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The authors have declared that no competing interests exist.

The spiking activity of principal cells in mammalian hippocampus encodes an internalized neuronal representation of the ambient space—a cognitive map. Once learned, such a map enables the animal to navigate a given environment for a long period. However, the neuronal substrate that produces this map is transient: the synaptic connections in the hippocampus and in the downstream neuronal networks never cease to form and to deteriorate at a rapid rate. How can the brain maintain a robust, reliable representation of space using a network that constantly changes its architecture? We address this question using a computational framework that allows evaluating the effect produced by the decaying connections between simulated hippocampal neurons on the properties of the cognitive map. Using novel Algebraic Topology techniques, we demonstrate that emergence of stable cognitive maps produced by networks with transient architectures is a generic phenomenon. The model also points out that deterioration of the cognitive map caused by weakening or lost connections between neurons may be compensated by simulating the neuronal activity. Lastly, the model explicates the importance of the complementary learning systems for processing spatial information at different levels of spatiotemporal granularity.

The reliability of our memories is nothing short of remarkable. Synaptic connections between neurons appear and disappear at a rapid rate, and the resulting networks constantly change their architecture due to various forms of neural plasticity. How can the brain develop a reliable representation of the world, learn and retain memories despite, or perhaps due to, such complex dynamics? Below we address these questions by modeling mechanisms of spatial learning in the hippocampal network, using novel algebraic topology methods. We demonstrate that although the functional units of the hippocampal network—the place cell assemblies—are unstable structures that may appear and disappear, the spatial memory map produced by a sufficiently large population of such assemblies robustly captures the topological structure of the environment.

Functioning of the biological networks relies on synaptic and structural plasticity processes taking place at various spatiotemporal timescales [

The paper is organized as follows. We start with a general outline of the key ideas behind the topological approach and describe a schematic model of a transient cell assembly network. We then study the statistics of its connections’ turnover, the resulting dynamics of the network as a whole and of the spatial map encoded by this network (

1. Dynamics of transient connections. 2. Dynamics of the transient network represented by a “flickering” simplicial complex. 3. Dependence of the network’s dynamics on the transience rate, _{k}, studied in several setups, including “quenched and random complexes. 4. Dependence of the results on the parameters of place cell activity—the mean ensemble firing rate _{c}, and the number of active cells, _{c}.

Our approach is based on recent experimental results [

Our model of the hippocampal network is based on a schematic representation of the information supplied by a population of spiking place cells [_{0}, _{1}, …, _{n} is represented by an abstract simplex _{0}, _{1}, …, _{n}]—a basic object from algebraic topology that may be viewed geometrically as a

This process of accumulation of the topological information can be represented by the dynamics of the coactivity complex. At the beginning of navigation, the complex

A: Simulated place field map _{min} required to eliminate the spurious loops can serve as a theoretical estimate of the minimal time needed to learn path connectivity of the environment. C: If the simplexes may not only appear but also disappear, then the structure of the resulting “flickering” coactivity complex

The topological structure of a steadily growing coactivity complex can be described using persistent homology theory methods [_{k}(_{min} required by the coactivity complex to produce the topological barcode of the underlying environment,

The construction of the coactivity complexes may be adopted to reflect physiological aspects of the hippocampal network. For example, the simplexes of the coactivity complex may represent not just arbitrary combinations of coactive cells, but the neuronal assemblies—groups of cells that jointly elicit spiking activity in the downstream neurons. As mentioned in the Introduction, these assemblies are unstable, transient structures that are recycled, according to different estimates, at the timescale between minutes to hundreds of milliseconds [

In order to represent this transience, the simplexes of the coactivity complex are allowed to appear and to disappear, i.e., “flicker,” following the appearances and disappearances of the corresponding cell assemblies. As a result, certain parts of the resulting “flickering” coactivity complex

