Robust spatial memory maps encoded by networks with transient connections
[['How does the decay of the connections affect the net structure of the flickering complex F τ ?', 'As shown on Fig 5A , the numbers of links N2 ( t ) and of the triple connections N3 ( t ) rapidly grow at the onset of the navigation and begin to saturate in about ts ≈ 4 minutes ( i . e . , by the time when a typical link had time to make an appearance ) , reaching their respective asymptotic values in ta ≈ 7 minutes .', 'To put the size of the resulting flickering complex into a perspective , note that the number of simplexes in a decaying complex F τ < ∞ can never exceed the number of simplexes that would have existed in absence of decay , i . e . , in the “perennial” coactivity complex , F ∞ ≡ T . Thus , the size of the complex at a moment t , F τ ( t ) , can be characterized by the proportion of simplexes that happened to be actualized at that moment .', 'As illustrated on Fig 5A , these numbers fluctuate around 60% for the second order simplexes and around 40% for the third order simplexes , with the relative variances ΔN2/N2 ≈ 12% and ΔN3/N3 ≈ 17% respectively .', 'In other words , the perennial coactivity complex F ∞ ( t ) loses about a half of its size due to the flickering of the simplexes , and fluctuates within about 15% margins from the mean .', '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 ti , but are missing at a later moment tj , normalized by the size of F τ ( t i ) , i . e . , d i j ( k ) = N k ( F τ ( t i ) \\ F τ ( t j ) ) / N k ( F τ ( t i ) ) , k = 2 , 3 .', 'As shown on Fig 5B , these numbers , which we refer to , respectively , as the second and third asymmetric distances between F τ ( t i ) and F τ ( t j ) , rapidly grow as a function of temporal separation |ti − tj| .', 'In fact , after approximately the effective decay time τ e ( 2 ) , the difference between F τ ( t i ) and F τ ( t j ) becomes comparable to the sizes of either F τ ( t i ) or F τ ( t j ) , which implies that the pool of simplexes in the simplicial complex is replenished at the effective decay timescale .', 'However , the shape of the coactivity complex changes slowly: Fig 5C demonstrates that nearly 100% of the connections are shared at two consecutive moments , i . e . , the changes in flickering complex from one moment of time to the next are marginal .', 'Over longer periods , the flickering complex can change significantly .', 'For example , the proportion of simplexes that are present at t* = 9 minutes , when F τ is particularly inflated , and at other moments , varies around N 2 ( F τ ( t * ) ∩ F τ ( t ) ) ≈ 82 % for second and N 3 ( F τ ( t * ) ∩ F τ ( t ) ) ≈ 64 % for the third order simplexes ( Fig 5C ) .', 'Despite the rapid recycling of the individual simplexes , the large-scale topological characteristics of the flickering complex remain relatively stable .', 'As demonstrated on Fig 5D , after the initial stabilization period of about two minutes ( which biologically may be interpreted as the initial learning period ) , F τ contains only one zero-dimensional and a single one-dimensional topological loop—as the simulated environment E . Some topological fluctuations appear around t* ≈ 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 F τ retain their physical values b 0 ( F τ ) = b 1 ( F τ ) = 1 ( Fig 5E ) .', 'Since Zigzag homology theory allows tracing individual loops in F τ continuously across time , these persistent topological loops can be viewed as ongoing representations of the simply connected environment E and of the physical hole in it .', 'Thus , the coactivity complex F τ preserves , for the most time , not only its approximate size , but also its topological shape—despite transience at the “microscale” , i . e . , at the individual simplex level .', '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 τ e ( 2 ) -period , the topology of the cognitive map encoded by the cell assembly network remain stable .', 'In other words , a transient cell assembly network can encode stable topological characteristics of the ambient space , despite transience of the connections .', 'We investigated the topological stability for a set of proper decay times τ ranging from one to five minutes .', 'As one would expect , the number of simplexes in the flickering complex increases with growing τ: in the studied map , the number of links raises from about 40% of the maximal value at τ = 75 secs to just under 60% for τ = 200 secs , whereas the number of the third order connections raises from 60% to about 80% ( Fig 6A ) .', 'The distributions of the effective lifetimes for the short-lived ( fluttering ) connections retain their exponential shapes ( see Suppl . Materials ) with the means that are approximately proportional to the proper decay times , τ e ( 2 ) ≈ 2 τ and τ e ( 3 ) ≈ τ ( Fig 6B and 6C ) .', 'The contribution of the surviving simplexes also steadily grows with τ ( see Suppl . Materials ) ; as a result , the net