Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function

Michael W. Reimann; Max Nolte; Martina Scolamiero; Katharine Turner; Rodrigo Perin; Giuseppe Chindemi; Paweł Dłotko; Ran Levi; Kathryn Hess; Henry Markram

doi:10.3389/fncom.2017.00048

Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function

Reimann, Michael W.; Nolte, Max; Scolamiero, Martina; Turner, Katharine; Perin, Rodrigo; Chindemi, Giuseppe; Dłotko, Paweł; Levi, Ran; Hess, Kathryn; Markram, Henry 2017-06-12 00:00:00 ORIGINAL RESEARCH published: 12 June 2017 doi: 10.3389/fncom.2017.00048 Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function 1 † 1 † 2 2 Michael W. Reimann , Max Nolte , Martina Scolamiero , Katharine Turner , 3 1 4‡ 5‡ 2 ‡ Rodrigo Perin , Giuseppe Chindemi , Paweł Dłotko , Ran Levi , Kathryn Hess * and 1, 3 ‡ Henry Markram * 1 2 Blue Brain Project, École Polytechnique Fédérale de Lausanne, Geneva, Switzerland, Laboratory for Topology and Neuroscience, Brain Mind Institute, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, Laboratory of Neural Microcircuitry, Brain Mind Institute, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, DataShape, INRIA Saclay, Palaiseau, France, Institute of Mathematics, University of Aberdeen, Aberdeen, United Kingdom The lack of a formal link between neural network structure and its emergent function has hampered our understanding of how the brain processes information. We have now come closer to describing such a link by taking the direction of synaptic transmission into account, constructing graphs of a network that reﬂect the direction of information ﬂow, and analyzing these directed graphs using algebraic topology. Applying this approach to a local network of neurons in the neocortex revealed a remarkably intricate and previously Edited by: unseen topology of synaptic connectivity. The synaptic network contains an abundance Paul Miller, Brandeis University, United States of cliques of neurons bound into cavities that guide the emergence of correlated activity. Reviewed by: In response to stimuli, correlated activity binds synaptically connected neurons into Cees van Leeuwen, functional cliques and cavities that evolve in a stereotypical sequence toward peak KU Leuven, Belgium complexity. We propose that the brain processes stimuli by forming increasingly complex Andreas Knoblauch, Hochschule Albstadt-Sigmaringen, functional cliques and cavities. Germany Keywords: connectomics, topology, directed networks, structure-function, correlations, Betti numbers *Correspondence: Henry Markram henry.markram@epﬂ.ch 1. INTRODUCTION Kathryn Hess kathryn.hess@epﬂ.ch † How the structure of a network determines its function is not well understood. For neural networks These authors have contributed speciﬁcally, we lack a unifying mathematical framework to unambiguously describe the emergent equally to this work. behavior of the network in terms of its underlying structure (Bassett and Sporns, 2017). While Co-senior author. graph theory has been used to analyze network topology with some success (Bullmore and Sporns, 2009), current methods are usually constrained to analyzing how local connectivity inﬂuences local Received: 11 March 2017 Accepted: 18 May 2017 activity (Pajevic and Plenz, 2012; Chambers and MacLean, 2016) or global network dynamics (Hu Published: 12 June 2017 et al., 2014), or how global network properties like connectivity and balance of excitatory and Citation: inhibitory neurons inﬂuence network dynamics (Renart et al., 2010; Rosenbaum et al., 2017). One Reimann MW, Nolte M, Scolamiero M, such global network property is small-worldness. While it has been shown that small-worldness Turner K, Perin R, Chindemi G, optimizes information exchange (Latora and Marchiori, 2001), and that adaptive rewiring during Dłotko P, Levi R, Hess K and chaotic activity leads to small world networks (Gong and Leeuwen, 2004), the degree of small- Markram H (2017) Cliques of Neurons worldness cannot describe most local network properties, such as the diﬀerent roles of individual Bound into Cavities Provide a Missing neurons. Link between Structure and Function. Algebraic topology (Munkres, 1984) oﬀers the unique advantage of providing methods to Front. Comput. Neurosci. 11:48. doi: 10.3389/fncom.2017.00048 describe quantitatively both local network properties and the global network properties that Frontiers in Computational Neuroscience | www.frontiersin.org 1 June 2017 | Volume 11 | Article 48 Reimann et al. Cliques, Cavities, Structure, and Function emerge from local structure, thus unifying both levels. More reconstruction provide a statistical and biological range of recently, algebraic topology has been applied to functional microcircuits for analysis. networks between brain regions using fMRI (Petri et al., 2014) We found a remarkably high number and variety of high- and between neurons using neural activity (Giusti et al., 2015), dimensional directed cliques and cavities, which had not been but the underlying synaptic connections (structural network) seen before in neural networks, either biological or artiﬁcial, were unknown. Furthermore, all formal topological analyses have and in far greater numbers than those found in various null overlooked the direction of information ﬂow, since they analyzed models of directed networks. Topological metrics reﬂecting the only undirected graphs. number of directed