An efficient implementation of the coactivity complex is based on a classical “cognitive graph” model of the hippocampal network [_{i} corresponds to a vertex _{i} of a graph _{ij} = [_{i}_{j}_{1}, _{2}, …, _{n} (“synaptically interconnected networks” in terminology of [_{1}, _{2}, …, _{n}

This construction provides a suitable ground for modeling a population of dynamical cell assemblies. Specifically, one can use a coactivity graph with appearing and disappearing (flickering) links to describe the appearing and disappearing connections in the hippocampal network. The topological shape of the corresponding flickering coactivity complex, will then represent the net topological information encoded by this network. This constitutes a simple phenomenological model that connects the information provided by individual dynamical place cell assemblies and their physiological properties (e.g., the rate of their transience) to the structure of the large-scale topological maps encoded by the cell assembly network as a whole. We implemented this model using the following basic assumptions.

_{ij} between cells _{i} and _{j} can disappear with the probability
_{ij} defines its mean decay time. The decay times of the higher order cliques in the coactivity graph (i.e., of the higher order cell assemblies in the hippocampal network) are then defined by the corresponding links’ half-lives.

In a physiological cell assembly network, the decay times _{ij} are distributed around a certain mean _{ij} =

_{ij} in the graph _{i} and _{j} become active within a _{i}, _{j}] either reinstate the link _{ij} (if it has disappeared by that moment) or rejuvenates it (i.e., its decay restarts). As a result, the links’ actual or _{e} may differ from the proper decay time _{ij} that appeared at a moment _{1}, does not disappear by the moment _{2} when it reactivates, then its net expected lifetime becomes _{2} − _{1} + _{3}, then its net expected lifetime is _{3} − _{1} +

A priori, one would expect that if

To start the simulations, we reasoned that in order for the flickering complex _{2}〉 = 50 times during the _{tot} = 25 min navigation period, i.e., the mean activation frequency is _{2} ≈ 1/30 Hz (_{tot}/15.

A: Histogram of the number of times that a given connection _{2} activates, computed for the place field map illustrated on

A histogram of the time intervals ^{th} consecutive birth (_{i}) and death (_{i}) of a link _{ς} ≈ _{tot}

A: The histogram of the time intervals between the consecutive births (_{i}) and deaths (_{i}) of the links, _{2} and _{3}, correspond to the means shown on panels A and B. The histograms of the number of times the link and the triple connections activate during the navigation period are shown on the panels E and F. The exponential fits are shown by solid lines, with the means shown at the top of the panels. The distributions of total existence times for the second (panel G) and third (panel H) order simplexes, with the averages that are approximately equal to the product of the mean effective lifetime and the mean number of appearances, Δ_{e,ς} ≈ _{e,ς}τ_{e,ς}

In other words, the net structure of the lifetimes’ statistics suggests that the coactivity complex contains a stable “core” formed by a population of surviving simplexes, enveloped by a population of rapidly recycling, “fluttering,” simplexes. The _{2} ≈ 4 minutes (

The rejuvenation of simplexes also affects the frequency of their (dis)appearances. As shown on _{ς} = Σ_{i}Δ_{ς,i}_{e,ς} ≈ _{e,ς}τ_{e,ς}

How does the decay of the connections affect the net structure of the flickering complex _{2}(_{3}(_{s} ≈ 4 minutes (i.e., by the time when a typical link had time to make an appearance), reaching their respective asymptotic values in _{a} ≈ 7 minutes. To put the size of the resulting flickering complex into a perspective, note that the number of simplexes in a decaying complex _{2}/_{2} ≈ 12% and Δ_{3}/_{3} ≈ 17% respectively. In other words, the perennial coactivity complex

A: The number of links _{2}(_{3}(_{e} timescale. C: The proportions of second and third order connections shared by the coactivity complexes at the consecutive moments, computed for links (top light-green line) and for the triple connections (top light-blue line) closely follow the 100% mark, which implies that _{*} ≈ 9 min) that are also present at another moment _{*} = 9 minutes. During other periods, _{*} = 9 min, but retain their physical values

