Dynamic core-periphery structure of information sharing networks in the entorhinal cortex and the hippocampus
Nicola Pedreschi, Christophe Bernard, Wesley Clawson, Pascale Quilichini, Alain Barrat, Demian Battaglia
DDynamic core periphery structure of informationsharing networks in entorhinal cortex andhippocampus
Nicola Pedreschi , Christophe Bernard , Wesley Clawson , Pascale Quilichini , AlainBarrat , and Demian Battaglia Aix Marseille Univ, Universit ´e de Toulon, CNRS, CPT, Turing Center for Living Systems, Marseille, France Aix Marseille Univ, Inserm, INS, Institut de Neurosciences des Syst `emes, Marseille, France Tokyo Tech World Research Hub Initiative (WRHI), Institute of Innovative Research, Tokyo Institute of Technology.Japan
ABSTRACT
Neural computation is associated with the emergence, reconfiguration and dissolution of cell assemblies in the context ofvarying oscillatory states. Here, we describe the complex spatio-temporal dynamics of cell assemblies through temporalnetwork formalism. We use a sliding window approach to extract sequences of networks of information sharing among singleunits in hippocampus and enthorinal cortex during anesthesia and study how global and node-wise functional connectivityproperties evolve along time and as a function of changing global brain state (theta vs slow-wave oscillations). First, we findthat information sharing networks display, at any time, a core-periphery structure in which an integrated core of more tightlyfunctionally interconnected units link to more loosely connected network leaves. However the units participating to the core orto the periphery substantially change across time-windows, with units entering and leaving the core in a smooth way. Second,we find that discrete network states can be defined on top of this continuously ongoing liquid core-periphery reorganization.Switching between network states results in a more abrupt modification of the units belonging to the core and is only looselylinked to transitions between global oscillatory states. Third, we characterize different styles of temporal connectivity thatcells can exhibit within each state of the sharing network. While inhibitory cells tend to be central, we show that, otherwise,anatomical localization only poorly influences the patterns of temporal connectivity of the different cells. Furthermore, cellscan change temporal connectivity style when the network changes state. Altogether, these findings reveal that the sharing ofinformation mediated by the intrinsic dynamics of hippocampal and enthorinal cortex cell assemblies have a rich spatiotemporalstructure, which could not have been identified by more conventional time- or state-averaged analyses of functional connectivity.
Since its early definitions , the notion of cell assembly , loosely defined as a group of neurons with coordinated firingwithin a local or distributed circuit, has been associated to information processing. According to a widely acceptedview (see e.g. ), neuronal representations and, more generally, computations are constructed via the dynamicintegration of the information conveyed by the spiking activity of different cells. The recruitment of a cell assemblygoes thus well beyond the mere co-activation of an ensemble of cells frequently firing together. It correspondsindeed to the instantiation of an actual transient functional network allowing information to be shared between theinvolved neurons and ultimately fed into novel informational constructs suitable for further processing .In this sense, it appears quite natural to describe the flexible materialization, transformation and dissolutionof groups of information-exchanging neurons as a dynamic network whose nodes and edges evolve along time,reflecting the recruitment (or the dismissal) of neurons into (or out of) the current integrated assembly. Nevertheless,cell assemblies at the level of neuronal microcircuits —and, particularly, in the hippocampal formation, involved inspatial navigation and episodic memory — have been most often characterized in terms of sets of nodes frequentlyco-activating in time or repeatedly activating in sequences . Such “static” catalogues of patterns of firingpartially fail at highlighting that the temporally coordinated firing of nodes gives rise to a dynamics of functionallinks , i.e., to a temporal network .The temporal network framework has recently emerged in order to take into account that for many systems, astatic network representation is only a first approximation that hides very important properties. This has been madepossible by the availability of temporally resolved data in communication and social networks in particular :studies of these data have uncovered features such as broad distributions of contact or inter-contact times (burstiness)1 a r X i v : . [ q - b i o . N C ] J a n etween individuals , multiple temporal and structural scales , and a rich array of intrinsically dynamicalstructures that could not be unveiled within a static network framework . Taking into account temporality hasmoreover been shown to have a strong impact in processes taking place on networks, in particular the propagationof diseases or of information . In the neuroscience field, emphasis have been recently put on the need toupgrade “connectomics” into “chronnectomics”, to disentangle temporal variability from inter-subject and inter-cohort differences and thus achieve a superior biomarking performance . However a majority of dynamicnetwork studies have been so far considering large-scale brain-wide networks of interregional connectivity —-see,e.g. for a study of link burstiness— and only fewer have addressed the dynamics of information sharing networksat the level of micro-circuits, often in vitro or in silico and even more rarely in vivo .Here, we propose to fully embrace a temporal network perspective when describing dynamic informationsharing within and between cell assemblies. Concretely, we analyze high-density electrophysiological recordingsin the hippocampus and medial entorhinal cortex of the rat, allowing to follow in parallel the spiking activity ofseveral tens of single units across different laminar locations in different brain regions. We focus on recordingsperformed during anesthesia, guaranteeing long and stable recordings of intrinsic cell assembly dynamics overseveral hours . Indeed, even during anesthesia, the spatiotemporal complexity of firing is not suppressed but awide repertoire of co-firing ensembles and associated information processing modes can be found . Furthermore,with the used anesthesia protocol, we observe a characteristic stochastic alternation between two global brainoscillatory states, respectively dominated by Slow Oscillations (SO) and Theta (THE) oscillations, reminiscent of aslowed-down version of natural sleep, with its alternation between non-REM and REM epochs (see Methods ). Suchexperimental condition is therefore particularly suitable to probe the dependence of cell assembly dynamics on thecurrently active global oscillatory state , which is expected to be a major modulator of information processing inthe hippocampus formation and cortical circuits in general . Using a sliding window approach and estimatingfunctional connectivity via an analysis of the pairwise mutual information between the spike trains of distinct singleunits —following —, we extract temporal networks of information sharing and we develop methods to investigatehow their connectivity properties evolve in time.At the level of whole network organization, we pay attention to whether connectivity structure changescontinuously —as in the case of many social or communication networks — or rather undergoes switchingbetween different discrete network states, —as recently observed for instance in an animal social network —,possibly in relation with transitions between global oscillatory states. We find that switching between discretenetwork states does spontaneously occur during anesthesia. Remarkably, we identify a multiplicity of states ofconnectivity between single neurons, with a rich switching dynamics ongoing even in absence of a change inthe global oscillatory state. The sharing network connectivity, however, is never frozen, but keeps fluctuatingeven within each of the network states. More specifically, at any time, the instantaneous information sharingnetwork displays a core-periphery organization , in which a limited number of neurons form a tightly mutuallyconnected core, while a majority of other neurons are more peripheral. Individual neurons flexibly modulate alongtime their degree of integration within the sharing network and may “float” between the core and the periphery,transiently leaving or getting engaged into the core, giving rise to what we call a liquid core-periphery architecture.At the level of single nodes, the neighborhood of each neuron is changing smoothly within network statesand more abruptly across a state switching. New connections can be formed or old connections disappear andthe number of neighbors can vary in time. Most importantly, neurons who have at the time aggregated levelsimilar static connectivities, having connections for a comparable overall amount of time, can strongly differ in theirtemporal connectivity profile. For instance, some neurons may form links that remain active for a limited amountof time but in an uninterrupted way. Other neurons may instead repeatedly connect and disconnect to others,sharing information in an intermittent and sporadic fashion. We thus define for each neuron its specific temporalconnectivity profile, which summarizes its dynamic patterns of attachment in the evolving core-periphery sharingnetwork. We then use an unsupervised classification approach to identify temporal connectivity style archetypes andshow that neurons can adopt different styles in different network states, possibly associated to variations of theirrole in information processing (see Discussion ).Going beyond averaged network analyses with suitable time-resolved metrics allows thus an unprecedentedprecision in characterizing the functional organization of information sharing. Cell assemblies are not anymore seenas rigidly defined groups of cells but as dynamic networks, restlessly exchanging flows of information between coreand periphery and continuously modifying their extent and reach toward cells in different anatomical locations. Inother words, the adoption of a temporal network framework makes it possible to witness and seize the inner life ofcell assemblies while it unfolds and gives rise to emergent computations.
