Transient cell assembly networks encode persistent spatial memories
TTransient cell assembly networks encode persistent spatial memories
Andrey Babichev , and Yuri Dabaghian , ∗ Jan and Dan Duncan Neurological Research Institute,Baylor College of Medicine,Houston, TX 77030, Department of Computational and Applied Mathematics,Rice University, Houston, TX 77005 ∗ e-mail: [email protected] (Dated: October 17, 2018)While cognitive representations of an environment can last for days and even months,the synaptic architecture of the neuronal networks that underlie these representations con-stantly changes due to various forms of synaptic and structural plasticity at a much fastertimescale. This raises an immediate question: how can a transient network maintain a stablerepresentation of space? In the following, we propose a computational model for describingemergence of the hippocampal cognitive map ina network of transient place cell assembliesand demonstrate, using methods of algebraic topology, that such a network can maintain arobust map of the environment. a r X i v : . [ q - b i o . N C ] J un I. INTRODUCTION
The mammalian hippocampus plays a major role in spatial cognition, spatial learning and spa-tial memory by producing an internalized representation of space—a cognitive map of the envi-ronment [1–4]. Several key observations shed light on the neuronal computations responsible forimplementing such a map. The first observation is that the spiking activity of the principal cells inthe hippocampus is spatially tuned. In rats, these neurons, called place cells, fire only in restrictedlocations—their respective place fields [5]. As demonstrated in many studies, this simple principleallows decoding the animal’s ongoing trajectory [6, 7], its past navigational experience [8], andeven its future planned routs [9–11] from the place cell’s spiking activity.The second observation is that the spatial layout of the place fields—the place field map—is“flexible”: as the environment is deformed, the place fields shift and change their shapes, whilepreserving most of their mutual overlaps, adjacency and containment relationships [12–15]. Thus,the sequential order of place cells’ (co)activity induced by the animal’s moves through morphingenvironment remains invariant within a certain range of geometric transformations [16–20]. Thisimplies that the place cells’ spiking encodes a coarse framework of qualitative spatiotemporalrelationships, i.e., that the hippocampal map is topological in nature, more similar to a schematicsubway map than to a topographical city map [17].The third observation concerns the synaptic architecture of the (para)hippocampal network:it is believed that groups of place cells that demonstrate repetitive coactivity form functionallyinterconnected “assemblies,” which together drive their respective “reader-classifier” or “readout”neurons in the downstream networks [21, 22]. The activity of a readout neuron actualizes thequalitative relationships between the regions encoded by the individual place cells, thus definingthe type of spatial connectivity information encoded in the hippocampal map [23].A given cell assembly network architecture appears as a result of spatial learning, i.e., itemerges from place cell coactivities produced during an animal’s navigation through a particu-lar place field map, via a “fire-together-wire-together” plasticity mechanism [24, 25]. However,a principal property of the cell assemblies is that they may not only form, but also or disband asa result of a depression of synapses caused by reduction or cessation of spiking activity over asu ffi ciently long timespan [26]. Some of the disbanded cell assemblies may later reappear duringa subsequent period of coactivity, then disappear again, and so forth. Electrophysiological studiessuggest that the lifetime of the cell assemblies ranges between minutes [27, 28] to hundreds of mil-liseconds [29–33]. In contrast, spatial memories in rats can last much longer [35–37], raising thequestion: how can a large-scale spatial representation of the environment be stable if the neuronalstratum that computes this representation changes on a much faster timescale?The hypothesis that the hippocampus encodes a topological map of the environment allows thisquestion to be addressed computationally. Below, we propose a phenomenological model of atransient hippocampal network and use methods of algebraic topology to demonstrate that a large-scale topological representation of the environment encoded by this network may remain stabledespite the transience of neuronal connections. II. THE MODEL
In [38, 39], we proposed a computational approach to integrating the information provided bythe individual place cells into a large-scale topological representation of the environment, based onseveral remarkable parallels between the elements of hippocampal physiology and certain notionsof algebraic topology. There, we regarded a particular collection of coactive place cells c , c ,..., c n as an abstract “coactivity simplex” σ = [ c , c , ..., c n ], which may be visualized (for n ≤ n ≥
5) as an ( n − T , which provides a link betweenthe cellular and the net systemic level of the information processing. Just like simplexes, theindividual cell groups provide local information about the environment, but together, as a neuronalensemble, they represent space as whole—as the simplicial complex. Numerical experiments [38–40] backed up by a remarkable theorem due to P. Alexandrov [42] and E. ˇCech’s [43] point outthat T correctly represents the topological structure of the rat’s environment and may serve as aschematic model of the hippocampal map [23]. For example, the paths traveled by the rat arerepresented by the “simplicial paths”—chains of simplexes in T that capture certain qualitativeproperties of their physical counterparts (see [44, 45] and Fig. 1A).Of course, producing a faithful representation of the environment from place cell coactivityrequires learning. In the model, this process is represented by the dynamics of the coactivity com-plex’s formation. At every moment of time, the coactivity complex represents only those placecell combinations that have exhibited (co)activity. As the animal begins to explore the environ-ment, the newly emerging coactivity complex is small, fragmented and contains