average lifetimes , computed for the entire population of simplexes , grow faster Δ t ς 2 ≈ 3 τ and Δ t ς 3 ≈ 2 τ .', 'As τ increases , the Betti numbers rapidly reduce to their physical values , b 0 ( F τ ) = b 1 ( F τ ) = 1: the lower is the connection decay rate , the smaller are the topological fluctuations generated in the flickering complex ( Fig 6D and Suppl . Materials ) .', 'This is a natural result: the longer the simplexes survive , the closer the topological shape of F τ is to the topological shape of the environment E . Physiologically , it implies that the lower is the cell assembly decay rate , the more stable is the cognitive map’s topological structure .', 'As shown on Fig 6D , a stabilization of topological barcode is achieved around τ ∼ 2 minutes .', 'This value can also be naturally interpreted: for such τ , the rat moving at the mean speed of about 25 cm/sec has time to visit most of the environment and reactivate connections in all parts of F τ before they may decay , which allows the induced coactivity complex to contract the spurious topological loops , to assume and to retain the correct topological shape .', 'Note however , that this is only a qualitative argument since the expected lifetimes of over 63% of links is smaller than τ and the lifetimes of 15% of them live longer than 2τ .', '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', 'p ( t ) = { 1 if t = τ 0 if t ≠ τ , ( 2 )', 'while keeping the other parameters of the model unchanged .', 'The results shown on Fig 7A demonstrate that due to the rejuvenation effects , the range of the effective lifetimes widens and becomes qualitatively similar to the histograms induced by the decay distribution ( 1 ) .', 'As before , there appear two distinct populations of links: the short-lived links whose lifetimes concentrate around the singular proper lifetime τ , and the “survivor” links , whose lifetimes approach Ttot .', 'However , the topological structure of the “fixed-lifetime” coactivity complex F τ * differs dramatically from that of the decaying complex F τ .', 'As shown on Fig 7C , F τ * contains a large number of short-lived , spurious topological loops even for the values of τ that reliably produce physical Betti numbers in the case of the exponentially distributed lifetimes .', 'For example , at τ = 100 secs , the zeroth Betti number of F τ * hovers at the average value of 〈b0〉 ≈ 40 , reaching at times b0 ∼ 100 , with nearly unchanged b1 = 1 , which indicates that F τ * is split into a few dozens of disconnected , contractible islets .', 'As the proper decay time increases , the population of survivor links grows and the disconnected pieces of F τ * begin to pull together: at τ = 200 secs , the Betti numbers b k ( F τ * ) retain their physical values most of the time , yielding occasional splashes of topological fluctuations ( Fig 7C and 7D ) .', 'These differences between the topological properties of F τ and F τ * indicate that the tail of the exponential distribution ( 1 ) , i . e . , the statistical presence of long-lasting connections is crucial for producing the correct topology of the flickering complex .', 'Physiologically , this implies that the statistical spread of the connections’ lifetimes plays important role: without it , the dynamical cell assembly network fails to represent the topology of the environment reliably .', '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 N2 ( t ) of links in the decaying coactivity graph G τ ( t ) for τ = 100 sec at every discrete moment of time t ( see Methods ) , and randomly selected the same number of links from the maximal available pool , i . e . , from the graph G ∞ ( t ) that would have formed by that moment without links’ decay ( Fig 8A ) .', 'The collections of links randomly selected at consecutive moments of time can be viewed as instances of a random connectivity graph G r ( t ) , i . e . , as a graph whose links can randomly appear and disappear , in contrast with the decaying links of G τ ( t ) ( compare Figs 8B and 3B ) .', 'As it turns out , the random and the decaying graphs G r ( t ) and G τ ( t ) , as well as their respective clique complexes F r ( t ) and F τ ( t ) exhibit a number of similarities .', 'First , the histogram of the net lifetimes of the links in G r ( t ) shown on Fig 8C is bimodal , with an exponential component characterized by the mean 〈T2〉 = 124 sec , and a component representing a population of surviving connections , similar to the histograms shown on Fig 4G and 4H .', 'Second , the Betti numbers of the random coactivity complex F r converge to the Betti numbers of the environment in about 3 minutes—about as quickly as the Betti numbers of its decaying counterpart F τ ( Fig 8D ) .', 'However , in contrast with the decaying flickering complex F τ , the random flickering complex F r keeps producing occasional one-dimensional loops over the entire navigational period at a low rate ( about 3% of the time , see Suppl . Materials ) .', 'Thus , according to the model , the topological properties of the map encoded by a network with randomly formed and pruned connections are similar to the properties of a map produced by a network with decaying connections , as long as the net probability of the links’ existence are same .', 'In either case , rapidly rewiring connections do not preclude the appearance of a stable topological map , which once again demonstrates that the latter is a generic phenomenon .', '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 [47] .', 'In particular , these processes require a balanced contribution of both “learning” and “forgetting” components , i . e . , of forming and pruning connections [11 , 12] .', 'The imbalances and pathological alterations in the corresponding synaptic mechanisms are observed in many neurodegenerative conditions , e . g . , in the Alzheimer’s disease , which is known to affect spatial cognition [48] .', 'However , interpreting the physiological meaning of these alterations is a challenging task , in particular because certain changes in neuronal activity may not be direct consequences of neurodegenerative pathologies .', 'For example , it is believed that neuronal ensembles may increase the spiking rates of the active neurons in order to compensate for the reduced synaptic efficacies [49–55] .', 'Such considerations motivate deep brain stimulation and other treatments that help to improve cognitive performance in animal models of Alzheimer’s diseases and in Alzheimer’s patients , by increasing the electrophysiological activity of hippocampal cells [56 , 57] .', 'Previous studies , carried out for the models of perennial cell assembly networks [58] , provided a certain theoretical justification for these approaches .', 'It was demonstrated that a place cell ensemble that fails to produce a reliable topological map of the environment due to an insufficient number of active neurons might be forced to produce a correct map by increasing the active place cells’ firing rates .', 'Similarly , the reduction in the firing rates or poor spatial selectivity of spiking may sometimes be compensated by increasing the number of active cells and so forth .', 'Since the current model allows modeling networks with transient connections , we wondered whether it might indicate a theoretical possibility to compensate for the reduced cell assemblies’ lifetimes by altering the place cell spiking parameters .', 'To that end , we varied the mean firing rate f and the number of cells N in the simulated place cell ensemble and studied the topological properties of the resulting coactivity complex as a function of the links’ proper half-life , τ .', 'The results shown on Fig 9 demonstrate that indeed , increasing neuronal activity helps to suppress topological fluctuations in the flickering coactivity complex for a wide range of the connections’ decay times .', 'Moreover , these changes also increase the proportion of trials in which the place cell ensemble captures the correct signature of the environment ( see Suppl . Materials ) .', '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 ( Fig 9C and 9E ) as compared to the number of active place cells ( Fig 9A ) suggests that targeting the active neurons’ spiking may provide a better strategy for designing clinical stimulation methods .'], ['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 , ” ϖ , and building the coactivity graph G ϖ from the most frequently cofiring pairs of neurons [58] .', 'The accumulation of topological information within each ϖ-period , was then described using persistent homology theory techniques .', 'The results indicate that if ϖ extends over 4-6 minutes or more , the topological fluctuations in the flickering complex are suppressed and the topological shape of F ϖ becomes equivalent to the shape of the environment .', 'In the current model , enabled by a much more powerful Zigzag persistent homology theory [34–36] , we employ an alternative approach , in which the links of the coactivity graph appear instantly following pairwise place cell coactivity events .', 'Thus , in contrast with the model discussed in [58] , the current model involves no selection of the “winning” coactivity links , which one might hold responsible for stabilizing the shapes of the flickering coactivity complexes .', 'Nevertheless , this model demonstrates the same effect: the large-scale topological shapes of resulting coactivity complexes stabilize , given that the connections decay sufficiently slowly and have sufficiently broadly distributed lifetimes .', 'The connections’ lifetimes required to achieve such stabilization in the “latency free” model are longer than in the input integration model ( τ ≈ 100 sec vs . τ ϖ ≈ 10 sec ) , which indicates that physiological networks may integrate spiking information over a certain extended period ϖ and optimize the network’s architecture over this information .', 'However , the fact that stable topological