Results
Single unit recordings of neuronal activity were acquired simultaneously from the CA1 region in the Hippocampusand in the mEC (Figures 1.A and S1) for 16 rats under anaesthesia (18 recording for 16 rats). Following , weconstructed time-resolved weighted networks of functional connectivity, adopting a sliding window approach.Within each 10s long time-window, we took connection weights (functional links) between pairs of neurons (networknodes) to be proportional to the amount of shared information between their firing rates (see Methods ). We thenslide the time window by 1s, in order to achieve a 90% overlap between consecutive windows. In Figure 1.B, werepresent the temporal network construction procedure. Based on the data segments in each of three windowscentered at times t a , t b and t c , we extract a N × N matrix for each time window, where N is the number of neuronsand in which the element ( i , j ) corresponds to the shared information between nodes i and j . Each such matrix, innetwork terms, is interpreted as the adjacency matrix of a weighted graph G of N nodes. Even if the used functionalconnectivity metric is in principle pseudo-directed (because of the presence of a time-lag between putative senderand receiver node, see Methods ), we found that asymmetries between reciprocal connections were very small (see
Methods ), especially for the stronger connections, and chose therefore to symmetrize the adjacency matrix formost analyses. This procedure thus maps each multi-channel recording of length T seconds to a time series of T network representations, obtaining finally a temporal network of information sharing among neurons, formed bythe temporal succession of these T network snapshots. Cartoon representations of the temporal network snapshots G ( t a ) , G ( t b ) and G ( t c ) in the three highlighted time windows are shown in Figure 1.C. Some actual network framesof a specific recording, together with a diagram describing emergence and disappearance of links ( edge activity plot )can be seen in Figure 2.On the top of Figure 1.B we also present the characteristic switching between global oscillatory states observedin our recordings. Analysis of the local field potentials recorded simultaneously to single unit activity allowedidentifying a spontaneous stochastic-like alternation between epochs belonging to a first SO global state (lightblue color) spectrally dominated by < i , how much its neighborhood changedbetween successive time windows . To this aim we computed for each i and at each time t the cosinesimilarity Θ i ( t ) between the neighborhoods of i (the subgraphs composed only by the edges involving i ) at time t − t . To analyze the unweighted temporal networks, we instead used the Jaccard index J i ( t ) betweenthese successive neighborhoods (see Methods for precise definitions). Values of these quantities close or equal to 1suggest that the node has not changed neighbors in successive time windows: hence its neighborhood shows low liquidity (elsewhere, it would be said that the node shows high “loyalty” ). On the contrary, values close or equalto 0 mean that the neuron has completely changed neighbors between subsequent times: its neighborhood is highlyliquid. At each time t , the set of cosine similarity values Θ i ( t ) , i ∈ [ N ] and the Jaccard index values J i ( t ) , i ∈ [ N ] (for the unweighted case) form the time-dependent feature vectors Θ ( t ) and J ( t ) , each of dimension N (Figure 1.D).In order to answer the second question and probe for the presence of specific network architectures, weconsidered the core-periphery organization of the graph. This way of characterizing the information sharing networksnapshots was suggested to us by the visual inspection of their spatial embeddings, some of which are representedin Figure 2. We thus computed the coreness coefficient C i ( t ) of each node i in each snapshot t , using the definitionof coreness introduced by (see Methods ). In a static, unweighted, undirected network, the coreness C i of node i ∈ [ N ] is a real number between 0 and 1, interpreted as follows: when C i ∼ core of thenetwork, i.e., a set of tightly connected nodes; when C i (cid:39) periphery , i.e., s only loosely linked to the rest of the network; if C i = k i =
0. We thus obtain a time dependent vector C ( t ) of dimension N by computing at each time t the coreness C i ( t ) for each node i ∈ [ N ] in the network of time-window t . The computation of coreness coefficientcan also be performed for weighted networks , possibly yielding, however, values larger than 1. Therefore, wenormalize the whole time series of vectors of weighted coreness coefficients by the maximum observed value inorder to obtain a time series of vectors { C w ( t ) | t ∈ [ T ] } (with C w ( t ) = { C iw ( t ) , i ∈ [ N ] } ) with the same range ofvalues for the unweighted and weighted coreness features (Figure 1.D). In order to investigate the core-periphery organization of the information sharing networks, we looked at thedistributions of the instantaneous coreness values C i ( t ) over all neurons and time-frames. A weighted corenessdistribution from a representative recording is shown in Figure 3.A (see Supplementary Figure S2 for equivalentunweighted coreness analyses). Figure 3.B moreover displays instantaneous distributions of the weighted coreness,for the same recording and for several time frames. We found that, within each time frame, a majority of neuronshad low coreness values, i.e., they were peripheral nodes in the instantaneous sharing network (red color in thecartoons of Figure 3.A), while fewer neurons had high coreness values. Interestingly, in most recordings there wasnot a sharp separation between core and periphery. On the contrary, we generically observed the presence of asmooth distribution spanning all possible coreness values. The transition between core and periphery was thussmooth, without gaps but with neurons displaying gradually less tight links with the core, without yet being fullyperipheral. We note that such smooth distributions are actually encountered in many systems , a strict distinctionbetween a very central core and a very loose periphery being only a schematic idealized vision and real networksdisplaying typically hierarchies of scales and local connectivities.The analyses of Figure 3.A and 3.B indicate that at any time-frame the sharing network has a soft core-peripheryarchitecture, but does not inform us about how individual neurons evolve in time within this architecture. In order tofollow dynamic changes in the coreness of individual neurons, we studied the time-evolution of this feature for eachneuron of each recording. In Figure 3.C we plot the coreness C iw ( t ) vs time, for each node i ∈ [ N ] of a representativerecording. The two highlighted lines in the figure represent the coreness evolution of two particular nodes. In lightgreen, we show the instantaneous coreness of the node with maximum average coreness (cid:104) C iw ( t ) (cid:105) T (averaged overthe recording length T ). The figure shows clearly that this neuron’s instantaneous coreness is always large: thecorresponding neuron is persistently part of the network’s core throughout the whole recording. This contrasts withthe purple line, which displays the instantaneous coreness of the neuron with largest coreness standard deviation( σ ( (cid:104) C iw ( t ) (cid:105) T ) ): the curve fluctuates from high to low coreness values, indicating that the corresponding neuronswitches several times between central core positions in the network and more peripheral ones. The contrastbetween these two behaviors is highlighted in the cartoon at the bottom of Figure 3.C.The continuous range of observed instantaneous coreness values and the fluctuations in individual corenessvalues indicate that the set of most central neurons changes in time. We thus examined whether some regionswere contributing more than others to this core. To this aim, we define the core, at each time-frame, as the set ofneurons whose instantaneous coreness lies above the 95th percentile of the distribution (in histograms such as thosein Figure 3.B). In Figure 3.D we then plot the core filling factors of the CA1 and mEC layers (top and center plots,respectively). We define the core filling factor of each region as the percentage of the overall number of neurons ofthe recording located in that region that belong to the core. We plot the time-evolution core filling factors separatelyfor neurons located in different hippocampal CA1 layers (light and dark blue lines, top panel) and for neurons indifferent medial entorhinal cortex (mEC) layers (red, orange and yellow lines, center panel). The figure illustratesthet the core-filling factors vary substantially along time. In the example shown here (corresponding to the samerecording as in Figure 3.C), the core-filling factor of CA1 Stratum Pyramidale (SP) neurons belonging to the coreincreases from ∼
2% to near 7% during the recording.The results of Figure 3.D indicate that the core is not restricted to neurons belonging to a specific region, but isgenerally composed of both neurons belonging to EC and neurons belonging to CA1. We remind indeed that ournetworks are networks of functional connectivity and do not have to reflect necessarily the underlying anatomicalconnectivity (for which it would be unlikely that our recordings pick up mono-synaptically connected cells betweendifferent regions). However, the participation of CA1 and EC neurons to the core is changing through time and, as aresult, the core is sometimes “more on the EC side” or “more on the CA1 side” (see lower cartoons in Figure 3.D).To visualize the relative fractions of core neurons belonging to the two different regions, we computed and show inthe bottom panel of Figure 3.D the normalized core filling regional fractions : the fraction of core nodes belongingeither to EC (orange) or CA1 (blue) – the sum of these two fractions adding up to 1. The orange and blue bands hange thickness along time, reflecting in this recording a progressive shift from a low to a higher involvement ofCA1 neurons in the core. This variation of the multi-regional core composition may reflect changes in the way thedifferent regions control information integrative processes (see