many holes, mostof which do not correspond to physical obstacles or to the regions that have not yet been visitedby the animal. These “spurious” structures tend to disappear as the pool of place cell coactivitiesaccumulates. Numerical simulations show that, if place cells operate within biological param-eters [38], the topological structure of T becomes equivalent to the topological stricture of theenvironment within minutes. The minimal time T min required to produce a correct topologicalrepresentation of the environment can be used as an estimate for the time required to learn spatialconnectivity (Fig. 1B, [38–40]). Coactivity complexes . A specific algorithm used to implement a coactivity complex may bedesigned to incorporate physiological information about place cell cofiring at di ff erent levels ofdetail. In the simplest case, every observed group of coactive place cells contributes a simplex; theresulting coactivity complex T (referred to as the ˇCech coactivity complex in [38, 39]) makes noreference to the structure of the hippocampal network or to the cell assemblies, and gives a purelyphenomenological description of the information contained in the place cell coactivity. In a moredetailed approach, the maximal simplexes of the coactivity complex (i.e., the simplexes that arenot subsimplexes of any larger simplex) may be selected to represent ignitions of the place cellassemblies, rather than arbitrary place cell combinations. The combinatorial arrangement of themaximal simplexes in the resulting “cell assembly coactivity complex,” denoted T CA , schemati-cally represents the network of interconnected cell assemblies [23, 46] (Fig. 2A).The specific algorithm of constructing the complex T CA may also reflect how neuronal coac-tivity is processed by the readout neurons. If these neurons function as “coincidence detectors,”i.e., if they react to the spikes received within a short coactivity detection period w (typically, w ≈ −
250 milliseconds [39, 47]), then the maximal simplexes in the corresponding coinci-dence detection coactivity complex (denoted T ∗ ) will appear instantaneously at the moments ofthe cell assemblies’ ignitions [46, 48]. Alternatively, if the readout neurons integrate the coactivityinputs from smaller parts of their respective assemblies over an extended coactivity integration period (cid:36) [49, 50], then the appearance of the maximal simplexes in the corresponding input in-tegration coactivity complex (denoted as T + ) will extend over time, reflecting the dynamics ofsynaptic integration. The time course of the maximal simplexes’ appearance a ff ects the rate atwhich large-scale topological information is accumulated and hence controls the model’s descrip-tion of spatial learning. Computational implementation of the coactivity complexes T ∗ and T + is as follows. The FIG. 1:
Schematic representation of place cell coactivity in the cell assembly complex T . ( A ) Bottom: Sim-ulated place map covering a small planar environment E that contains a square hole in the middle. The clusters ofcolored dots represent the place fields, and a typical path traversed by the animal is shown by the black loop γ . Thecoactivity complex T induced from the coactivity of the place cells (on the top) provides a topological representationof environment. The hole in the middle of T (red dashed line) corresponds to the central hole in the environment E .The non-contractible closed chain of simplexes—a simplicial path Γ —corresponds to the non-contractible physicalpath γ . In general, the complex can be fragmented in pieces and contain gaps (holes encircled by blue loops), whichrepresent topological noise, rather than information about the environment. ( B ) Timelines of zero-dimensional (0 D ),one-dimensional (1 D ) and two-dimensional (2 D ) topological loops encoded in a coactivity complex T which repre-sents the environment E . For as long as a given 0 D loop persists, it indicates that the coactivity complex containsthe corresponding connectivity component. A persisting 1 D loop represents a noncontractible hole in T (see red holein T on the left panel). A persistent 2 D loop represents a noncontractible 2 D sphere in T . In the illustrated case,the 0 D spurious loops disappear in about 1.5 minutes, indicating that the coactivity complex T fuses into one pieceat that time. The 1 D spurious disappear when all the spurious holes in T close up in about 2.8 min, and the 2 D loops disappear by T min = . T becomes equivalent to thetopology of the environment. maximal simplexes of the coincidence detection coactivity complex T ∗ are selected from the poolof the most frequently appearing groups of simultaneously coactive place cells [46]. To modelan input integrator coactivity complex T + , we first built a connectivity graph G that representspairwise place cell coactivities observed within a certain period (cid:36) . Then we build the associatedclique complex T ς ( G ), i.e., we view the maximal, fully interconnected subgraphs of the graph G ,which are its cliques ς , as simplexes of T ς (for details see Methods in [46] and [51]). The processof assembling the fully interconnected cliques from pairwise connections is designed to modelthe process of integrating the spiking inputs in the cell assemblies, so that the resulting cliquecoactivity complex T ς serves as a model of the input integration cell assembly network.Numerical simulations show that, for a given population of place cells, the clique complex T ς is typically larger and forms faster than the coincidence detector ( ˇCech) complex T ς , and,as a result, T ς reproduces the topological structure of the environment more reliably [23, 46].Moreover, the coincidence detection coactivity complexes can be viewed as a specific case ofthe input integration coactivity complexes: as the integration period shrinks and approaches thecoactivity period (cid:36) → w , the input integration coactivity complex T ς reduces to the coincidencecomplex T ς . For these reasons, in the following we will model only the input integration, i.e.,clique coactivity complexes. FIG. 2:
Place cell assemblies and flickering coactivity complexes . ( A ) Functionally interconnected groups ofplace cells (place cell assemblies) are schematically represented by fully interconnected cliques. The place cells (smalldisks) in a given assembly ς are synaptically connected to the corresponding readout neuron n ς (pentagons below).An assembly ς ignites (red clique / tetrahedron in the middle) when its place cells elicit jointly a spiking response fromthe readout neuron n ς (active cells have red centers). A cell assembly may be active at a certain moment of time, thendisactivate, then become active again, and so forth. If a certain cell assembly ceases to ignite and another combinationof place cells begins to exhibit frequent coactivity, the old cell assembly is replaced by new one. ( B ) The formationand disbanding of the cell assemblies is schematically represented in the “flickering” coactivity complex, in whichthe maximal simplexes appear and disappear, representing appearance and disappearance of the cell assemblies in thehippocampal network. Instability of the cell assemblies . In our previous work [46], the frequencies of the cell as-semblies’ appearances, f ς , were computed across the entire navigation period T tot , i.e., the cellassemblies were presumed to exist from the moment of their first appearance for as long as thenavigation continued. In order to model cell assemblies with finite lifetimes, these frequenciesshould be evaluated within shorter periods (cid:36) ς < T tot . Physiologically, (cid:36) ς can be viewed as theperiod during which the readout neuron n ς may connect synaptically to a particular combinationof coactive place cells, i.e., form a cell assembly ς , retain these connections, and respond to subse-quent ignitions of ς . In a population of cell assemblies, the integration periods can be distributedwith a certain mode (cid:36) and a variance ∆ (cid:36) . However, in order to simplify the approach, we willmake two assumptions. First, we will describe the entire population of the readout neurons interms of the integration period of a typical readout neuron, i.e., describe the ensemble of readoutneurons with a single parameter, (cid:36) . Second, we will assume that the integration periods of allneurons are synchronized, i.e., that there exists a globally defined coactivity integration windowof width (cid:36) during which the entire population of the readout neurons synchronously processescoactivity inputs from their respective place cell assemblies. In such case, (cid:36) can be viewed as aperiod during which the cell assembly network processes the ongoing place cell spiking activity.Below we demonstrate that these restrictions result in a simple model that allows describing apopulation of finite lifetime cell assemblies and show that the resulting cell assembly network, fora su ffi ciently large (cid:36) , reliably encodes the topological connectivity of the environment. Computational model of the transient cell assembly network . A network of rewiring cellassemblies is represented by a coactivity complex with fluctuating or “flickering” maximal sim-plexes. To build such a complex, denoted F , we implement a “sliding coactivity integration win-dow” approach. First, we identify the maximal simplexes that emerge within the first (cid:36) -period FIG. 3:
Topological shapes defined in terms of topological loops . ( A ) Any two zero-dimensional (0 D ) loops (i.e.,points, yellow dots) in the two-dimensional space E navigated by the simulated rat (see Fig. 1A) can be matched withone another via continuous moves. This implies that all 0 D loops are topologically equivalent to a single “representa-tive” loop, i.e., that zeroth Betti number of E is b ( E ) =
1. The one-dimensional (1 D ) loops are of two types: somecontract to a point (e.g., the black loop in the corner) and others (e.g., the red loop in the middle) are non-contractible,signaling the existence of a topological obstruction—the central hole, which is the main topological feature of E .Thus, b ( E ) =
1. Since the entire space E can be contracted to the 1 D rim of the central hole, there are no higher di-mensional noncontractible topological loops in E , b n > ( E ) =
0. The net barcode of E is therefore b ( E ) = (1 , , , , ... ).( B ) On a two-dimensional (2 D ) sphere S , every 1 D loop can be contracted to a point (hence b ( S ) = S into a single representative 0 D loop (hence b ( S ) = D loop (hence b ( S ) = b n > ( S ) =
0. Thus, the topological barcode of a sphere is b ( S ) = (1 , , , , , ... ). ( C ) Atwo-dimensional torus T contains two inequivalent types of noncontractible 1 D cycles, represented by the red andthe blue loops, implying that b ( T ) =
2. The other Betti numbers in the T case are the same as in the S case, b ( T ) = b ( T ) = b n > ( T ) =
0. Thus, the topological barcode of T is b ( T ) = (1 , , , , , ... ). after the onset of the navigation (cid:36) based on the place cell activity rates evaluated within that win-dow, f ς ( (cid:36) ), and construct the corresponding input integration coactivity complex F ( (cid:36) ). Thenthe algorithm is repeated for the subsequent windows (cid:36) , (cid:36) ,... which are obtained by shifting thestarting window (cid:36) over small time steps ∆ t . Since consecutive windows overlap, the correspond-ing coactivity complexes F ( (cid:36) ), F ( (cid:36) ), ... consist of overlapping sets of maximal simplexes. Agiven maximal simplex ς (defined by the set of its vertexes) may appear in a chain of consecutivewindows (cid:36) , (cid:36) , ..., (cid:36) k − then disappear at a step (cid:36) k (i.e., ς ∈ F ( (cid:36) k − ), but ς (cid:60) F ( (cid:36) k )), thenreappear in a later window (cid:36) l , then disappear again, and so forth (Fig. 2). The midpoint t k of thewindow in which the maximal simplex ς has (re)appeared defines the moment of ς ’s (re)birth, andthe midpoints of the windows were is disappears, are viewed as the times of its deaths. Indeed,one may use the left or the right end of the shifting integration window, which would a ff ect theendpoints of the navigation, but not the net results discussed below. As a result, the lifetime δ t ς, k of a cell assembly ς between its k -th consecutive appearance and disappearance can be as short as ∆ t (if ς appears within (cid:36) k and disappears at the next step, within (cid:36) k + , or as long as T tot - (cid:36) inthe case that ς appears at the first step and never disappears. However, a typical maximal simplexexhibits a spread of lifetimes that can be characterized