To quantify the changes in the complexes’ structure as a function of time, we evaluated the number of two- and three-vertex simplexes that are present at a given moment of time _{i}, but are missing at a later moment _{j}, normalized by the size of _{i} − _{j}|. In fact, after approximately the effective decay time _{*} = 9 minutes, when

Despite the rapid recycling of the individual simplexes, the large-scale topological characteristics of the flickering complex remain relatively stable. As demonstrated on _{*} ≈ 9 minutes, as indicated by an outburst of short-lived spurious loops, most of which last less than a minute. After this period, the first two Betti numbers of

Physiologically, these results indicate that the large-scale topological information significantly outlasts the network’s connections: although in the discussed case about a half of the functional links rewire within a

We investigated the topological stability for a set of proper decay times

A: The number of the two-vertex cliques (

As

To test how these results are affected by the spread of the link lifetimes, we investigated the case in which the lifetimes of all the links are fixed, i.e., the decay probability is defined by the function
_{tot}.

A: The effective timelines of links with the proper decay time of _{e} = _{tot} produces timelines shorter than _{0} (blue) and _{1} (green) is shown on panel D.

However, the topological structure of the “fixed-lifetime” coactivity complex _{0}〉 ≈ 40, reaching at times _{0} ∼ 100, with nearly unchanged _{1} = 1, which indicates that

As the proper decay time increases, the population of survivor links grows and the disconnected pieces of

These differences between the topological properties of

These observations led us to another question: might the topology of the flickering complex be controlled by the shape of the lifetimes’ distribution and the sheer number of links present at a given moment, rather than the specific timing of the links’ appearance and disappearance? To test this hypothesis, we computed the number _{2}(

A: The number of links in the stochastic coactivity graph

As it turns out, the random and the decaying graphs _{2}〉 = 124 sec, and a component representing a population of surviving connections, similar to the histograms shown on

The turnover of memories (encoding new memories, integrating them into the existing frameworks, recycling old memories, consolidating the results, etc.) is based on adapting the synaptic connections in the hippocampal network [

Previous studies, carried out for the models of perennial cell assembly networks [

To that end, we varied the mean firing rate

A: As the number of active cells increases, the number of spurious topological loops drops. To compactify the information, we use the sum of the first two Betti numbers, _{0} + _{1}), which describes the total number of 0_{k} as a function of the decay rate _{k} grow more rapidly at higher firing rates.

Physiologically, these results indicate that recruiting additional active cells and/or boosting place cell firing rates helps to overcome the effect of overly rapid deterioration of the network’s connections, i.e., increasing neuronal activity stabilizes the topological map. In particular, a higher responsiveness of the Betti numbers of the flickering coactivity complex to an increase of the mean firing rate (

The formation and disbanding of dynamical place cell assemblies at the short- and intermediate-memory timescales enable rapid processing of the incoming information in the hippocampal network. Although many details of the underlying physiological mechanisms remain unknown, the schematic approach discussed above provides an instrument for exploring how the information provided by the individual cell assemblies may combine into a large-scale spatial memory map and how this process depends on the physiological parameters of neuronal activity. In particular, the model demonstrates that a network with transient connections can successfully capture the topological characteristics of the environment.

Previously, we investigated this effect using an alternative model of transient cell assemblies, in which the connections were constructed by identifying the pool of cells that spike within a certain “coactivity window,”

In the current model, enabled by a much more powerful Zigzag persistent homology theory [

In all cases, the model reveals three principal timescales of spatial information processing. First, the ongoing information about local spatial connectivity is rapidly processed at the working memory timescale, which physiologically corresponds to rapid forming and disbanding of the dynamical place cell assemblies in the hippocampal network. The large-scale characteristics of space, as described by the instantaneous Betti numbers, unfold at the intermediate memory timescale. At the long-term memory timescale the topological fluctuations average out, yielding stable, qualitative information about the environment. While the former may take place in the hippocampus, the latter two might require involvement of the cortical networks. Thus, the model reaffirms functional importance of the complementary learning systems for processing spatial information at different timescales and at different levels of spatial granularity [

A: The trajectory covers a small planar arena

_{c} localized at the place field center _{c},
_{c} defines the place field’s size [

We consider a group of place cells _{0}, …, _{k} _{tot}] splits into 1/4 sec long time bins that define the discrete time steps _{1}, …, _{n}.