Discussion ). As previously mentioned and summarized in Figure 1.C, for each recording and each time window t , we computedfor each neuron i ∈ [ N ] several temporal network properties, tracking notably the “liquidity” of its neighbor-hood (Jaccard index and cosine similarity) and its position within the core-periphery architecture (weighted andunweighted instantaneous coreness values). To investigate how these properties change dynamically at the globalnetwork level, we computed for each of these four quantities the correlation between their values at differenttimes, obtaining four correlation matrices of size T × T . For instance, the element ( t , t (cid:48) ) of the unweighted liquiditycorrelation matrix is given by the Pearson correlation between the N values of the Jaccard coefficient computed at t { J i ( t ) , i ∈ [ N ] } and the N values computed at t (cid:48) { J i ( t (cid:48) ) , i ∈ [ N ] } (see Methods for definition). In Figure 4.A, weshow these four correlation matrices for a representative recording, two for the unweighted features (above, Jaccardindex and unweighted coreness), and two for the weighted ones (below, cosine similarity and weighted coreness).The block-wise structure of these correlation matrices suggests the existence of epochs in time where neurons’feature values are strongly correlated (red blocks on the diagonal). In the case of Figure 4.A, mostly diagonal blocksare observed, with low correlation values outside the blocks, meaning that the network configurations are similarduring each epoch but very different in different epochs. In other cases, we sometimes observe as well off-diagonalblocks, indicating that the network might return to a configuration close to one previously observed (we show anexample of this behavior in Supplementary Figure S3, as well as an example in which only one epoch is observed).Each block on the diagonal (epoch in which the node properties are strongly correlated) can be interpreted as anetwork connectivity configuration associated to specific liquidity and coreness assignments of the various neurons.We call network states these configurations.To quantitatively extract such discrete network states, we use the time-series of the feature vectors Θ ( t ) , J ( t ) , C w ( t ) and C ( t ) . We concatenate these vectors two by two at each time, obtaining two 2 N -dimensional featurevectors: the first one contains at each time the values of the unweighted liquidity and coreness of all nodes( { J ( t ) , C ( t ) } ), and the second one contains the corresponding weighted values ( { Θ ( t ) , C w ( t ) } ). We then perform ineach case (weighted and unweighted) an unsupervised clustering of these T N -dimensional feature vectors. As aresult of this clustering procedure, as shown in Figure 4.A, we obtain a sequence of states (temporal clusters of thefeature vectors) that the network finds itself in at different times (yellow state spectrum for the unweighted case,red for the weighted case). We also observe that these states are not a mere artificial construction. Our procedurewould segment the temporal network into states even if discretely separated clusters did not exist, but we explicitlychecked that, for all recordings but two, clustering was meaningful: Supplementary Figure S4.A shows indeed thatclustering quality was in a large majority of cases well above chance level (see Methods ).We compared the network states spectra found for the weighted and unweighted case by computing the mutualinformation between the two sequences of states for each recording, normalized by the largest entropy amongthe entropies of the two distinct sequences. Such relative mutual information is bounded in the unit interval andquantifies the fraction of information that a state sequence carries about the other (reaching the unit value when thetwo state sequences are identical, and being zero if the two sequences are statistically independent). We computethis quantity for each recording and display the distribution of values obtained as a light green boxplot on theleft of Figure 4.B. This boxplot shows that the mutual information values between the weighted and unweightednetwork states sequences of a recording are concentrated around a median approaching 0.8. Therefore, the spectraof network states extracted by the weighted and unweighted analyses are generally matching well, indicating therobustness of their extraction procedure. Most importantly, the high degree of matching between weighted andunweighted analyses confirms that network state changes correspond to actual connectivity re-organizations (asrevealed by unweighted analyses) and not just to weight modulations on top of a fixed connectivity.As previously discussed, the system undergoes switching between two possible global brain states duringthe anesthesia recordings: these global states are associated to different oscillatory patterns, dominated by eitherTheta (THE) or slow (SO) oscillations. We studied therefore what is the relation occurring between changes in thenetwork state and global oscillatory state switching. To this aim, we computed the normalized mutual informationbetween network state sequences (weighted or unweighted) and global state sequences. The distributions of thevalues obtained are shown as boxplots on the right of Figure 4.B, for both weighted (red) and unweighted (yellow)network states sequences. In both cases, we detect positive, although low, values of the relative mutual informationwith global oscillatory states, with a median value close to ∼ lobal oscillatory state and network state switching exists but that oscillatory state switching does not well explainnetwork state switching. A very simple reason for this poor correlation is that, while there are just two main globaloscillatory states (see however Discussion ), the number of network states is not a priori limited. In fact, the statisticsof the number of network states in the different recordings, shown in Figure 4.C, indicates that in many recordingswe could extract at least three network states and sometimes up to seven. Therefore, network state switching canoccur within each oscillatory global state.Nevertheless, it is possible that each given network state would tend to occur mostly within one specific globaloscillatory state. To check whether this is the case, we computed for each network state the fraction of times that thisstate occurred during THE or SO epochs. We show in Figure 4.D the histograms of these time fractions, measuredover the set of all network states. The light blue histogram corresponds to the fractions of time a network statemanifested itself during the THE state (the dark blue histogram gives the same information but for the SO state).Both histograms are markedly bimodal, indicating that a majority of states occur a large fraction of times duringeither the THE or the SO states, but not in both. In other words, network states are to a large degree oscillatorystate specific. Therefore, the global oscillatory states do not fully determine the observed coreness and liquidityconfigurations (there may be several network states for each of the oscillatory states) but most network states can beobserved only during one specific global oscillatory state and not during the other.
To refine our analysis, we now investigate and characterize the temporal network properties at the level of singleneurons within each of the detected network states. In order to do so, we computed, for each node and in each state,a set of dynamical features averaged over all time frames assigned to the specifically considered state. We focus hereon the weighted features, since the weighted and unweighted analysis provide similar results. The state-specific connectivity profile of a given neuron i in a given state included first: • its state-averaged weighted coreness value; • its state-averaged cosine similarity value.Note that analogous time-resolved features were already used for network state extraction, but that we considerhere state-averaged values. We also computed four additional network state specific features, defined as follows foreach node i ∈ [ N ] in a state h spanning the set of times T h (see explanatory cartoons in Figure 5.A): • the state-averaged strength (cid:104) s i ( t ) (cid:105) t ∈ T h ≡ (cid:104) ∑ j w ij ( t ) (cid:105) t ∈ T h , hence the state-averaged total instantaneous weightof the connections of i ; • the activation number n ia : it is the number of times that the strength of node i changes from 0 to a non-zerovalue within the state, hence it gives the number of time that i changes its connectivity from being isolated tobeing connected to at least one another neuron; • the total connectivity time τ i : it is the number of time frames within T h in which i is connected to at least oneanother neuron; • the Fano factor Φ i ≡ σ ( (cid:104) ∆ t i (cid:105) h ) (cid:104) ∆ t i (cid:105) h : each of the n ia periods in which i is connected to at least another neuron has acertain duration ∆ t i (the sum of these durations is τ i ), and Φ i , the ratio of the variance and the average of thedifferent connectivity durations of i over the state h , quantifies whether these durations are of similar value orvery diverse.Each neuron’s temporal properties within a given state are thus summarized by the values of these overall sixfeatures, which define the neuron’s state-specific connectivity profile as a six-dimensional vector. Normalizing allthe features to have values between 0 and 1, connectivity profiles can be visually represented as radar plots (Figure5.B) in which the value of each feature is plotted on the corresponding radial axis.After computing the connectivity profile of each node, in each network state and in each recording, we performedan unsupervised clustering (using K-means clustering) over all these six-dimensional connectivity profiles in orderto identify categories of these profiles, which we call connectivity styles . With this approach we uncovered theexistence of four general connectivity styles that a neuron can manifest within a network state: • Core style (or “streamers”) , a class of nodes of high average coreness, average strength, average cosine-similarity,total connectivity time and low activation number n a and Fano factor Φ : overall, a class of rather central eurons with numerous stable connections that are persistently connected within a state (blue representativepolygone in Figure 5.B). A behavior similar to that of a speaker of an assembly, continuously conveyinginformation to the same, and many, people: a “streamer” of information; • Peripheral style (or “callers”) , nodes with high activation number and total connectivity time but low strength,Fano factor and coreness: a class of peripheral nodes that are periodically connected in numerous events ofsimilar connectivity durations and low weights within a state, whose connections are not completely liquidnor completely stable (red representative polygone in Figure 5.B). A behavior similar to a customer or guestregularly making short calls to trusted core members to be updated on the latest news: aka, a “caller”. • Bursty and