by a half-life, as we will discuss below.It is natural to view the coactivity complexes F ( (cid:36) i ) as instances of a single flickering coactivitycomplex F (cid:36) , F (cid:36) ( t i ) = F ( (cid:36) i ), having appearing and disappearing maximal simplexes (see Fig. 2Band [52]). In the following, we will use F (cid:36) as a model of transient cell assembly network andstudy whether such a network can encode a stable topological map of the environment on themoment-by-moment basis. The large-scale topology . The topological structure of a space X can be described in termsof the topological loops that it contains, i.e., in terms of its non-contractible surfaces countedup to topological equivalence. A more basic topological description of X is provided by sim-ply counting the topological loops in di ff erent dimensions, i.e., by specifying its Betti numbers b n ( X ) [53]. The list of the Betti numbers of a space X is known as its topological barcode, b ( X ) = ( b ( X ) , b ( X ) , b ( X ) , ... ), which in many cases captures the topological identity of topo-logical spaces [54]. For example, the environment E shown at the bottom of Fig. 1A has thetopological barcode b ( E ) = (1 , , , ... ), which implies that E is topologically (homotopically)equivalent to an annulus (Fig. 3A). Other familiar examples of topological shapes identifiable viatheir topological barcodes are a two-dimensional sphere S and the torus T with the barcodes b ( S ) = (1 , , , , ... ) and b ( T ) = (1 , , , , ... ) respectively (Fig. 3B,C). For the mathematicallyoriented reader, we note that the matching of topological barcodes does not always imply topolog-ical equivalence but, in the context of this study, we disregard e ff ects related to torsion and othertopological subtleties.In the following, we compute the topological barcode of the flickering coactivity complex ateach moment of time, b ( F (cid:36) ( t i )), and compare it to the topological barcode of the environment, b ( E ). If, at a certain moment t i , these barcodes do not match, the coactivity complex F (cid:36) ( t i ) and E are topologically distinct, i.e., the coactivity complex F (cid:36) misrepresents E at that particular mo-ment. In contrast, if the barcode of F (cid:36) ( t i ) is “physical,” i.e., coincides with b ( E ), then the coactiv-ity complex provides a faithful representation of the environment. More conservatively, one maycompare only the physical dimensions of the barcodes b ( F (cid:36) ) and b ( E ), i.e., 0 D , 1 D , 2 D loops, orthe dimensions containing the nontrivial 0 D and 1 D loops for the environment shown on Fig. 1A.Using the methods of persistent homology [54–56], we compute the minimal time required to pro-duce the correct topological barcode within every integration window, which allows us to describethe rate at which the topological information flows through the simulated hippocampal networkand discuss biological implications of the results. III. RESULTS
Flickering cell assemblies . We studied the dynamics the flickering cell assemblies producedby a neuronal ensemble containing N c =
300 simulated place cells. First, we built a simulatedcell assembly network (see Methods and [46]) that contains, on average, about N ς ≈
320 finitelifetime—transient—cell assemblies (Fig. 4A). As shown in Fig. 4B, the order of the maximalsimplexes that represent these assemblies ranges between | ς | = | ς | =
14, with the mean ofabout | ς | =
7, implying that a typical simulated cell assembly includes | ς | = ± δ t ς, k as a function of their dimensionalityshows that higher-dimensional simplexes (and hence the higher-order cell assemblies) are shorterlived than the low order cell assemblies (Fig. 4C). The histogram of the mean lifetimes t ς = (cid:104) δ t ς, k (cid:105) k is closely approximated by the exponential distribution (Fig. 4D), which suggests that the durationof the cell assemblies’ existence can be characterized by a half-life τ . The individual lifetimes δ t ς, k ,the number of appearances N ς , and net existence time ∆ T ς = (cid:80) k δ t ς, k of the maximal simplexes andof pairwise connections are also exponentially distributed (see Fig. 4E and Fig. S1). As expected,the mean net existence time approximately equals to the product of the mean lifetime and the meannumber of the cell assembly’s appearance (cid:104) ∆ T ς (cid:105) ≈ (cid:104) N ς (cid:105)(cid:104) δ t ς, k (cid:105) .Fig. 4F shows how these parameters depend on the width of the integration window. As (cid:36) widens, mean lifetime t ς of maximal simplexes (and hence its half-life and the net lifetime) growslinearly, whereas the number of appearances (cid:104) N ς (cid:105) remains nearly unchanged. The latter result is FIG. 4:
Fluctuating simplexes . ( A ) Maximal simplex timeline diagram: each strike represents a timeline of aparticular maximal simplex ς , computed for the coactivity window (cid:36) = N ς =
320 maximalsimplexes at every given timestep (first 200 are enumerated along the y -axis), whereas the total number of maximalsimplexes observed during the entire navigation period is about 11,000. The color of the timelines marks the order of ς (colorbar on the right). Notice that the simplexes of lower orders generally persist over longer intervals. ( B ) Numberof maximal simplexes as a function of their order has a Gaussian shape with the mean d = ∆ d ≈ ±
2, suggesting that a typical cell assembly contains about seven neurons and about two neurons may appear ordisappear in it at a given moment. ( C ) Average existence time of the maximal simplexes tends to decay with increasingorder. An exception is provided by the lowest order (1 D ) connections, which rarely appear as independently andquickly become absorbed into higher order maximal simplexes. ( D ) Histogram of the maximal simplexes’ individualaverage lifetimes τ ς fits with the exponential distribution with mean τ = (cid:36) . ( E ) Histogram of the maximal simplexes’ lifetimes t ς, k , i.e., histogram of the lengthsof all intervals between consecutive appearance and disappearance of the maximal simplexes, the histogram of thenumber of simplex-births N ς and the histogram of the total existence periods T ς fit with their respective exponentialdistributions. The mean number of simplex’ appearances (cid:104) N ς (cid:105) ≈ . (cid:104) T ς (cid:105) ≈ .
