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The authors have declared that no competing interests exist.

Analyzed the data: YD FM LF GC. Contributed reagents/materials/analysis tools: YD FM LF GC. Wrote the paper: YD FM. Edited the paper: YD FM LF GC.

An animal's ability to navigate through space rests on its ability to create a mental map of its environment. The hippocampus is the brain region centrally responsible for such maps, and it has been assumed to encode geometric information (distances, angles). Given, however, that hippocampal output consists of patterns of spiking across many neurons, and downstream regions must be able to translate those patterns into accurate information about an animal's spatial environment, we hypothesized that 1) the temporal pattern of neuronal firing, particularly co-firing, is key to decoding spatial information, and 2) since co-firing implies spatial overlap of place fields, a map encoded by co-firing will be based on connectivity and adjacency, i.e., it will be a topological map. Here we test this topological hypothesis with a simple model of hippocampal activity, varying three parameters (firing rate, place field size, and number of neurons) in computer simulations of rat trajectories in three topologically and geometrically distinct test environments. Using a computational algorithm based on recently developed tools from Persistent Homology theory in the field of algebraic topology, we find that the patterns of neuronal co-firing can, in fact, convey topological information about the environment in a biologically realistic length of time. Furthermore, our simulations reveal a “learning region” that highlights the interplay between the parameters in combining to produce hippocampal states that are more or less adept at map formation. For example, within the learning region a lower number of neurons firing can be compensated by adjustments in firing rate or place field size, but beyond a certain point map formation begins to fail. We propose that this learning region provides a coherent theoretical lens through which to view conditions that impair spatial learning by altering place cell firing rates or spatial specificity.

Our ability to navigate our environments relies on the ability of our brains to form an internal representation of the spaces we're in. The hippocampus plays a central role in forming this internal spatial map, and it is thought that the ensemble of active “place cells” (neurons that are sensitive to location) somehow encode metrical information about the environment, akin to a street map. Several considerations suggested to us, however, that the brain might be more interested in topological information—i.e., connectivity, containment, and adjacency, more akin to a subway map— so we employed new methods in computational topology to estimate how basic properties of neuronal firing affect the time required to form a hippocampal spatial map of three test environments. Our analysis suggests that, in order to encode topological information correctly and in a biologically reasonable amount of time, the hippocampal place cells must operate within certain parameters of neuronal activity that vary with both the geometric and topological properties of the environment. The interplay of these parameters forms a “learning region” in which changes in one parameter can successfully compensate for changes in the others; values beyond the limits of this region, however, impair map formation.

In order for an animal to be able to navigate a space, remember its route, find shortcuts, and so forth, it must have a fairly sophisticated internal representation of the spatial environment. This internal map is made possible by the activity of pyramidal neurons in the hippocampus known as place cells. Place cells are so named because of their striking spatial selectivity: as an animal (in experiments, typically a rat) explores a given environment, different place cells will fire a series of action potentials in different, discrete regions of the space. Each region, referred to as that cell's “place field,” is defined by the pattern of neuronal firing (most intense at the center and attenuated toward the edges of the field) (

(a) As a rat explores a given environment, various place cells will fire in spatially discrete locations. Here, for the sake of simplicity, we depict three place fields as they might arise from spike trains from three place cells, as in the next panel. (b) Schematic representation of spike trains fired from three different place cells as a rat explores an environment. Note that there is contemporaneous spiking activity, or co-firing. (c) The place fields derived from the three place cells in (b): the co-firing patterns indicates areas of overlap of the place fields. When the rat makes a straightforward trajectory through an explored environment, different place cells will be activated and their place fields can overlap.

How does the brain convert the pattern of neuronal firing into an approximation of the surrounding space? And what information is most important to navigation and spatial memory? In theory, the mental map could represent metric information (distances and angles), affine aspects (colinearity or parallels), or topological information (connectedness, adjacency, containment). The reigning paradigm is that the maps encode geometric information: in fact, most efforts to analyze cognitive maps derived from place fields are based explicitly on the geometry of both the place fields and the environment

If we restrict ourselves to cell spiking activity, the temporal features of the firing pattern become paramount: in particular, if spatial location is the primary determinant of each place cell's firing, then contemporaneous activity or co-firing of several place cells implies that the corresponding place fields overlap. It is, in fact, generally assumed that neurons downstream of the hippocampus interpret place cell spiking patterns based on co-firing. What is not often appreciated, however, is that if place cell co-firing implies spatial overlap of place fields, then the map formed by co-firing is going to be based on connectivity, adjacency and containment—in other words, it will be a topological, rather than a geometric, map.

Indeed, the way place fields cover an environment calls to mind a basic theorem of algebraic topology: if one covers a space

Here we investigate whether a topological connectivity map can be effectively and reliably derived from neuronal spiking patterns using computational tools recently developed in the field of algebraic topology. We show that there exist certain requirements for the firing activity to produce a stable topological map and that the experimentally observed characteristics of firing activity likely satisfy these requirements.