Regular core-skin style (or “free-lancer helpers” and “staff helpers”) , two classes of connectivity profilesboth characterized by nodes with intermediate coreness and strength and high values of cosine similarityand total connectivity time, whose difference lies in the values of the Fano factor Φ : the former (yellowrepresentative polygone in Figure 5.B) displaying high values of Φ can therefore be interpreted as a class ofnodes that have stable connections, that are active for a long time yet with highly varying connectivity times;the latter (purple representative polygone in Figure 5.B) is characterized instead by low values of the Fanofactor, hence the connectivity durations of these nodes do not fluctuate much. The behaviors of these neuronscan be assimilated to the one of external experts assisting core staff in a company, either with regular workschedules (the regular core-skin neurons could then be seen as “staff helpers” ) or sporadically and irregularly(the bursty core-sking neurons could then be seen as “free-lancer helpers” ).Note that we also identified (and subsequently discarded) an additional “Junk” cluster with relatively fewerelements (9.7% of the total number of connectivity profiles computed for all neurons in all recordings) and smallvalues of all features. We thus removed these cases, as usual in unsupervised clustering applications , to betterdiscriminate the remaining “interesting” classes listed above.Overall each connectivity profile (one for each neuron in each possible network state in the associated recording)was categorized as belonging to one of the above connectivity styles, according to the output cluster label assignedby the unsupervised clustering algorithm. However, a substantial diversity of connectivity profiles subsists withineach of the clusters. We therefore considered as well a soft classification scheme, which quantifies the degree ofrelation of each individual connectivity profile with the tendencies identified by each of the different connectivitystyle clusters. Concretely, we trained a machine learning classifier to receive as input a connectivity profile andpredict the connectivity style assigned to it by this unsupervised clustering. In this way, after training, the classifierassigned to each connectivity profile a four-dimensional vector whose elements represented the probabilities ofbelonging to any one of the four possible connectivity styles (see Methods for details). In Figure 5.C, each connectivityprofile is represented as a dot with as coordinates the soft classification labels produced by the classifier, i.e., theprobabilities that each given connectivity profile belongs to the periphery, core-skin (summing the probabilities forthe bursty and regular subtypes) or core connectivity styles. This plot reveals, on the one hand, the existence of a gapbetween core (blue) and periphery (red) connectivity profiles: in other words, connectivity profiles that are likelyto be classified as of the “streamer” type are very unlikely to be classified as being of the “caller” type, stressingthe radical difference between these two connectivity styles. On the other hand, both the core and peripheryconnectivity styles display some mixing with the core-skin style, as made clear by the almost continuous paths ofconnectivity profiles from the core (blue) to the core-skin (yellow) and from the core-skin to the periphery (red).This means that there is a continuum spectrum of connectivity profiles interpolating between “streamers” and“helpers” on one side and “helpers” and “callers” on the other. The polygones shown in Figure 5.B are on thecontrary archetypal (in the sense introduced by ). These archetype profiles manifest in an extreme manner thetendencies inherent to their connectivity style. This is reflected by the fact that they lie at the vertices of the boundedsoft membership space represented in Figure 5.C. They display thus strong similarity to just one connectivity style,which they epitomize even better than the centroids of the associated connectivity style cluster, having near zerochance of being misclassified (cluster centroids are shown for comparison in Supplementary Figure S4.B).We finally checked whether the different connectivity styles were adopted more or less frequently by neuronsin specific anatomical locations or of specific types (excitatory or inhibitory). In Figure 5.D, we plot the numberof connectivity profiles assigned to each style (colors as in Figure 5.B), separating them by anatomical layer andbrain region. However, since we recorded unequal number of cells in the different layers (see SupplementaryFigure S1), we also accounted for the different numbers of cells and recordings per layer and estimated chance-levelexpectations for the connectivity style counts in each layer: this allowed us to detect significant over- or under-representations of certain styles at different locations. The numbers of “streamers” (core), “staff helpers” (regularcore-skin) and “callers” (periphery) profiles were compatible with chance levels at all the recorded locations. We nly detected over-representations (green upward triangles) of “free-lancer helpers” (bursty core-skin) in StratumRadiatum (SR) of CA1 and Layer II of medial Enthorinal Cortex, and an under-representation (red downwardtriangle) of this style in Layer III of medial Enthorinal Cortex (see Discussion for possible interpretations). Thesemoderate deviations from chance levels suggest that the connectivity styles adopted by different neurons (and thustheir centrality in the core-periphery architecture of information sharing networks) are only poorly affected by theiranatomical location in the hippocampal formation circuit, in apparent contrast with the widespread belief that the“hubness” of neurons should be strongly determined by structural and developmental factors .We found however a stronger inter-relation between cell type and connectivity styles (Figure 5.D, bottom). Westill found representatives of any of the connectivity styles among both excitatory and inhibitory neurons. Howeverwe found that the fraction of inhibitory (resp., excitatory) neurons among the core neurons was significantly above(resp., below) chance level. Conversely, the fraction of inhibitory (resp., excitatory) neurons among the peripheralneurons was significantly below (resp., above) chance level (see Discussion ). We have computed connectivity profiles per neuron and per network state, in order to enable the detection of apossible network-state dependency of the temporal properties of the neurons connectivity. However, the state-specificity of this computation does not prevent a priori a neuron to always assume the same connectivity profileacross all possible network states to which it participates. It is thus an open question, whether connectivity profilesare only node-dependent (for a given neuron, the same in every state) or, more generally, state-dependent (for agiven neuron, possibly different across different network states).To answer this question, we checked whether network state transitions are associated or not to connectivity stylemodifications at the level of individual neurons. We found that changes in the connectivity style of a neuron upon achange of state are the norm rather than the exception. We computed for every neuron the index η , quantifying thediversity of connectivity styles that a neuron assumes across the different network state transitions occurring duringa recording. Such an index is bounded in the range 0 ≤ η ≤ Methods ). The distribution of the observed values ofthis η index is shown in Figure 6.A. This distribution is bimodal, with a first peak occurring around η ∼ η ∼ η ).Figure 6.B is a graph-representation of the transition matrix between connectivity styles, computed over all theobserved connectivity style transitions. We plot here a weighted, directed graph, in which the two connectivitystyles “regular” and “bursty core-skin” have been merged for simplicity into a single category. The width of eachcolored edge corresponds to the value of the transition rate from the class of the same color. The transition rate isdefined as the probability that a node characterized by one of the three connectivity styles in one network-state,switches to one of the other two connectivity styles in the successive network-state. Consistently with Figure 5.C,we find that edges of large weight connect the core-skin to the periphery and to the core in both directions: hightransition rates are found between these styles. The edges connecting core and periphery are sensibly smaller. Wecan thus conclude that it is largely more likely that neurons of a core connectivity profile switch to a core-skin profilein the following network state than to a peripheral profile, and vice versa: direct transitions between core profiles andperiphery profiles are not very likely to occur, although they are not impossible. More complete transition graphsand tables (including as well the core-skin class separation into “bursty” and “regular”, and the “junk” classes) areshown in Figure S5. Individual neurons can thus float through the core-periphery architecture, descending fromcore towards periphery and ascending back into the core, via crossing the core-skin styles.The overall behavior of neurons switching styles between states is illustrated in Figures 6.C and 6.D for a specificrecording. The former is a matrix whose element ( i , s ) has the color of the connectivity style exhibited by node i in state s . In this plot we highlight the row corresponding to the state-wise evolution of a specific selected node,whose successive connectivity profiles in the successive network state ares shown in Figure 6.D. In our recordings we observed a diversity of firing rates between different single units. While the median firing ratewas close to ∼ ariations of temporal connectivity features compiled in the state-specific connectivity profiles of different neuronscould simply be explained by these firing rate variations, we constructed scatterplots of state-specific coreness,liquidity, strength, number of activations, Fano factors and total time of activation for the different neurons againsttheir firing rate, averaged over the corresponding state. These scatter plots are shown in Supplementary FigureS4.C.The degree of correlation between firing rate and connectivity features was at best mild. Cells with larger thanmedian firing rate tended to have slightly less liquid neighborhoods on average, but the range of variation wasbroad and largely overlapping with the one of liquidity values for lower firing rate neurons. Other features, suchas total time of activation, displayed an even weaker dependence on firing rate, while some others, such as theconnectivity number, tended to be inversely correlated with firing, but, once again, in a rather weak manner (thecorrelation coefficient is ρ = − We focused in this study on the dynamics of links of information sharing. Information sharing can be seen as ageneralized measure of cross-correlation between the firing activity of two different single units (see
Methods ). In ,we analyzed as well active information storage , a complementary measure, which can be seen as a generalized auto-correlation of the firing activity of a single node (or, equivalently, to a self-loop of information sharing connectivity).While information sharing quantifies the amount of information in common between the present activity of a neuronand the past activity of another, information storage is meant to capture the amount of information in the presentactivity of a neuron which was already conveyed by its past activity (see Methods ). In this sense, neurons with a largevalue of information storage would functionally act as “memory buffers” repeating in time the same informationcontent (cf.
Discussion ). In we found that information storage, as information sharing, is state-dependent andthat some cells, that we called therein “storage hubs” display particularly large values of storage. We checked herewhether high storage cells manifest some preferential connectivity style.We show in Figure S6, distributions of information storage values, separated according to the connectivitystyle assumed by different neurons in each different network state. We found that high storage is not associatedexclusively to specific styles of connectivity. On the contrary, cells with high storage could be found for any of theconnectivity styles. The overall larger storage values where found for core streamers cells. However, we could findhigh storage cells even among the core-skin helpers and the periphery callers. Interestingly, distributions of storagefor core-skin and periphery styles were bimodal, including a peak at high-storage values (see Discussion for possibleinterpretations).