57s is approximately equal to the product of the mean lifetime and the mean number of appearances (cid:104) T ς (cid:105) ≈ (cid:104) N ς (cid:105)(cid:104) t ς, k (cid:105) . ( F ) As the size of the memory window (cid:36) increases, the lifetimes, half-lives, and net existenceperiods of the maximal simplexes grow linearly with (cid:36) . natural since the frequency with which the cell assemblies ignite is defined by how frequently theanimal visits their respective cell assembly fields, i.e., the domains where the corresponding sets ofplace fields overlap [46]). This frequency does not change significantly if the changes in (cid:36) do notexceed the characteristic time required to turn around the maze and revisit cell assembly fields, inthis case ca. 1 − τ ≈ Flickering coactivity complex . We next studied the flickering coactivity complex F (cid:36) formedby the pool of fluctuating maximal simplexes. First, we observed that the size of F (cid:36) does not fluc-tuate significantly across the rats’ navigation time. As shown in Fig. 5A, the number of maximalsimplexes N ς ( F (cid:36) ( t )) fluctuates within about 4% of its mean value. The fluctuations in the numberof coactive pairs N ( F (cid:36) ( t )) is even smaller: 3% of the mean, and the variations in number of thethird order simplexes N ( F (cid:36) ( t )) are about 7% of the mean. To quantify the structural changes in F (cid:36) , we computed the number of maximal simplexes that are present at time t i and missing at time t j , yielding the matrix of asymmetric distances, d i j = N ς ( F (cid:36) ( t i ) F (cid:36) ( t j )) for all pairs t i and t j (seeMethods and Fig. 5B). The result suggests that as temporal separation | t i − t j | increases, the di ff er-ences between F (cid:36) ( t i ) and F (cid:36) ( t i ) rapidly accumulate, meaning that the pool of maximal simplexesshared by F (cid:36) ( t i ) and F (cid:36) ( t i ) rapidly thins out. After about 50 timesteps ( | i − j | >
50) the di ff erenceis about 95% (Fig. 5B).Since the coactivity complexes are induced from the pairwise coactivity graph G as cliquecomplexes, we also studied the di ff erences between the coactivity graphs at di ff erent moments oftime by computing the normalized distance between the coactivity matrices (see Methods). Theresults demonstrate that the di ff erences in G , i.e., between G ( t j ) and G ( t i ), accumulate more slowlywith temporal separation than in F (cid:36) : after about two minutes the connectivity matrices di ff er byabout 10 −
15% (Fig. 5C).The Fig. 5D shows the asymmetric distance between two consecutive coactivity complexes F (cid:36) ( t i ) and F (cid:36) ( t i + ), and the asymmetric distance between the starting and a later point F (cid:36) ( t )and F (cid:36) ( t i ), normalized by the size of F (cid:36) ( t ) as a function of time. The results suggest that,although the sizes the coactivity complexes at consecutive time steps do not change significantly,the pool of the maximal simplexes in F (cid:36) is nearly fully renewed after about two minutes. In otherwords, although the coactivity complex changes its shape slowly, the integrated changes acrosslong periods are significant (compare Fig. 5E with Fig. 2B). Biologically, this implies that thesimulated cell assembly network, as described by the model, completely rewires in a matter ofminutes. Topological analysis of the flickering coactivity complex exhibits a host of di ff erent behav-iors. First, we start by noticing that the 0th and the higher-order Betti numbers always assumetheir physical values b = b n > =
0, whereas the intermediate Betti numbers b , b , b and (forsmall (cid:36) s) b may fluctuate (Fig. 6A and Fig. S2). Thus, despite the fluctuations of its simplexes,the flickering complex F (cid:36) does not disintegrate into pieces and remains contractible in higherdimensions ( D > D loops, 2 D surfaces and 3 D bubbles. For example, an occur-rence of b = D loop that surroundsa spurious gap in the cognitive map (Fig. 1A). On the other hand, at the moments when b =
0, all1 D loops in F (cid:36) are contractible, i.e., the central hole is not represented in the simulated hippocam-pal map. The moments when b n > > F (cid:36) containsnon-physical, non-contractible multidimensional topological surfaces. One can speculate about0 FIG. 5:
Behavior of the flickering coactivity complex computed for the memory window width (cid:36) = ∆ t = . A ) The number of maximal simplexes in F (cid:36) (blue trace) fluctuates within 4%of the mean value of N ς = D simplexes N ( F (cid:36) ) (red trace) and the number of the 1 D simplexes appearing in consecutive windows (i.e., links shared by F (cid:36) ( t i ) and F (cid:36) ( t i − ), green trace) fluctuate withina 3% bound. The fluctuations in the number of 2 D subsimplexes ( N ( F (cid:36) ), light blue trace) and the number of 2 D simplexes shared by two consecutive windows (purple trace) do not exceed 7% of the mean. N ( F (cid:36) ) and N ( F (cid:36) )are scaled down by a factor of 10 to fit the scale of the figure. ( B ) The asymmetric distance between F (cid:36) ( t i ) and F (cid:36) ( t j ) is defined as the number of the maximal simplexes at moment t i which are missing at the moment t j for allpairs ( t i , t i ). As the temporal separation | t i − t j | grows, the asymmetric distance between F (cid:36) ( t i ) and F (cid:36) ( t j ) rapidlyincreases. ( C ) The matrix of similarity coe ffi cients r i j between the weighted coactivity graphs at di ff erent momentsof time. For close moments t i and t j the di ff erences between G ( t i ) and G ( t j ) are small, but as time separation grows,the di ff erences accumulate, though not as rapidly as with the coactivity complexes. ( D ) At each t i the blue lineshows the proportion of maximal simplexes present at the previous time, t i − . The green line shows the proportionof maximal simplexes contained at the start (in F (cid:36) ( t )) that remain in the coactivity complex at the later time F (cid:36) ( t i ).The population of simplexes changes by about 0 .