cliques and cavities not only distinguished We developed a mathematical framework to analyze both the the reconstructions from all null models, they also revealed subtle structural and the functional topology of the network, integrating diﬀerences between reconstructions based on biological datasets local and global descriptions, enabling us to establish a clear from diﬀerent animals, suggesting that individual variations in relationship between them. We represent a network as a directed biological detail of neocortical microcircuits are reﬂected in the graph, with neurons as the vertices and the synaptic connections repertoire of directed cliques and cavities. When we simulated directed from pre- to postsynaptic neurons as the edges, which microcircuit activity in response to sensory stimuli, we observed can be analyzed using elementary tools from algebraic topology that pairwise correlations in neuronal activity increased with the (Munkres, 1984). The structural graph contains all synaptic number and dimension of the directed cliques to which a pair connections, while a functional graph is a sub-graph of the of neurons belongs, indicating that the hierarchical structure of structural graph containing only those connections that are active the network shapes a hierarchy of correlated activity. In fact, we within a speciﬁc time bin (i.e., in which a postsynaptic neuron found a hierarchy of correlated activity between neurons even ﬁres within a short time of a presynaptic spike). The response to within a single directed clique. During activity, many more high- a stimulus can then be represented and studied as a time series of dimensional directed cliques formed than would be expected functional graphs. from the number of active connections, further suggesting that Networks are often analyzed in terms of groups of nodes correlated activity tends to bind neurons into high-dimensional that are all-to-all connected, known as cliques. The number of active cliques. neurons in a clique determines its size, or more formally, its Following a spatio-temporal stimulus to the network, we dimension. In directed graphs it is natural to consider directed found that during correlated activity, active cliques form cliques, which are cliques containing a single source neuron and a increasingly high-dimensional cavities (i.e., cavities formed by single sink neuron and reﬂecting a speciﬁc motif of connectivity increasingly larger cliques). Moreover, we discovered that while diﬀerent spatio-temporal stimuli applied to the same circuit (Song et al., 2005; Perin et al., 2011), wherein the ﬂow of information through a group of neurons has an unambiguous and the same stimulus applied to diﬀerent circuits produced direction. The manner in which directed cliques bind together diﬀerent activity patterns, they all exhibited the same general can be represented geometrically. When directed cliques bind evolution, where functional relationships among increasingly appropriately by sharing neurons, and without forming a larger higher-dimensional cliques form and then disintegrate. clique due to missing connections, they form cavities (“holes,” “voids”) in this geometric representation, with high-dimensional cavities forming when high-dimensional (large) cliques bind 2. RESULTS together. Directed cliques describe the ﬂow of information in the network at the local level, while cavities provide a global measure 2.1. The Case for Directed Simplices of information ﬂow in the whole network. Using these naturally Networks of neurons connected by electrical synapses (gap arising structures, we established a direct relationship between junctions) can be represented as undirected graphs, where the structural graph and the emergent ﬂow of information in information can ﬂow in both directions. Networks with response to stimuli, as captured through time series of functional chemical synapses, which impose a single direction of synaptic graphs. communication from the pre- to the postsynaptic neuron We applied this framework to digital reconstructions of rat (Figures 1B2,B3), are more accurately represented as directed neocortical microcircuitry that closely resemble the biological graphs. Sub-sampling networks of neurons experimentally has tissue in terms of the numbers, types, and densities of neurons revealed small motifs of synaptic connectivity, but not large and their synaptic connectivity (a “microconnectome” model cliques of neurons (Song et al., 2005; Perin et al., 2011). Knowing for a cortical microcircuit, Figures 1A,B; see Markram et al., the complete directed network of neurons, as we do in the case 2015; Reimann et al., 2015). Simulations of the reconstructed of the reconstructed microcircuit, enables us to detect all cliques, microcircuitry reproduce multiple emergent electrical behaviors directed, and otherwise (Figure 1). found experimentally in the neocortex (Markram et al., When the direction of connections is not taken into account, 2015). The microcircuit, formed by ∼8 million connections a great deal of information is lost. For example, in the undirected (edges) between ∼31,000 neurons (vertices), was reconstructed case, there is only one possible conﬁguration for a clique of from experimental data, guided by biological principles of four fully connected neurons (Figure 2A1, left). However, in the organization, and iteratively reﬁned until validated