We have here described the internal organization of assemblies of neurons dynamically exchanging informationthrough time, using a temporal network framework and developing adequate tools for their analysis. In linewith our previous study , we found a coexistence of organization, such as the existence of discrete networkstates, and freedom, such as the liquid continuous reorganization of network neighborhoods within each state.The rich information sharing dynamics we revealed in our anesthesia recordings in hippocampus and enthorinalcortex cannot be explained merely in terms of transitions between global brain states (here, alternations between“REM-sleep”-like THE epochs and SO epochs). On the contrary, we found that each of the global states givesaccess to wide repertoires of possible networks of information sharing between neurons. These network statesrepertoires are largely global state-specific and are robustly identified using both weighted and unweighted networkcharacterizations.We extracted here functional connectivity networks evaluating information sharing —i.e., time-lagged mutualinformation— between the firing of pairs of single units. This choice was motivated by the fact that mutualinformation is relatively simpler to evaluate than other more sophisticated and explicitly directed measures ofinformation transfer , which require larger amounts of data to be properly estimated. Using a simpler measurewas thus better compliant with our need to estimate connectivity within short windows, to give rise to a temporalnetwork description. Although mutual information is a symmetric measure, the introduction of a time-lag makesour metric “pseudo-directed” because information cannot causally propagate from the future to the past. Therefore,sharing of information between the past activity of a node i and the present activity of another node j could beindicative of a flow of information from i to j . When computing time-lagged mutual information between alldirected pairs of single units in our recording, we found however only a very mild degree of asymmetry. As shown n Figure S5, most strong information sharing connections were very close to symmetric, indicative of time-laggedmutual information peaking near zero-lag for these connections. This finding corresponds to the early intuitionthat members of a cell assembly share information via their tightly synchronized firing . Figure S5 also showsthat some of the more weakly connected pairs of neurons displayed a varying degree of directional asymmetry,with the strongest unbalance found for cells with a core connectivity style. However, the strengths of nodes withunbalanced pseudo-directed sharing were orders of magnitude smaller than for balanced nodes. Our choice toignore asymmetries appears thus well justified, as well as the choice to use lagged mutual information rather thanmore complex directed metrics.A striking architectural property of the measured information sharing networks was their core-peripheryarchitecture. Such architecture was preserved at every time-frame (as shown in Figure 3.B), although the corenessof individual neurons changed smoothly through time even within states. This preservation of a global functionalarchitecture despite the variation of the specific realizations and participant neurons is reminiscent of “functionalhomeostasis” observed in highly heterogeneous circuits that must nevertheless perform in a stable and efficientmanner a crucial function . What is important is that, at any time, a functional separation between “streamers”,“helpers” or “callers” exist, however which neurons specifically assume these roles and the exact location of theseneurons within the anatomical circuit (cf. Figure 5.D) appear to be less important.Core-periphery organizations have been found in many social, infrastructure, communication and informationprocessing networks, with various types of coreness profiles, i.e., strong or gradual separations between theinnermost core and the most peripheral nodes . The dynamics of this type of structures has however barelybeen tackled in temporal networks . In neuroscience, such architecture has been identified in larger-scale networksof inter-regional functional connectivity, during both task and rest , in line with the general idea that cognitionrequires the formation of integrated coalitions of regions, merging information streams first processed by segregatedsub-systems . Here, we find that a similar architecture is prominent at the completely different scale of networksbetween single units within local micro-circuits in the hippocampal formation. The core-periphery architecture ofinformation sharing thus reflects dynamic information integration and segregation. Cells belonging to the coreform an integrated ensemble, within which strong flows of information are continually streamed and echoed. Atthe opposite end of the hierarchy of connectivity styles, peripheral neurons are segregated and perform transient“calls” toward “streamer” neurons in the core to get updated and share their specific information contents.Following , a “syntax of information processing” would be enforced by the spatio-temporal organization ofneuronal firing. Our dynamic core-periphery architecture is also ultimately determined by the coordinated firingpatterns of many neurons. The fact that these coordinated firing patterns invariantly translate into networks witha core-periphery architecture can be seen as a proxy for the existence of a syntactic organization of informationflows. In this self-organized syntax, peripheral neurons are in the ideal functional position to act as “readers” of thecontents streamed by the integrated cell assembly formed by core neurons. Transitions between network states mayact as the equivalent of bar lines, parsing the flow of information into “words” or longer sentence blocks, spelledwith an irregular tempo. In the vision of , roles such as the one of reader or chunking into informational wordswould be a by-product of neuronal firing organization. We stress thus here once again that even in our case the real“stuff” out of which sharing functional networks are made are spiking patterns at the level of the assembly. Thenetwork representation provides however a natural and intuitive visualization of the dynamics of these patterns.Coordinated firing events —and not single neuron firing properties, cf. the poor correlations in Figure S4— aremapped into network structures and changes of firing pattern into changes of connectivity style within the network.The logical syntax of assembly firing is thus translated into a dynamic topological architecture of sharing networks,which is easier to study and characterize.In order to interpret the functional role of this emergent core-periphery organization, one should elucidate theinformation processing roles played by individual neurons within this network organization. An entire spectrum ofpossible roles may exist, mirroring the smooth separation subsisting between core and periphery. Communicationbetween core and periphery can be helped by core-skin neurons, which sporadically connect to the core, transientlyexpanding its size. Speculatively, such breathing of the integrated core may be linked to fluctuating needs in termsof information processing bandwidth, analogous to dynamic memory allocation in artificial computing systems.The possible algorithmic role of neurons with different connectivity styles could be investigated using metricsfrom the partial information decomposition framework . Beyond pseudo-directed sharing, it may be possibleto identify: neurons acting as active memory buffers displaying large information storage ; neurons activelytransfering information to others, associated to large transfer entropy ; or, yet, neurons combining received inputsinto new original representations, associated to positive information modification values . As previously said, wecould at least quantify information storage. In particular, as shown in Figure S6, sub-groups of core-skin helpers nd periphery callers existed with high storage values, approaching the ones of storage hubs within the core.Therefore, at least some of the core-skin or periphery cells transiently connected to the core may play the role ofancillary memory units, “flash drives” sporadically plugged-in when needed to write or read specific informationsnippets. Unfortunately, the number of coordinated firing events observed in our recordings was not sufficientfor a reasonable estimation of transfer or modification, beyond storage. We can nevertheless hypothesize thatcore neurons are the work-horses of information modification, shaping novel informational constructs via theirintegrated firing.We found that the bursty core-skin style —the “freelancer helper” role— was over-represented in SR of CA1 andLayer 2 of EC. SR receives inputs from hippocampus CA3 which are believed to be linked to retrieval of previouslyencoded associations . CA3, an associative memory module crucial for retrieval, receives inputs on its turn fromlayer 2 of EC. The over-representation of the bursty core-skin style in SR and layer 3 of EC may thus be compliantwith our intuition of “helpers” as additional on-demand storage resources, plugged to the core when needed to readthe specific contents buffered by their spiking. On the contrary, layer 3 of EC, which sends inputs to SLM of CA1,would rather mediate encoding than retrieval . Correspondingly, in layer 3 of EC the over-representation offreelancer helpers observed in layer 2 is replaced by an under-representation, consistently with the complementaryfunctions that inputs from these EC layers are postulated to play.We also found an excess of inhibitory interneurons among core style cells (Figure 5.D). Inteneurons tended toexhibit larger firing rates than excitatory cells, but we showed that coreness is not significantly affected by firingrate (Figure S4.C). Therefore, this excess is rather to be linked to a key functional role of interneurons in mediatingcell assembly formation. Their central position within the core of the information sharing network may allow themto efficiently control the recruitment of new neurons into the integrated core and orchestrate their coordinatedfiring, as already discussed in the literature .Remarkably, however, in a majority of cases, the connectivity style adopted by a neuron was only poorly affectedby its anatomical localization or by its cell type. Apart from the few exceptions just discussed above, the distributionof the different styles through the different anatomical regions and layers was indeed close to chance levels. Inparticular, the fact of being localized within a specific layer of CA1 or EC did not affect in a significant way theprobability of belonging to the core or to the periphery. Furthermore, a majority of neurons switched betweendifferent styles when network state changed (Figure 6), indicating that connectivity styles and, in particular, coremembership are not hardwired. In contrast, it is often thought that the function that a neuron plays is affectedby its individual firing and morphology properties, as well as by details of its synaptic connections within thecircuit . In this dominating view, functional hubness would thus be the garland reserved to a few elite cells,selected because of their extreme technical specialization, largely determined by their developmental lineage .Here —further elaborating on — we propose a more “democratic” view in which core membership and, morein general, connectivity style would be dynamically appointed, such that the total number of neurons that areelected into the core at least once is much larger than the currently active core members at specific times. Thecore composition can indeed be radically reorganized when the network state switch and can fluctuate betweenalternative majorities of hippocampal or enthorinal cortex neurons (Figure 3.D).Such a democratic system implies a primacy of collective dynamics at the neuronal population level, flexiblyshaping coordinated firing ensembles, on technical specialization and “blood” origins at the single neuron level.Switching network states may reflect transitions between alternative attractors of firing dynamics implementingalternative firing correlations. Discrete state switching coexist however with more liquid fluctations of coreattachment, which would rather suggest a complex but not random dynamics at the edge of instability . Statetransitions and flexible core-periphery reorganization are also poorly determined by global oscillations, despitethe important role played by oscillations in information routing . Even if global oscillatory modes do not“freeze” information sharing patterns they nevertheless affect them, with different