95% in about 2 min. ( E ) A schematic illustration of the changes ofthe coactivity complex’s shape: the fluctuations induce permanent restructurings. No skeletal structure, similar to thatin the “expected” scenario shown in Fig. 2B, is preserved. the biological implications of these fluctuations, see Fig. S4.As the coactivity window increases, the fluctuating topological loops become suppressed andvice versa. As the integration window shrinks, the fluctuations of the topological loops intensify(Fig. 6). This tendency could be expected, since the cell assembly lifetimes reduce as the integra-tion window shrinks and increase as the coactivity integration window grows (Fig. 4F). However,a nontrivial result suggested by Fig. 6 is that the topological parameters of the flickering complex1 FIG. 6:
Stability of large-scale topological information . ( A ) The low-dimensional Betti numbers b , b , b as a function of discrete time, computed for three coactivity integration windows, (cid:36) = (cid:36) = (cid:36) = b =
1, remains stable at all times and is therefore not shown. At su ffi ciently largecoactivity windows, (cid:36) ∼ − F (cid:36) is about 10 secs (Fig. 4C). As the integration window narrows, the topological fluctuations intensify (Fig. S2).( B ) The variation in the time required to extract the topological information increases as the coactivity integrationwindow narrows. At (cid:36) = F (cid:36) fails to produce the correct topological information in 24% of thecases (convergence score ξ = . (cid:36) = (cid:36) = T min to be thetime required to establish the correct topology only in the dimensions that may contain physical obstructions, 0 D and1 D . Therefore, the points where T min diverges are marked by appearances of spurious 1 D loops (encapsulated intored dashed boxes across panels). The points where the learning time rapidly changes are often accompanied by theappearance or disappearance of higher dimensional topological loops (blue dashed boxes). can stabilize completely, even though its maximal simplexes keep appearing and disappearing, or“flickering.” At (cid:36) ≈ F (cid:36) remain unchanged (Fig. 6A), whereas thelifetime of its typical simplex is about 10 seconds (Fig. 4F). Biologically, this implies that a stablehippocampal map can be encoded by a network of transient cell assemblies, i.e., that the ongoingsynaptic plasticity in the hippocampal network does not necessarily compromise the integrity ofthe large-scale representation of the environment. Local learning times . If the information about the detected place cell coactivities is retainedindefinitely, the time required for producing the correct topological barcode of the environment T min may be computed only once, starting from the onset of the navigation, and used as the low-bound estimate for the learning time [38, 39]. In the case of a rewiring (transient) cell assemblynetwork, the pool of encoded spatial connectivity relationships is constantly renewed. As a result,the time required to extract the large-scale topological signatures of the environment from placecell coactivity becomes time-dependent and its physiological interpretation also changes. T min ( t k )2 FIG. 7:
Stability the large-scale topological information . ( A ) A schematic illustration of the growing coactivitywindow (cid:36) , superimposed over a fragment of the maximal simplex timeline diagram. ( B ) The learning times T min ( (cid:36) k )computed within the growing coactivity window. The learning times computed within narrow coactivity windowseither diverge ( T min ( (cid:36) k ) > (cid:36) ) or converge barely ( T min ( (cid:36) k ) ≈ (cid:36) ). As (cid:36) exceeds a certain critical value (cid:36) c (forthe simulated place cell ensemble, (cid:36) c ≈ − T min ( (cid:36) k ) stops increasing and begins tofluctuate around a certain mean value T min = (cid:104) T min ( (cid:36) k ) (cid:105) k . This value is independent of the coactivity window widthand hence represents a parameter-free characterization of the mean time required to extract topological informationfrom place cell coactivity. ( C ) The low-dimensional Betti numbers b , b , b and b as a function of the coactivityintegration window width (cid:36) . As (cid:36) exceeds a critical value (cid:36) c , the Betti numbers b n stabilize, indicating suppressionof the topological fluctuations in F (cid:36) . now defines the period over which the topological information emerges from the ongoing spikingactivity at every stage of the navigation.As shown in Fig. 6B, the proportion ξ of “successful” coactivity integration windows, i.e.,those windows in which T min assumes a finite value, depends on their width (cid:36) . For small (cid:36) ,the coactivity complex frequently fails to reproduce the topology of the environment (Fig. 6A).As (cid:36) grows, the number of failing points, i.e., those for which T min ( t k ) > (cid:36) , reduces due tothe suppression of topological fluctuations. The domains previously populated by the divergentpoints are substituted with the domains of relatively high but still finite T min ( t k ). For su ffi cientlylarge coactivity windows ( (cid:36) > T min ( t k ) exhibits abrupt rises and declines, with characteristic 45 ◦ slantsin-between. The rapid rises of T min ( t k ) correspond to appearances of obstructions in the coactivitycomplex F (cid:36) (and possibly higher-dimensional surfaces) that temporarily prevent certain spuriousloops from contracting. As more connectivity information is supplied by the ongoing spikingactivity, the coactivity complex F (cid:36) may acquire a combination of simplexes that eliminates theseobstructions, allowing the unwanted loops to contract and yielding the correct topological barcode.Thus, Fig. 6B suggests that the dynamics of the coactivity complex is controlled by a sequence ofcoactivity events that produce or eliminate topological loops in F (cid:36) , while the 45 ◦ slants in T min ( t k )represent “waiting periods” between these events (since with each window shift over ∆ t , the locallearning time decreases by exactly the same amount).It should be noticed that the network’s failure to produce a topological barcode at a particularmoment, i.e., within a particular integration window (cid:36) k , is typically followed by a period ofsuccessful learning. This implies that the rudimentary forgetting mechanism incorporated into themodel, whereby the removal of older connectivity relationships from F (cid:36) as newer relationships