against directed case, there are 3 = 729 possible conﬁgurations, as each a battery of independent anatomical and physiological data of the six connections can be in one of three states (i→ j, j← i, obtained from experiments. Multiple instantiations of the or i↔ j connection types; Figure 2A1 right). Frontiers in Computational Neuroscience | www.frontiersin.org 2 June 2017 | Volume 11 | Article 48 Reimann et al. Cliques, Cavities, Structure, and Function FIGURE 1 | (A) Thin (10 μm) slice of in silico reconstructed tissue. Red: A clique formed by ﬁve pyramidal cells in layer 5. (B1) Full connection matrix of a reconstructed microcircuit with 31,146 neurons. Neurons are sorted by cortical layer and morphological type within each layer. Pre-/postsynaptic neurons along the vertical/horizontal axis. Each grayscale pixel indicates the connections between two groups of 62 neurons each, ranging from white (no connections) to black (≥8% connected pairs). (B2) Zoom into the connectivity between two groups of 434 neurons each in layer 5, i.e., 7 by 7 pixels in (A), followed by a further zoom into the clique of 5 neurons shown in (A). Black indicates presence, and white absence of a connection. (B3) Zoom into the somata of the clique in (A) and representation of their connectivity as a directed graph. A clique with reciprocal connections contains two or Directed cliques have the highest net directionality among all more cliques consisting only of uni-directional connections cliques (Figure S1; Section 4.1.4). A clique that contains cycles (Figure 2A2). When only uni-directional connections are always decomposes into directed cliques with the same number considered, there are 2 possible conﬁgurations of four fully of neurons or fewer, at the very least any single connection connected neurons, which are of two types: those that contain between two neurons forms a 2-clique. A cyclical clique of three cycles (40 conﬁgurations; Figure 2A3 left; Section 4.1.3) and neurons therefore decomposes into three 2-cliques. Following the those that do not (24 conﬁgurations; Figure 2A3 right). Directed conventions in algebraic topology, we refer to directed cliques of cliques are exactly the acyclic cliques. The net directionality n neurons as directed simplices of dimension n-1 or directed (n-1)- of information ﬂow through any motif can be deﬁned as the simplices (which reﬂects their natural geometric representation sum over all neurons of the squares of the diﬀerences between as (n-1)-dimensional polyhedra) (see Figure S2; Section 4.1.3). their in-degree and their out-degree (see Equation 2, Figure S1). Correspondingly, their sub-cliques are called sub-simplices. Frontiers in Computational Neuroscience | www.frontiersin.org 3 June 2017 | Volume 11 | Article 48 Reimann et al. Cliques, Cavities, Structure, and Function FIGURE 2 | (A1) A 4-clique in the undirected connectivity graph has one of 729 conﬁgurations in the directed graph. (A2) Conﬁgurations containing bidirectional connections are resolved by considering all sub-graphs without bidirectional connections. (A3) Without bidirectional connections, 64 possible conﬁgurations remain, 24 of which are acyclic, with a clear sink-source structure (directed simplices, in this case of dimension 3). (B) Number of simplices in each dimension in the Bio-M reconstruction (shaded area: standard deviation of seven statistical instantiations) and in three types of random control networks. (C) Examples of neurons forming high-dimensional simplices in the reconstruction. Bottom: Their representation as directed graphs. (D) (Left) Number of directed simplices of various dimensions found in 55 in vitro patch-clamp experiments sampling groups of pyramidal cells in layer 5. (Right) Number of simplices of various dimensions found in 100,000 in silico experiments mimicking the patch-clamp procedure of (B). 2.2. An Abundance of Directed Simplices many as 80 million directed 3-simplices (Figure 2B; blue). This is 2.2.1. Reconstructed Neocortical Microcircuitry the ﬁrst indication of the existence of such a vast number of high- dimensional directed simplices in neocortical microcircuitry, or We analyzed 42 variants of the reconstructed microconnectome, grouped into six sets, each comprised of seven statistically in any neural network. varying instantiations (Markram et al., 2015; Section 4.3). The ﬁrst ﬁve sets were based on speciﬁc heights of the six layers 2.2.2. Control Models of the neocortex, cell densities, and distributions of diﬀerent To compare these results with null models, we examined how the cell types experimentally measured in ﬁve diﬀerent rats (Bio1- numbers of directed simplices in these reconstructions diﬀered from those of artiﬁcial circuits and from circuits in which some 5), while the sixth represents the mean of these measurements (Bio-M). Individual instantiations within a set varied with the of the biological rules of connectivity were omitted (see Section outcome of the stochastic portions of the reconstruction process. 