oscillatory states giving rise toalternative repertoires of possible information sharing networks (Figure 4.D). Computational modelling of spikingneural circuits may help in the future to reach a mechanistic understanding of how ongoing collective oscillationsinteract with discrete attractor switching, metastable transients and plasticity to give rise to liquid core-peripheryarchitectures of information sharing.We have started here applying a temporal network language to describe the internal life of cell assembliesalong their emergence, expansion and contraction and sudden transformations. For methodological convenience— possibility to use long time windows for network estimation— we focused however on anesthesia, which is acondition in which intrinsic information processing is not suppressed but less functionally relevant than duringbehavior. Our method could however be extended in perspective to recordings in pathological conditions — e.g.,epilepsy, in which intrinsic assembly dynamics is altered —, relating temporal network properties alterations to he degree of cognitive deficits or even, in perspective, during actual tasks. To cope with the much faster behavioraltime-scales, information sharing networks could be estimated first in a state-resolved manner, by pooling togetherfiring events based on the similarity of the conditions in which they occur — e.g., transient phase relations andsynchrony levels or co-activation patterns — rather than strict temporal contiguity. State-specific networkframes could then be reallocated to specific times, depending on which “state” the system is visiting at differenttimes, reconstructing thus an effective temporal network with the same time-resolution as the original recordings(see e.g. for an analogous approach used at the macro-scale of fMRI signals). In this way it would become possibleto link temporal network reconfiguration events to actual behavior, probing hence their direct functional relevance. We used in this work a portion of the data (12 of 18 experiments) initially published by , which includes local fieldpotentials (LFPs) and single-unit recordings obtained from the dorsomedial Enthorinal Cortex of anesthetized rats.Seven additional simultaneous recordings in both mEC and dorsal HPC under anesthesia were included in thisstudy, previously analyzed by . Details of anatomical locations and numbers of cells included can be found inSupplementary Figure S1.All experiments were in accordance with experimental guidelines approved by the Rutgers University (Insti-tutional Animal Care and Use Committee) and Aix-Marseille University Animal Care and Use Committee. Weperformed experiments on 12 maleSprague-Dawley rats (250 to 400 g; Hilltop Laboratory Animals) and 7 maleWistar Han IGS rats (250 to 400 g; Charles Rivers Laboratories). We performed acute experiments anesthetizingthese rats with urethane (1.5 g/kg, intraperitoneally) and ketamine/xylazine (20 and 2 mg/kg, intramuscularly),with additional doses of ketamine/xylazine (2 and0.2 mg/kg) being supplemented during the electrophysiologicalrecordings to avoid recovery from anesthesia. The body temperature was monitored and kept constant withaheating pad. The head was secured in a stereotaxic frame (Kopf) and the skull was exposed and cleaned. Twominiature stainless steel screws, driven into the skull, served as ground and reference electrodes. To reach the mEC,we performed one craniotomy from bregma: − + − + Extracellular signal recorded from the silicon probes was amplified (1000 x ), bandpass-filtered (1 Hz to 5 kHz), andacquired continuously at 20 kHz with a 64-channel DataMax System (RC Electronics ora 258-channel Amplipex) orat 32 kHz with a 64-channel DigitalLynx (NeuraLynx at 16-bit resolution). We preprocessed raw data using a custom-developed suite of programs . After recording, the signalswere downsampled to 1250 Hz for the LFP analysis. Spikesorting wasperformed automatically using KLUSTAKWIK (http://klustakwik.sourceforge.net ]), followed bymanual adjustment of the clusters, with the help of autocorrelogram, cross-correlogram (CCG), and spike waveformsimilarity matrix (KLUSTERS software package; https://klusta.readthedocs.io/en/latest/ ) Afterspike sorting, we plotted the spike features of units as a function of time and discarded the units with signs ofsignificant drift over the period of recording. Moreover, we included in the analyses only units with clear refractoryperiods and well-defined clusters.Recording sessions were divided into brain states of THE and SO periods. The epochs of stable theta (THE)or slow oscillations (SO) were visually selected from the ratios of the whitened power in the THE band ([3 6] Hz n anesthesia) and the power of the neighboring bands of EC layer 3 LFP, which was a layer present in all the 18anesthesia recordings.We determined the layer assignment of the neurons from the approximate location of their somata relative tothe recording sites (with the largest amplitude unit corresponding to the putative locationof the soma), the knowndistances between the recording sites, and the histological reconstruction of the recording electrode tracks. Toassess the excitatory or inhibitory nature of each of the units, we calculated pairwise crosscorrelograms betweenspike trains of these cells. We determined the statistical significance of putative inhibition or excitation (troughor peak in the [+ ] ms interval, respectively) using well-established nonparametric test and criteria used foridentifying monosynaptic excitations or inhibitions , in which each spike of each neuron was jittered randomlyand independently on a uniform interval of [ − ] ms a thousand times to form 1000 surrogate datasets and fromwhich the global maximum and minimum bands at 99% acceptance levels were constructed. We estimated time-resolved information sharing networks following the same procedures as . In particular,we used time lagged Mutual Information between spike trains as a measure of the amount of shared information .Information sharing between each pair of neurons ( i , j ) i , j = N is defined as the time lagged Mutual Information MI [ i ( t ) , j ( t − λ )] between the firing patterns extracted for the relative neurons in a 10 seconds long time-window: I shared ( j −→ i ) = ∑ λ MI [ i ( t ) , j ( t − λ )] (1) I shared ( i −→ j ) = ∑ λ MI [ j ( t ) , i ( t − λ )] (2)where the time-lag λ varies in the range 0 ≤ λ ≤ T θ , T θ being the phase of the THE cycle (THE oscillationshave higher frequency than slow oscillations). The information sharing network is defined by an adjacency matrix A ( i , j ) i , j = N , whose element A ( i , j ) corresponds to I shared ( i −→ j ) . It is therefore a weighted, directed network. Thetime-window is then shifted by 1 second to extract the information sharing network at the next time-step, assuringa 90% overlap with the previous time-window for the computation of the information theoretic measures.In order to study the effect of asymmetries in the adjacency matrices of our information sharing networks wecompute the asymmetry coefficient ∆ S % for each neuron in each state h . defined as follows: ∆ S % ih = s i , hout − s i , hin s i , hout + s i , hin , ∀ i =
1, . . . , N . (3) ∆ S % ih thus represents the ratio between the amount of information that neuron i outputs to other neuronsduring state h ( s i , hout = ∑ t = T h ∑ j w ij ) and the amount of information that is conveyed to neuron i by its neighborsduring state h ( s i , hin = ∑ t = T h ∑ j w ji ). ∆ S % ih = i in state h , s i , hin =
0, therefore it is a perfect sender of information, whereas if ∆ S % ih = − s i , hout = receiver . If ∆ S % ih = s i , hout = s i , hin and therefore the amount of information that neuron i conveys to its neighbors during the state h isequal to the amount of information that it receives. As shown in Supplementary Figure S7, the asymmetries in theadjacency matrices of the information sharing networks have a negligible effect. We thus neglect the aforementionedasymmetries and consider the networks to be undirected. Information sharing can be seen as a generalized form of cross-correlation where the time-lagged mutual informationcapture all types of linear and nonlinear correlations. Analogously, one can evaluate a generalized auto-correlationfunctional, known under the name of active information storage . Following , we evaluated active informationstorage of a given neuron, within a given time-window as: I storage ( i ) = ∑ λ MI [ i ( t ) , i ( t − λ )] (4)where the time-lag λ varies once again in the range 0 ≤ λ ≤ T θ . The values of storage whose distributions areshown in Figure S6 are averages for each neuron over all the time-windows whose associated connectivity profilemuch a specific connectivity style (see later for definitions of connectivity profiles and styles). .5 Network liquidity To quantify how much, or how little, the connections of a the node’s neighborhood change from a time-step tothe next, we compute the cosine similarity (for weighted networks) and the Jaccard index (for unweighted ones)between the successive neighborhoods of the node. Both measures have been intensively used to study the stabilityor instability of connections in temporal networks . The cosine similarity between a node i ’s neighborhoodsat times t − t is defined as follows: Θ i ( t ) ≡ ∑ j w ij ( t − ) w ij ( t ) (cid:113) ∑ j w ij ( t − ) (cid:113) ∑ j w ij ( t ) (5)where w ij ( t ) is the weight of the link between nodes i and j at time t .The local Jaccard index for node i at time t (Figure 1.C, on the left) is defined as: J i ( t ) ≡ | ν i ( t − ) ∩ ν i ( t ) || ν i ( t − ) ∪ ν i ( t ) | , J i ( t ) ∈ [
0, 1 ] (6)where ν i ( t − ) and ν i ( t ) are the neighborhoods of node i at times t − t , respectively. J i ( t ) = ν i ( t − ) and ν i ( t ) are two disjoint sets of nodes, while J i ( t ) = J i ( t ) ∼ J i ( t ) ∼ t the outcome of the computation are two vectors of dimension N : Θ ( t ) = ( Θ ( t ) , . . . , Θ N ( t )) and J ( t ) = ( J ( t ) , . . . , J N ( t )) . We consider the definition of coreness C i of node i in a network introduced by . To compute the coreness of neuronsat each time frame we used the implementation of the method available at https://core-periphery-detection-in-networks.readthedocs.io/en/latest/index.html .When computed for a static, unweighted, undirected network, the coreness C i of node i ∈ [ N ] is a real numberbetween 0 and 1, interpreted as follows: when C i ∼ core of the network; when C i (cid:39) periphery ; if C i = t ∈ [ T ] we compute the coreness for each node i ∈ [ N ] for both the unweighted and weighted networks: at eachtime step we therefore obtain two N -dimensional feature vectors, one for the weighted network { C iw ( t ) ; i = · · · , N } and one for the unweighted one { C i ( t ) ; i = · · · , N } . The two pairs of feature vectors computed at each time t , two vectors for the unweighted case and two for theweighted one, are merged into two 2 × N -dimensional vectors, one containing the unweighted features (Jaccardand unweighted coreness) and the other formed by the weighted ones (cosine similarity and weighted coreness).For each recording we have therefore two series of length T ( T varies for different recordings), of weighted andunweighted feature vectors. In order to investigate the existence of recurring epochs of stable (correlated) liquidityand topology configuration of the network (the network states ) we perform a K -means clustering on the featurevector time series. As a result of the clustering we obtain for each recording two sets (one for the weighted analysis,one for the unweighted one) of network states of specific liquidity (Jaccard index, for unweighted networks, orcosine similarity, for weighted networks) and core-periphery organization (unweighted or unweighted coreness) ofthe system (Figure 4.A). As shown in Figure 5.B, we represent the connectivity behaviour of a neuron in a network state with a radarplot whose radial axis correspond to the six features defined in section Results, which are averages of temporalproperties over the network state. We therefore compute one six-dimensional connectivity profile for each neuron, ineach network state, for each recording. We investigate the existence of typical connectivity profiles across differentrecordings: this corresponds to searching for common radar plot shapes for neurons of different network statesof different recordings. We therefore perform a K-means clustering on all the connectivity profiles computed for ach neuron, in each network state, in each recording: we obtain four cluster of connectivity profiles, that we name connectivity styles . These connectivity styles are shown in Figure 5.B and interpreted in the Results and Discussionsections.