areacquired, allows correcting some of the accidental connections that that may have been responsiblefor producing persistent spurious loops at previous steps. In other words, a network capable of not3only accumulating, but also forgetting information, exhibits better learning results.Thus, the process of extracting the large-scale topology of the environment should be quantifiedin terms of the mean learning time T min = (cid:104) T min ( t k ) (cid:105) k and its variance ∆ T min / T min , which does notexceed 40% (typically ∆ T min / T min ≈ T min provides a statistically soundcharacteristics of the information flow across the simulated cell assembly network.To better understand how the learning time depends on the memory width, we tested the depen-dence of T min on the size of the coactivity integration window (cid:36) . We fixed the position of severalcoactivity integration windows (cid:36) k and expanded their right side, (cid:36) (1) k > (cid:36) (2) k > ... > (cid:36) ( q ) k (Fig. 7and Fig. S3). As one would expect, small values of (cid:36) generated many failing points, whereasthe learning times T min ( t k ) computed for the successful trials remained nearly equal to (cid:36) , i.e., thewidth of the narrow integration windows was barely su ffi cient for producing the correct barcode b ( E ). However, as (cid:36) grows further, T min stops increasing and, as (cid:36) exceeds a certain critical value (cid:36) c (typically about five or six minutes), the learning time begins to fluctuate around a mean value T min = (cid:104) T min ( t k ) (cid:105) of about two minutes. In other words, for su ffi ciently large coactivity windows (cid:36) > (cid:36) c , the learning times become independent of the model parameter (cid:36) and therefore providesa parameter-free characterization of the time required by a network of place cell assemblies torepresent the topology of the environment, whereas (cid:36) c defines the time necessary to collect therequired spiking information (Fig. S4). IV. DISCUSSION
Fundamentally, the mechanism of producing the hippocampal map depends on two key con-stituents: on the timing of the action potentials produced by the place cells and by the way inwhich the spiking information is processed by the downstream networks. A key determinant forthe latter is the synaptic architecture of the cell assembly network, which changes constantly dueto various forms of synaptic and structural plasticity: place cell assemblies may emerge in cellgroups that exhibit frequent coactivity or disband due to lack thereof. The latter phenomenon isparticularly significant: since the hippocampal network is believed to be one of the principal mem-ory substrates, frequent recycling of synaptic connections may compromise the integrity of its netfunction. For example, the existence of many-to-one projections from the CA3 to the CA1 regionof the hippocampus suggests that the CA1 cells may serve as the readout neurons for the assem-blies formed by the CA3 place cells [8, 57]. Electrophysiological studies suggest that the recurrentconnections within CA3 and the CA3-CA1 connections rapidly renew during the learning processand subsequent navigation [58, 59]. On the other hand, it is also well known that lesioning theseconnections disrupts the animal’s performance in spatial [60–62] and nonspatial [63, 64] learningtasks, which suggests that an exceedingly rapid recycling of functional cell groups may impair thenet outcome of the hippocampal network, which is the hippocampal spatial map [65–68].The proposed model allows investigating whether a plastic, dynamically rewiring network ofplace cell assemblies can sustain a stable topological representation of the environment. Theresults suggest that if the intervals between consecutive appearance and disappearance of the cellassemblies are short, the hippocampal map exhibits strong topological fluctuations. However, ifthe cell assemblies rewire su ffi ciently slowly, the information encoded in the hippocampal mapremains stable despite the connectivity transience in its neuronal substrate. Thus, the plasticityof neuronal connections, which is ultimately responsible for the network’s ability to incorporatenew information [69–72], does not necessarily degrade the information that is already stored inthe network. These results present a principal development of the model outlined in [38, 39, 46]from both a computational and a biological perspective.4 Physiological vs. schematic learnings . The schematic approach proposed in [23] allowsdescribing the process of spatial learning from two perspectives: as training of the synaptic con-nections within the cell assembly network—referred to as physiological learning in [23]—or asthe process of establishing large-scale topological characteristics of the environment, referred toas “schematic,” or “cognitive,” learning. The di ff erence between these two concepts is particularlyapparent in the case of the rewiring cell assembly network, in which the synaptic configurationsmay remain unsettled due to the rapid transience of the connections. On the other hand, schematiclearning is perfectly well defined since the large-scale topological characteristics of the environ-ment can be achieved reliably.In fact, the model outlines three spatial information processing dynamics at the short-term,intermediate-term, and long-term memory timescales [73]. First, local spatial connectivity in-formation is represented in transient cell assemblies within several seconds. This timescale cor-responds to the scope of memory processes that involve temporary maintenance of informationproduced by the ongoing neural spiking activity, commonly associated with short-term memory[73, 74]. The short-term memory capacity is around seven (7 ±
2, [75]) items, corresponding inthe model to the order of the simulated cell assemblies (Fig. 4B). The information about the large-scale connectivity of the environment is acquired and updated—the (mean) learning time T min ,Figs. 6 and 7—is on the order of minutes, corresponding to intermediate-term memory timescale[76, 77]. The persistent topological information, represented by the stable Betti numbers, mayrepresent long-term memory. V. METHODS
The rat’s movements were modeled in a small planar environment, similar to the arenas usedin electrophysiological experiments (bottom of Fig. 1A). The trajectory covers the environmentuniformly, without artificial favoring of one segment of the environment over another.