We will first outline the key concepts underlying our approach; more precise mathematical explanations are provided in the

In algebraic topology, the topological features of a space _{0}(X)_{1}(X)_{2}(X)_{i}_{j}_{ij}_{ij}_{jk}_{ki}

Drawing on this concept of the nerve simplicial complex

Given a certain experimental, phenomenological or theoretical description of place cell firing, it should be possible to trace the accumulation of topological information with

The parameters that might be taken into account to define place cell activity are numerous and complex: there are biophysical variables (firing rates, spike amplitude, etc.), behavioral variables (the animal's running speed, etc.), out-of-field firing (not all place cell firing is for spatial encoding purposes), and so forth. For the sake of simplicity, at least for this first attempt to model place cell ensemble behavior, we zeroed in on just a few key parameters that will still enable us to ask key questions.

First we had to decide how to define temporal overlap between spike trains. There is some conjecture in the field that each theta cycle—the basic EEG cycle in the hippocampus, with a frequency of

For this initial analysis, we ignored the details of the spike train structure, such as spike bursting

Thus, for an N-cell ensemble this approach produces 3N independent parameters, _{1}, _{2}, …, _{N}_{x,1}, s_{x,2},…, s_{x,N}, s_{y,1}, s_{y,2},…, s_{y,N}_{i}_{i}

A typical place cell fires at a rate of ∼10–20 Hz and place fields typically range from 10 to 30 cm across. These experimentally derived distributions serve as realistic constraints on our simulated data by providing proportionality coefficients

Given these starting assumptions and simplifications, the individual firing rates _{i}_{i}

Given the temporal nature of our map formation model, we will adopt one more simplifying assumption, namely, that all the instances of co-activity that occur between

The foregoing considerations led to the following (very simplified) working model of place cell activity:

Place cell firing activity is a stationary Poisson process described by the rate model

Two cells are considered to be co-active if they fire within two consecutive periods of theta oscillations, i.e., within ∼1/4 sec. We expect shorter time windows would require longer periods for map formation, so this value helps us establish a lower bound on the length of time required to extract connectivity information.

The firing rate amplitudes _{i}_{x,1}_{y,1}

Retained memory assumption: all firing events occurring up to time

Our analysis is based on the dynamics of “cycles,” objects that can be used to count the number of topological holes within the temporal complex

Top and bottom graphs show which 0

The time required for the correct number of bars (cycles) to appear in every dimension is, by design, the time required to extract the correct topological signature of the environment, which can thus be interpreted as the minimal time

We simulated map formation times using different place cell parameters and three separate planar 2×2 meter areas with 1 or 2 holes (

The

The trajectories were simulated to be: 1) sufficiently ergodic to represent non-preferential exploratory spatial behavior (i.e., there was no artificial circling or other ad hoc favoring of one segment of the environment over another). The spatial occupancy of the immediate vicinities of the holes and of the corners was therefore higher than the average, which is similar to patterns of spatial occupancy in the open field and linear track experiments. 2) The mean and the maximal speed were kept within the range of typical experimental values (based on our experience; the mean speed was chosen to be slightly higher than a typical experimental mean value in order to get a lower estimate for the learning time _{sim}_{exp}

We asked whether, and for which ensembles, the place cell spiking signals would be able to produce a temporal simplicial complex with the correct number of topological loops (Betti numbers; see

The results are shown in

These data illustrate, first, that the firing activity of smaller place cell ensembles (

These 2D sections are based on the point cloud data in

It is noteworthy that the points with intermediate sizes, representing the partially failing ensembles, tend to diffuse out from the center of

It is noteworthy that at the core of

For each environment, the graph shows how much time is required to cover a certain percentage of the 3×3 cm spatial bins. This ergodic time scale shows that it takes approximately ten minutes for a rat to cover 80% of the environment; by comparison, the topological map formation time for stable regimes is much lower.

_{1}), so that two persistent loops have to be extracted from a set of non-persisting loops. At the same time, it is also more geometrically complex than the quasi-linear environment (

It is also important to note that the mean map formation time

Finally, in order to single out the hippocampal states for which the mean map formation time

In summary, it is usually assumed that an ensemble of cells with spatially selective firing will naturally encode a spatial map. Our results demonstrate that the spatial selectivity of firing does not, by itself, guarantee a reliable mapping of the actual environment. The geometric shape of the learning region

We have examined the dynamics of hippocampal spatial map formation beginning with arbitrary place cell activity regimes, both those that resemble biological cells and those that do not. We created a computational program to simulate map formation with three independent variables: the firing rate of the place cells, the size of the place field, and the number of cells. We then tested the model on three different scenarios (which included two topological configurations and two different geometries), and repeated the simulation in each scenario 10 times prior to statistical analysis. Our simulations show that in order to form a reliable topological map of the environment, the place cell ensemble must operate within certain parameters—outside these parameters, place cells can be spatially specific but will not be able to produce a reliable map. It is noteworthy that the parameters for place cell firing and place field size that produced a robust map formation region

Our current model relies on a simple spike train structure based on a Gaussian firing rate (_{model}