We use the K-nearest-neighbor unsupervised classification method, trained on the connectivity style labels of allthe connectivity profiles, to obtain a soft classification of each neuron’s connectivity profile. Each neuron’s state-averaged connectivity profile’s soft label is represented by a 4 − dimensional normalized vector: each component ofthis vector represents the probability of the connectivity profile to belong to the corresponding connectivity style(core, periphery, bursty core-skin, regular core-skin). The connectivity styles shown in Figure 5.B correspond to theconnectivity profiles of maximum probability to belong to the connectivity style of the corresponding colour, i.e.of maximum soft label component of the corresponding connectivity style. The soft classification of connectivityprofiles is graphically represented in Figure 5.C by plotting the connectivity profiles’ vectors of connectivity stylesoft labels in a 3 − D space. In order to achieve a clearer visual representation, the probabilities of belonging to theregular core-skin or the bursty core-skin connectivity styles have been summed into a single component of the softlabel vector. The cross-network-state connectivity profile transition rate η i of neuron i is defined as: η i ≡ pro f ile transitions o f node imax ( global classes , network − states ) . (7)It quantifies the tendency of neuron i to switch, often or rarely, between connectivity profiles belonging to differentconnectivity styles in different states: when η i =
1, neuron i switches connectivity style at every change of networkstate; when η i =
0, neuron i is always in the same connectivity style. Acknowledgments
This project has received funding from the European Union’s Horizon 2020 research and innovation programmeunder the Marie Skłodowska-Curie grant agreement No. 713750. Also, it has been carried out with the financialsupport of the Regional Council of Provence- Alpes-Côte d’Azur and with the financial support of the A*MIDEX(ANR-11-IDEX-0001-02), funded by the Investissements d’Avenir project funded by the French Government,managed by the French National Research Agency (ANR). CB and WC were supported by the M-GATE project(European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grantagreement no. 765549). PQ acknowledges support from FRM, FFRE, and CURE Epilepsy Taking Flight Award. DBand AB acknowledge support by the CNRS "Mission pour l’Interdisciplinarité" INFINITI program (BrainTime).
Author contributions
Project was formulated by DB and AB, based on data and experiments by PQ and CB and previous informationtheoretical analyses by WC. NP, AB and DB formalised the study quantitative approaches and NP implementedand ran all the novel temporal network analyses. All authors wrote the manuscript.
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Recordings and feature vectors. A)
Approximate recording locations in mEC (orange) and CA1 (blue)during anesthesia. B) An example of LFP signal recorded by the channels in CA1 and in mEC; below the LFPs aretwo examples of single unit activity from the same recording. The intervals t a , t b and t c are examples of timewindows in which the corresponding network time-frames are constructed. The horizontal bar above therecordings represents the Global Oscillations states: in light blue the Slow Oscillations and in darker blue the
ThetaOscillations . C) Illustrative sketches of the two features computed for each node at each time-frame: on the left, inthe green box, we illustrate the
Jaccard index , which quantifies the overlap between two sets. Here we consider thelocal Jaccard indices between the neighborhoods of each node i ∈ [ N ] in successive time frames (e.g., J i ( t a ) is theJaccard index for node i between times t a − t a ): these quantities carries information on how little or howradically the neighborhoods of i changes across time and we call them liquidity . The center and right panels are aschematic representation of the core-periphery organization of the network and the intuitive explanation of the coreness C values of the nodes: core (blue) nodes have high values of coreness C ∼
1, whereas peripheral (red) nodeshave low values, C ∼ core-skin , i.e., nodes whose coreness values arehigher than 0 and lower than 1. When a node is disconnected from the rest of the network (when it has noneighbours), its coreness is exactly zero. D) At each timeframe, we obtain feature vectors of the coreness andliquidity values of all neurons i ∈ [ N ] , schematized by the two colourful vectors (yellow, blue and red forcoreness and green for liquidity). The analysis were performed both for unweighted networks, in which links areeither present or absent, and for weighted networks, in which the links are characterized by a weight given by theamount of shared information between the nodes. For weighted networks, liquidity is measured by the cosinesimilarity instead of the Jaccard index (see Methods ), and a weighted coreness is computed for each node. igure 2.
Temporal network visualization.