Place cell spiking activity is modeled as a stationary temporal Poisson process with a spatiallylocalized Gaussian rate characterized by the peak firing amplitude f c and the place field size s c [78].In the simulated ensemble of N c =
300 place cells, the peak firing amplitudes are log-normallydistributed with the mode f c =
14 Hz and the place field sizes are log-normally distributed withthe mode s c =
17 cm. The place cell spiking probability is modulated by the θ -component of theextracellular field oscillations (mean frequency of ∼ The activity vector of a place cell c is constructed by binning its spike trains into an array ofconsecutive coactivity detection periods w . If the time interval T splits into N w such periods, thenthe activity vector of a cell c over this period is m c ( T ) = [ m c ;1 , ..., m c ; N w ], where m c ; k specifies howmany spikes were fired by c into the k -th time bin [46]. The activity vectors of N c cells, combinedas rows of a N c × N w matrix, form the activity raster R . A binary raster B is obtained from theactivity raster R by replacing the nonzero elements of R with 1. Place cell spiking coactivity is defined as the firing that occurred over two consecutive θ -cycles, which is an optimal coactivity detection period w both from the computational [39] andfrom the physiological [47] perspective. A coactivity ρ of a pair of cells c and c can be computedas the formal dot product of their respective activity vectors ρ c c = m c ( T ) m c ( T ). Shifting coactivity window . The spiking activity confined within the k -th coactivity integra-tion window of size (cid:36) produces a local binary raster B k of size N c × N (cid:36)/ w , where N (cid:36)/ w = (cid:98) (cid:36)/ w (cid:99) .The coactivity integration window was shifted by the discrete time steps ∆ t = w ≈ . n s = (cid:36)/ ∆ t steps, the local rasters B k and B k + ns cease to overlap during the four-minute-long5coactivity integration window n s = Coactivity distances . For each window (cid:36) n , we compute the coactivities of every pair of cells ρ ni j = (cid:88) k B ni k B nj k , where B ni k is the “local” binary raster of coactivities produced within that window. To comparedi ff erent local rasters, we compute the similarity coe ffi cients between them r mn = (cid:88) i . j | ρ ni j − ρ mi j | / (cid:88) i . j | ρ ni j | , where indexes i , j run over all the cells in the ensemble, illustrated in Fig. 4C. The cell assemblies were constructed within each memory window using the Method II of[46], which is computationally more stable, produces less maximal simplexes and yields correctvertex statistics for the simulated hippocampal network.
Topological analyses were implemented using the JPlex package [80].
VI. ACKNOWLEDGMENTS
We thank R. Phenix for his critical reading of the manuscript. The work was supported by theNSF 1422438 grant and by the Houston Bioinformatics Endowment Fund.
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FIG. S1:
The statistics for the flickering connections in the cells assemblies . The statistics of the mean lifetimes t ς , individual lifetimes t ς , k , number of appearances N ς and the total existence periods T ς of the 1 D subsimplexesof the coactivity complex F (cid:36) , which is the links of the coactivity graphs G ( (cid:36) n ), representing connections in thesimulated cell assemblies. In all cases, the mean net existence period equals approximately to the product of the meanlifetime by the mean number of appearances (cid:104) T ς (cid:105) ≈ (cid:104) N ς (cid:105)(cid:104) t ς , k (cid:105) . FIG. S2:
Flickering coactivity complex as a function of time . ( A ) As the coactivity integration window (cid:36) increases, the topological fluctuations in the coactivity complex F (cid:36) are suppressed. ( B ) The corresponding learningtimes T min . Red dots mark the moments when the map acquires a non-physical topological barcode. As the coactivitywindow (cid:36) grows, the topological fluctuations are suppressed and the number of failures decreases. Notice that thelearning time remains high immediately after the areas as the failures are suppressed. At (cid:36) ≈ τ ς ≈
10 secs (Fig. 4F), the map retains a topologically correct shape atall times. ( C ) Variations in the size of the coactivity complex F (cid:36) reduce with increasing (cid:36) . FIG. S3:
Growing memory window . ( A ) If the coactivity window is placed at a timepoint with few topologicalfluctuations, the Betti numbers b , b , b and b , and the learning time quickly stabilize. The last panel indicates thatsize of the coactivity complex F (cid:36) grows as a function of (cid:36) and then acquires a stable size. ( B ) At a typical temporaldomain, the behavior of the “asymptotic” coactivity complex F (cid:36) exhibits stronger topological fluctuations and thelearning time fluctuates as a function of increasing (cid:36) . ( C ) In the locations in which the topological fluctuations arestrong, the Betti numbers of the flickering coactivity complex F (cid:36) take longer to stabilize and the learning time mayretain high values for longer periods, before returning to the typical regime shown on panel B. FIG. S4:
Topological fluctuations in the hippocampal map . As in the case illustrated in Fig. 6, we assume thatthe map forms one single piece at all times, hence its 0-th Betti number b = b = b = b n > =
0, the hippocampal map warps into a shape that istopologically equivalent to a torus (Fig. 3C). At the moment when b = b n > =
0, the map contains an extra1 D loop indicating an extra gap in F (cid:36) . At the times when b = b = b n > =
0, the map contains a 2 D bulgeand a non-contractible cycle fused together (since b = F (cid:36) coincides withthe topological type of the simulated rat’s environment, b = b n > ==