One could conceivably choose any valid set of parameters to define hippocampal states that produce a model-defined learning region _{model}_{bio}

The topological model predicts that: (1) the parameters describing the hippocampal place cell map in healthy animals should fall inside of the stable learning region _{model}

Despite its simplifications, the current model allows us to examine whether a particular set of place cell parameters can be used to map a given environment and vice versa, and to reason about the effect of the geometry and topology of an environment on place cell behavior. For example,

Although the current model does not describe the formation of place fields themselves, it provides some insight into the process of learning in novel environments. Place fields show considerable plasticity over the course of learning new environments, expanding in adaptation to large environments

Indeed, perhaps the most striking aspect of the current study is not that it supports the hypothesis that the hippocampus encodes topological information about the environment, but that the learning region

Numerous studies have documented spatial learning deficits and changes in place field characteristics in mice bearing specific genetic mutations, but the connection between behavioral changes and the changes in place field properties has been unclear. We suggest that significant alterations of place cell behavior result in hippocampal states hovering at or beyond the boundaries of _{bio}

We open this section by outlining the assumptions we made about place fields and place cells in this first attempt at a model of hippocampal spatial map formation. We then define key theoretical concepts from algebraic topology that motivated our particular computational approach, particularly relating to the relatively new tools of Persistent Homology theory.

Each experimental environment depicted in the top row of _{sim}_{exp}

For this initial analysis, we ignored the details of the spike train structure, such as spike bursting _{i}_{i}

In this model, we assume that: 1) Place fields are ellipsoid and omni-directional, as typically recorded in open field environments

Simplicial complexes are used to approximate the structure of topological spaces _{0}_{1}_{k}_{i}_{j}_{0}_{1}_{i−1}_{i+1},…,v_{k}^{m}^{m}^{n}

It can be shown that topological features, e.g., holes in the environment, correspond to loops in the simplicial complex, which can be detected through combinatorics of the simplices. It is possible to determine, for example, whether two points in the complex are connected by a sequence of edges or not. The simplicial complex produced by the overlaps between the place fields covering the environment is known in algebraic topology as the “nerve of the cover” or the “nerve simplicial complex”

The hypothesis that drives this project is that the hippocampus encodes a topological map. To begin our investigation we ask whether the topological map produced by the place cells captures the most basic topological features of the environment, namely, the number of holes in it. This question can be addressed using homology theory, which aims to detect homologous loops and to categorize holes in a space. Since the structure of the nerve simplicial complex approximates the structure of the environment, we can use homology theory to count the loops in the simplicial complex and therefore the number of holes in the environment.

There are numerous variants of homology: we use simplicial homology with

Let Σ denote a simplicial complex. Roughly speaking, the homology of Σ, denoted, _{k}_{k}_{k}_{0}_{0}_{1}_{1}

(a) A point is a 0

For each _{k}(Σ)_{0}_{k}_{0}_{k}_{k}

The boundary map is defined to be the linear transformation ∂: _{k}_{k−1}_{0}_{k}_{k}

_{k}_{k}_{k−1}

_{k}_{k+1}_{k}

A simple lemma demonstrates that ∂ ○ ∂ = 0; that is, the boundary of a chain has an empty boundary. It follows that _{k}_{k}_{k}_{k}_{k}

By arguments utilizing barycentric subdivision, one may show that the homology _{1}_{1}

Given a set of place fields {_{1},PF_{2},…,PF_{N}_{1}_{k}_{i}_{j}_{ij}_{i}_{j}_{ijk}_{i}_{j}_{k}_{1},PF_{2},…,PF_{N}

In the context of studying a hippocampal map formation, in which the analysis is based on temporal characteristics of place cell activity, the simplicial complex can be constructed using the notion of _{i}_{j}_{i}_{j}_{k}_{1}_{2}_{N}_{i}_{i}

This defines a “temporal simplicial complex”

This construction achieves the goal of providing us with a topological method that can tell us whether cells are indeed receiving all the information necessary for reconstructing the topology of the environment. The main question discussed in the paper is whether and to what extent different hippocampal states (as defined by variations in the mean firing rate

In theory, there are two ways in which one can build simplicial complexes in order to describe the topological information contained in place cell firing activity: use place field geometry or place cell spike trains. How are the corresponding simplicial complexes N and

It is often remarked that homology is functorial, by which it is meant that it faithfully represents topological information. To clarify this point, consider two simplicial complexes Σ and Σ′. Let f: Σ→Σ′ be a continuous simplicial map: _{#}_{k}_{k}_{#}_{*}_{*}_{k}_{k}_{*}_{*}_{*}

Given ε>0 and ^{+}_{i}_{i}

Note that by definition, for any simplex _{1}≤t_{2}≤…≤t_{n}_{n}_{n}

Edelsbrunner and colleagues, however, made the following observation _{k}_{k}_{k}

Such a system is called a

To analyze both simulated and experimental data we used jPLEX, a collection of MATLAB functions for computational topology that implements the concepts described above. It is freely available from

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We thank the anonymous reviewers and V. Brandt for their critical reading of the manuscript.