We use simplexes and simplicial complexes to represent combinatorially the topology of the neural activity. An abstract

A: A zero-dimensional (0_{i}; a one-dimensional (1_{i} and _{j}; a two-dimensional (2^{n}, of the corresponding order. B: A single simplex ^{n} is a contractible figure, i.e., it can be collapsed into one of its facets ^{n−1}, then to a facet of lower dimensionality ^{n−2} and eventually to a point ^{0}. Shown is a triangle contracting onto its bottom edge and then to the right vertex. C: A linear chain of simplexes bordering each other at a common face is also contractible. The shade of the triangles constituting the chain defines the order in which the triangles can be contracted (the lighter is the triangle, the sooner it contracts) and the arrows indicate the direction of the contractions. D: If a chain of simplexes loops onto itself and encircles a gap in the middle, then it is not contractible. Collapsing the triangles on the sides of such a closed chain produces an equivalent closed loop, which, ultimately, can be reduced to a non-contractible 1

In the constructions studied in this paper, our simplicial complexes consist of coactive place cells. If all cells {_{0}, …, _{k}} are coactive within a given time window, then so is any subset of them, meaning coactive simplexes form a complex. In fact, because coactivity is defined for a pair of cells, our simplexes are precisely the cliques in the coactivity graph. A simplex {_{0}, …, _{k}} is present if and only if all of its cells are pairwise coactive.

In flickering clique complexes, certain pairwise connections may decay over time, while others appear as time progresses. The effect on the simplicial complex is that some simplexes are removed from the complex, while others are added to it. So we get a sequence of “flickering complexes,” _{i}, connected by alternating inclusions:

A _{k}(_{k}. The image of ∂_{k+1} consists of the _{k}(_{k+1}.

Cycles count “_{k} ∘ ∂_{k+1} = 0. In other words, boundaries are cycles. This allows us to take a quotient, _{k}(_{k}(_{k}(_{k}(_{k}(

Given the sequence of flickering complexes above, we compute homology of each one. Inclusions between complexes induce maps between the homology vector spaces: the homology class of a cycle in the smaller complex maps to the homology class of the same cycle in the larger complex. Accordingly, we get a sequence of homology vector spaces, connected by linear maps:

This sequence, called

On the surface, zigzag persistent homology tracks how the Betti numbers of the flickering complexes change. But the maps that connect homology vector spaces provide extra information. It is possible to select a basis for each vector space in this sequence, so that the bases for adjacent vector spaces are compatible [_{k}(_{i}_{k}(_{i}_{i}. Furthermore, such collections are compatible in the sense that adjacent basis elements map into each other: if we have a map _{k}(_{i}_{k}(_{i±1}), then

It follows that the sequence of homology vector spaces can be decomposed into a barcode, where each bar represents the part of the sequence, where a particular basis element is non-zero. The bars capture when independent holes appear in the flickering complex, how long they persist, and when they eventually disappear. The authors will provide the software used for these computations upon request.

A: Histograms of the intervals between consecutive births (_{e}, computed for the under 16 minutes long intervals. The exponential fit to the histogram of the effective lifetimes of short-living triple connections (_{k}, is shown at the on each panel.

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A: Timelines of 0

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A. Four tests of the topological behavior of the random complex

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The six consecutive pairs of rows (colors alternate for illustrative purposes) correspond to the ensemble mean firing rate

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