maps can emerge in all these different types of transient networks ( including randomly flickering networks ) suggests that this is a generic effect that fundamentally may be responsible for the appearance of stable cognitive representations of the environment in the physiological neuronal networks with transient connections .', 'In other words , the emergence of stable topological maps may represent a common “umbrella” phenomenon that can be implemented via different physiological mechanisms .', '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 [47 , 59 , 60] .'], ['We consider a group of place cells c0 , … , ck coactive , if they produce spikes within two consecutive θ-periods [26 , 43] .', 'As a result , the time interval [0 , Ttot] splits into 1/4 sec long time bins that define the discrete time steps t1 , … , tn .', 'We use simplexes and simplicial complexes to represent combinatorially the topology of the neural activity .', 'An abstract simplex of dimensionality n is a set containing n + 1 elements .', 'A subset of a simplex is called its face .', 'A simplicial complex is a collection of simplexes closed under the face relation: if a simplex belongs to a simplicial complex , then so do all of its faces ( Fig 11 ) .', 'In the constructions studied in this paper , our simplicial complexes consist of coactive place cells .', 'If all cells {c0 , … , ck} 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 {c0 , … , ck} 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 , ” Xi , connected by alternating inclusions:', 'X 1 ⊆ X 2 ⊆ X 3 ⊇ X 4 ⊆ X 5 ⊇ … A k-dimensional chain is a set of k-dimensional simplexes ( Fig 11 ) that can be combined with suitable coefficients .', 'If the coefficients form an algebraic field , then the chains form a vector space .', 'Here we use the simplest algebraic field Z 2 , which consists of two Boolean values 0 and 1 .', 'A boundary of the simplex is the sum of its one-dimension-lower faces: ∂ k { c 0 , … , c k } = ∑ i = 0 k { c 0 , … , c i - 1 , c i + 1 , … , c k } .', 'The map extends linearly to the entire simplicial complex , X , mapping its k-dimensional chains to its ( k − 1 ) -dimensional chains .', 'The kernel of this map , i . e . , all the chains without a boundary , is the set of cycles of the complex , denoted by Zk ( X ) = ker ∂k .', 'The image of ∂k+1 consists of the k-dimensional chains that are boundaries of some ( k + 1 ) -dimensional chains , denoted by Bk ( X ) = im ∂k+1 .', 'Cycles count “k-dimensional holes” in the complex .', 'But not all such holes are independent of each other .', 'We consider two cycles equivalent , or homologous , if they differ by a boundary .', 'Algebraically , one can verify that boundaries themselves have no boundaries , ∂k ∘ ∂k+1 = 0 .', 'In other words , boundaries are cycles .', 'This allows us to take a quotient , Hk ( X ) = Zk ( X ) /Bk ( X ) , called the k-dimensional homology vector space .', 'By definition , it considers two cycles equivalent , if their difference is a boundary of some ( k + 1 ) -dimensional chain .', 'The dimension of this vector space , called the k-th Betti number , βk ( X ) = dim Hk ( X ) , counts the number of independent holes in the topological space .', '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:', 'H k ( X 1 ) → H k ( X 2 ) → H k ( X 3 ) ← H k ( X 4 ) → H k ( X 5 ) ← … This sequence , called zigzag persistent homology , generalizes ordinary persistent homology [36] , where all the maps between homology groups point in the same direction .', 'It is this generalization to the alternating maps that allows us to handle the flickering complexes .', '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 [34] .', 'Specifically , we can select a collection of elements { z i j } j for each vector space Hk ( Xi ) , such that the non-zero elements form a basis for the homology vector space Hk ( Xi ) —in other words , they represent a set of independent holes in Xi .', 'Furthermore , such collections are compatible in the sense that adjacent basis elements map into each other: if we have a map f: Hk ( Xi ) → Hk ( Xi±1 ) , then f ( z i j ) = z i ± 1 j , if z i j ≠ 0 .', 'The experiments in this paper use the algorithm of Carlsson et al . [35] to compute such compatible bases .', '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 .']]
['Results', 'Discussion', 'Methods']
['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 .']
['learning', 'cell physiology', 'action potentials', 'medicine and health sciences', 'neural networks', 'membrane potential', 'social sciences', 'electrophysiology', 'neuroscience', 'learning and memory', 'cognitive psychology', 'mathematics', 'algebra', 'computer and information sciences', 'animal cells', 'vector spaces', 'psychology', 'cellular neuroscience', 'linear algebra', 'cell biology', 'physiology', 'neurons', 'algebraic topology', 'topology', 'biology and life sciences', 'physical sciences', 'cellular types', 'cognitive science', 'neurophysiology']