different schemas may differ from one another, e.g. some of them may have stronger topologies than others. However, all memory spaces may be regarded as finitary topological spaces and hence can be considered on the same footing, irrespective of the specific set of rules according to which the information provided by individual place cells is combined in

Current understanding of hippocampal neurophysiology rests on the assumption that place cells' spiking “tags” cognitive regions. Such approach allows describing the information contained in the spike trains phenomenologically, without addressing the “hard problem” of how the brain can intrinsically interpret spiking activity as “spatial” (Chalmers,

Establishing a topological correspondence between the environment and the memory space requires a few definitions.

1. A _{i} ∈ Ω(

Basic notions of point set topology. _{r}. Another collection of elements (blue circles) may form another, “blue” neighborhood _{b} that may overlap with the red neighborhood _{r}, yet another set may form the green neighborhood _{g}, and so forth. _{0} spaces. In particular, all Alexandrov spaces are _{0}-separable. In the illustrated example, the topology base can “resolve” only 20 points, whereas all other elements of

2. A _{i} consists of a smaller set of “base” neighborhoods that can be combined to produce any other neighborhood _{i} of Ω. A key property of a topology base is that it is closed under the overlap operation: an intersection of any two base neighborhoods yields (or, more generally, contains) another base neighborhood. A topology base generates a unique topology for which it forms a base, and hence it is a convenient tool for studying topological spaces (a rough analogy is a set of basis vectors in a linear space, see Alexandrov,

_{E} of a Euclidean domain

_{i} and by augmenting this set with the regions obtained by all possible intersections _{i} ∩ _{j} ∩ …∩ _{k}. By construction, the resulting system of regions will be closed under the overlap operation and hence define a topology base 𝔅_{U}. To obtain a topological base that is as rich as the Euclidean base 𝔅_{E}, the collection of cover regions should be sufficiently large (certainly infinite). However, one can generate much more modest bases and topologies using finite covers. In particular, one can construct a topology of the environment starting from the place fields covering the environment

and build a discrete approximation to the Euclidean topology base from the place field domains and their intersection closure (Figures

Discrete topological spaces and place field maps. _{i}, correspond to the place fields, links σ_{ij}, to overlapping pairs, the triangles σ_{ijk} to simultaneously overlapping triples of place fields. Alternatively, one can view this as the coactivity complex

_{σ} of a simplex σ is formed by the set of simplexes σ_{m}, _{σ}, that include σ (Figure _{σ} = ∩_{m}_{σm}, is its minimal neighborhood. The minimal neighborhoods form a topology base in finitary space

A space

It is important to notice, that if the space

A continuous mapping of the environment into the memory space can be constructed as follows. Let us consider first the coactivity complex _{i} → π_{i}, and the firing pattern of a place cell combination σ into its simplex field _{σ}—the domain where all the cells in σ are active, _{σ}. Notice that simplex fields exist for all (not only maximal) simplexes of

Since simplexes of _{σ}—where _{σ} with any other region yields _{σ}) and that they are disjoint (

which may be viewed as the ultimate discretization of space produced by the given place field map.

Since each atomic element corresponds to a particular simplex σ of _{σ} of _{σ} maps into the corresponding point _{σ} of

A similar argument applies to the memory space generated by the cell assembly complex _{σ} form a cover

The intersection closure of the cell assembly cover yields the decomposition of the environment into the non-overlapping atomic regions 𝔞_{k}, which form a partition of the environment,

Since every point of the environment belongs to one atomic region that corresponds to a particular minimal neighborhood of the memory space, we have a continuous mapping from

Alternatively, one can establish continuity of

The numerical analyses of the finite memory spaces were carried out in terms of the Stong matrices. If a finite topological space _{1}, _{2},…, _{N}, then the topological structure on _{ij}, defined as following:

_{ii} = number of points that fall inside of the neighborhood _{i};

if _{i} is the immediate neighborhood of _{j}, _{ij} = 1 and _{ji} = −1;

_{ij} = 0 otherwise;

Conversely, every integer matrix satisfying the requirements 1-3 describes a finite topological space

For two finitary spaces

If minimal neighborhood _{i} is contained in a single immediate neighborhood _{k}, then it only adds granularity to the Alexandrov space _{i}. If, as a result of coarsening, the neighborhoods separating two points _{1} and _{2} disappear, then they fuse into a single point. This yields a “reduced” Alexandrov space

The numerical procedure implementing the Alexandrov space reduction is as follows. If a column _{i} of a Stong matrix contains only one non-zero element _{ik}, it is removed along with the corresponding row, then the

One can quantify difference between finite topologies Ω_{1} and Ω_{2} by estimating the norm of the difference between the corresponding Stong matrices _{1} and _{2}, minimized over the set

As a simpler option, one can evaluate the distance between the reduced row echelon forms of the Stong matrices,

illustrated Figure _{1} and _{2} are equivalent, i.e., if the corresponding memory spaces are homeomorphic.

Computational algorithms used to simulate the place cell activity are outlined in Dabaghian et al. (

YD conceived of the project, carried out computations, wrote the paper. AB carried out computations, produced materials for figures.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The work was supported in part by Houston Bioinformatics Endowment Fund, the W. M. Keck Foundation grant for pioneering research and by the NSF 1422438 grant.

The Supplementary Material for this article can be found online at:

Cores of the memory spaces in three environments. The figure demonstrates cores of the memory spaces obtained for 10 different place field maps in the three environments shown on Figure

Navigation in the memory space

Reduction of the memory spaces in the environments shown on Figure

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