In this figure we represent the evolution of the information sharingtemporal network computed for one of the recordings, using the visualization toolbox TACOMA( http://tacoma.benmaier.org/about.html ). The six networks on the top plots represent six snapshots ofthe network’s evolution at the corresponding times: each black dot represents a neuron whose activity has beenrecorded at the corresponding time. The bottom panel displays the temporal network’s edge activity plot : each rowof the plot represents the activity of one edge of the network, i.e., a black dot on a row means that the correspondingedge on the vertical axis is active (a non-zero element of the network’s adjacency matrix) at the corresponding timeon the horizontal axis. The edges are ordered on the vertical axis by increasing value of their first activation time . igure 3. Dynamic Core-Periphery structure. A)
Top: Cartoon representing the core-periphery organization of thenetwork with core nodes colored in blue and peripheral nodes in red; Bottom: histogram of the values of theinstantaneous weighted coreness C i ( t ) of each neuron at each time of a recording: the most frequent values of C iw ( t ) are close to 0 (red dot), hence coreness values typical of peripheral nodes, whereas coreness values of core nodes, C i ( t ) ∼ B) Histograms of instantaneous weighted coreness values of the neurons in severalindividual time-frames (every 60 seconds). The different histograms show that the picture of a majority of neuronswith low coreness value and few neurons with high coreness holds at all times. The core periphery organization ofthe information sharing network is thus persistent in time. C) Plot of the temporal evolution of the weightedcoreness C iw ( t ) of each node i in a specific recording: the red line highlights the evolution of the coreness of thenode with highest average coreness (cid:104) C w ( t ) (cid:105) max (time average over the whole recording); the orange linecorresponds to the node whose coreness fluctuates the most (node of maximum variance of C iw ( t ) ). We show belowthis plot a cartoon of the core-periphery organization of the network, where we illustrate how the former nodesteadily belongs to the core, while the latter node evolves several times from the core to the periphery and back. D) Plots of the core-filling factor (top and center panels) and of the core-filling fraction (bottom) of the different layers of hippocampus (HPC) and medial entorhinal cortex (mEC). The core-filling factor of each layer is the percentage ofnodes of each layer that are located in the core (nodes above the 95 th percentile of coreness values). In the bottompanel, the green line represents the separation between the fraction of core nodes located either in the mEC layers(orange) or in the HPC (blue): the core filling fraction . The cartoon below the core-filling fraction plot illustrates howthe neurons that belong to the core at different times can be located in different anatomical structures. igure 4. Liquidity and coreness network-states. A)
For each recording, the feature vectors of liquidity andcoreness described in Figure 1D are combined at each time, obtaining time-dependent vectors of dimension 2 N : { J ( t ) , C ( t ) } in the unweighted case, { Θ ( t ) , C w ( t ) } for the weighted case. K-means clustering of these vectors yieldsfor each recording a sequence of network states visited by the network (centre panels). We show on both sides of thenetwork state sequences the correlation matrices (C.M.) of the various feature vectors (unweighted coreness andJaccard, weighted coreness and cosine similarity): each element ( t , t ∗ ) of a matrix is the Pearson correlationcoefficient between the corresponding feature vectors computed at times t and t ∗ . For instance, the cosine similarityC.M. has as element ( t , t ∗ ) the Pearson correlation between the list of values of the cosine similarity (weightedliquidity) of nodes at times t and t ∗ , { Θ i ( t ) , i =
1, . . . , N } and { Θ i ( t ∗ ) , i =
1, . . . N } . B) Left panel: Distribution of thevalues of the mutual information between the weighted and unweighted network states spectra of all recordings:the mutual information between the two spectra is high, showing that the two spectra are highly correlated for allrecordings. Right panel: distribution of the values of mutual information between the weighted network statesspectrum and the global oscillating states spectrum (red boxplot) for all recordings (yellow boxplot: same for theunweighted network states spectra). C) Statistics of the number of network states in the different recordings. Inmany recordings this number is larger than 2, and it can reach values as large as 7. D) Histograms showing thepercentage of states having probability p to occur during a specific Global Oscillations state , slow (blue) or theta (lightblue). Both histograms are bi-modal, hence the majority of states are strongly global-state -specific; the blue histogramshows an abundance of states ( ∼ slow global oscillations. igure 5. Connectivity profiles. A)
Cartoons illustrating the features: the strength s i ( t ) of a node i at time t measures the global importance of node i ’s connections (the cartoon shows a comparison of low vs high values of s i ( t ) ); its liquidity, hence the cosine similarity (Jaccard index in the unweighted case) between successiveneighborhoods (the cartoon illustrates a change between the neighborhood of i at successive times); its coreness; theconnectivity number n ia and total connectivity time τ i of node i in a state: τ i represents the number of time framesin which neuron i is connected to at least one other neuron, and n ia is the number of times that i switches from beingdisconnected to being connected to at least one other neuron; the Fano factor quantifies the fluctuations of thedurations of the n ia periods in which i is connected. B) The connectivity profile of each neuron i in a state h is a6-dimensional vector whose components are its state-averaged features: average coreness (cid:104) C iw (cid:105) h , average liquidity (cid:104) Θ i (cid:105) h , average strength (cid:104) s i (cid:105) h , connectivity number n ia , total connectivity time τ i , Fano factor Φ i . Clustering of allthe connectivity profiles yields 4 connectivity styles that we represent here as radar plots. We identify four mainstyles: core (blue), periphery (red), bursty core-skin (yellow) and regular core-skin (magenta), with differentdistinctive values of the different features composing the connectivity profile. Shown here are radar plots for theconnectivity profile of archetypal cells in each of the connectivity styles (see Results for details). C)
3D plot of theconnectivity styles space: the three axis are the probabilities of the connectivity profile of a node in a network stateto belong to the periphery, core-skin or core, respectively. D) Histograms of the layer location (top plot) of theneurons exhibiting each connectivity style (identified by the same color as panel B), and histograms of cell type(inhibitory and excitatory, bottom plot) populations per connectivity style. A green upward arrow means that theconnectivity style of the corresponding color is statistically over-represented in the corresponding layer or cell type,whereas a red downward arrow means that the connectivity style is under-represented in that layer or cell type. igure 6.
Network-state specificity of connectivity profiles. A)
Histogram of the connectivity-style-switchingprobability η of nodes, in the weighted (red) and unweighted (yellow) cases, computed for each neuron in eachrecording. η is defined (see Methods ) as the ratio between the number of transitions between different connectivitystyles in subsequent states of a neuron during the whole recording and the maximum value between the totalnumber of connectivity styles and number of network states observed in the recording. B) Visualization as aweighted, directed graph of the cross-network-state connectivity style transition matrix. The color of an edgecorresponds to the color of the node from which it emanates. The width of an edge from connectivity style A toconnectivity style B is proportional to the probability of a neuron to switch from a connectivity profile ofconnectivity style A in a state to a profile of connectivity style B in the next state. C) Matrix representation of thechanges in connectivity styles of the neurons when the network changes state. Each element ( s , n ) of the matrix iscoloured according to the connectivity style to which node n ’s connectivity profile belongs in state s . D) Connectivity profile transitions of node n =
28, whose evolution is characterized by transitions between profilesbelonging to different styles, highlighting how a given node can display radically different dynamical behavioursin different states.
Supporting Information igure S1. A) Simultaneous mEC/CA1 recording setup. B) Number of neurons recorded for each layer of eachregion: a majority of recorded neurons were located in the medial Entorhinal Cortex layers. igure S2. A) Histogram of the instantaneous unweighted coreness of all neurons in all time-steps for the samerecording as in Figure 3 (unweighted analogous of Figure 3.A). B) Unweighted core-filling factors and core-fillingfractions of the same recording as in Figure 3. C) Histograms of instantaneous unweighted coreness values for 9different time-steps of the network’s evolution. D) Density plots of the values of instantaneous weighted (red) andunweighted (yellow) coreness of all neurons at all times, separately for each recording. igure S3.
Recordings with different types of feature vectors correlation matrices and network states spectra. Ineach case, we show as in Figure 4.A. the temporal sequence of network states extracted by the unsupervisedclustering of feature vectors, with on both sides the correlation matrices. Top plots correspond to unweightedfeatures, bottom plots to weighted features. A) Case with diagonal blocks in the correlation matrices, with nooff-diagonal blocks. The sequence of network states is in agreement with this structure, i.e. the network visits eachnetwork state only once. B) Case with chess-board-like correlation matrices. As seen from the network-state-spectrathis recording is indeed in periodic oscillation between two states. C) Case in which no state can be clearlyidentified: this recording can be interpreted as in an extremely liquid single state, where both the core-peripheryorganization of the network and the neighborhoods of neurons change continuously with no clear temporalstructure. igure S4. A) Silhouette plot: the red stars give for each recording the silhouette value for the Kmeans clusteringperformed on the connectivity profiles to retrieve the connectivity styles. These values are compared todistributions (boxplots) obtained for each recording by the following null model. We first reshuffle randomly thenetwork-state labels of the time-frames of the whole recording while conserving the total length of eachnetwork-state. We then compute the connectivity profiles on the randomized states, cluster them in order toretrieve the connectivity styles, and compute the new silhouette. The boxplots correspond to the distribution ofthese null model silhouette values, obtained for 200 realizations of the reshuffling. The real silhouette values arewell above the randomized distributions, suggesting that the definition of the discrete global states in the evolutionof the information sharing liquidity and core-periphery organization is crucial for the analysis on the connectivityprofiles and styles. B) Maximal connectivity profile of each connectivity style (lower opacity), as shown in Figure5.C and connectivity profile of the centroid of the Kmeans clustering result (higher opacity). We stress that thecentroid of the connectivity style does not correspond to the connectivity profile of any specific a neuron, butrepresents the average connectivity profile of the corresponding connectivity style. C) Scatterplots between eachone of the 6 features computed for each neuron in each network-state of each recording and the average firing rateof the same neuron in the same network state. There is no evident relation between the average firing of a neuron ina network state and the network properties that we computed, showing that the various connectivity styles are notsimply related to the neurons’ activity. igure S5. A) Transition matrix T ij between connectivity styles of a neuron in successive network states taking intoaccount the core, periphery, core-skin and junk connectivity styles, as well as network states in which the neuronsare not connected to the rest of the network. B) Transition matrix T ij between connectivity styles of a neuron insuccessive network states, taking into account the core, periphery, core-skin regular, core-skin bursty and junkconnectivity styles, as well as the disconnected state of neurons. For both matrices, each diagonal element T ( i , i ) represents the persistency rate of the corresponding connectivity style, i.e., the probability of a neuron to exhibit thesame connectivity style in two successive global network states. The non-diagonal matrix elements are normalizedon each row so that ∑ j (cid:54) = i T ( i , j ) = igure S6. Density plots of the logarithm of the network-state-aggregated storage of connectivity profiles of eachof the five (junk included) connectivity styles. igure S7. A) Distribution of values of ∆ S % for the connectivity profiles of each connectivity style. All thedistributions of are peaked around 0, with the extreme values ( + −
1) reached only for core connectivityprofiles. The corresponding population of perfect senders and receivers represents a small portion of the overallnumber of connectivity profiles. B) Scatterplot of the values of S tot vs. ∆ S %, the former representing the totalaggregate strength of a neuron during a state: S i , htot = s i , hin + s i , hout . The scatterplot shows that perfect sender/receiverneurons ( ∆ S % = ±
1) correspond to very low values of S tot , hence they correspond to very weakly connectedneurons., hence they correspond to very weakly connectedneurons.