4.4). For one control, we generated ﬁve Erdo˝s-Rényi random Surprisingly, we found that the reconstructions consistently graphs (ER) of equal size (∼31,000 vertices) and the same average contained directed simplices of dimensions up to 6 or 7, with as connection probability as the Bio-M circuit (∼0.8%; ∼8 million Frontiers in Computational Neuroscience | www.frontiersin.org 4 June 2017 | Volume 11 | Article 48 Reimann et al. Cliques, Cavities, Structure, and Function edges) (Figure 2B; dark green). For another, we constructed a neurons, only by inhibitory neurons, and only in individual circuit with the same 3D model neurons as the Bio-M circuit, but layers by both excitatory and inhibitory neurons. Restricting to connected the neurons using a random connectivity rule [“Peters’ only excitatory neurons barely reduces the number of simplices Rule” (Peters and Feldman, 1976), PR; Figure 2B, red]. For the in each dimension (Figure 3A1), while simplex counts in last control we connected the neurons in the Bio-M circuit inhibitory sub-graphs are multiple orders of magnitude smaller according to the distance-dependent connection probabilities (Figure 3A2), consistent with the fact that most neurons in the between the diﬀerent morphological types of neurons. Since microcircuitry are excitatory. Analyzing the sub-graphs of the this control is similar to deriving connectivity from the average layers in isolation shows that layers 5 and 6, where most of overlap of neuronal arbors (Shepherd et al., 2005), it retains the excitatory neurons reside (Markram et al., 2015), contain the general biological (GB) features of connectivity between the most simplices and the largest number of high-dimensional diﬀerent types of neurons (Reimann et al., 2015), excluding simplices (Figure 3A3). only explicit pairwise connectivity between individual neurons, The large number of simplices relative to the number of which is determined by the overlap of their speciﬁc arbors neurons in the microcircuit implies that each neuron belongs (Figure 2B, yellow). In all cases, the number of directed simplices to many directed simplices. Indeed, when we counted the of dimensions larger than 1 was far smaller than in the Bio-M number of simplices to which each neuron belongs across circuit. In addition, the relative diﬀerences between the Bio-M dimensions, we observed a long-tailed distribution such that and the null models increased markedly with dimension. a neuron belongs on average to thousands of simplices (Figure 3B). Both the mean maximal dimension and the number 2.2.3. In vitro of simplices a neuron belongs to are highest in the deeper Simplices of high dimensions (such as those depicted in cortical layers (Figure 3C). Neurons in layer 5 belong to the Figure 2C) have not yet been observed experimentally, as largest number of simplices, many spanning multiple layers doing so would require simultaneous intracellular recording (Figure 3D), consistent with the abundance of neurons with of large numbers of neurons. To obtain an indication of the the largest morphologies, which are connected to all layers. On presence of many high-dimensional directed simplices in the the other hand, layer 6 has the largest number of simplices actual neocortical tissue, we performed multi-neuron patch- that are fully contained in the layer (Figure 3A3), consistent clamp experiments with up to 12 neurons at a time in in vitro with the fact that layer 6 contains the most neurons. While the slices of the neocortex of the same age and brain region as the number of simplices that can form in the microcircuitry depends digitally reconstructed tissue (Section 4.5.1). Although limited by essentially on the number of neurons, the number of simplices the number of neurons we could simultaneously record from, we to which a single neuron belongs depends fundamentally found a substantial number of directed simplices up to dimension on its number of incoming and outgoing connections (its 3, and even one 4-dimensional simplex, in just 55 multi-neuron degree), which in turn depends on its morphological size recording experiments (Figure 2D, left). We then mimicked (Figure 3E). these experiments on the reconstructed microcircuit by repeating the same multi-neuron patch-clamp recordings in silico (Section 4.5.2) and found a similar shape of the distribution of 4-, 3-, and 2.3. Topology Organizes Spike Correlations 2-simplices, though in lower frequencies than in the actual tissue The presence of vast numbers of directed cliques across a (Figure 2D, right). These ﬁndings not only conﬁrm that high- range of dimensions in the neocortex, far more than in dimensional directed simplices are prevalent in the neocortical null models, demonstrates that connectivity between these tissue, they also suggest that the degree of organization in the neurons is highly organized into fundamental building blocks neocortex is even greater than that in the reconstruction, which of increasing complexity. Since the structural topology of the is already highly signiﬁcant (see Section 3). neural network takes into account the direction of information ﬂow, we hypothesized that emergent electrical activity of the 2.2.4. C. elegans microcircuitry mirrors its hierarchical structural organization. To test whether the presence of large numbers of high- To test this hypothesis, we simulated the electrical activity of dimensional directed simplices is a general phenomenon of the microcircuit under in vivo-like conditions (Markram et al., neural networks rather than a speciﬁc phenomenon found in this 2015). part of the brain of this particular animal and at this particular Stimuli, conﬁgured as nine diﬀerent spatio-temporal input age, we computed the numbers of directed simplices in the patterns (Figure 4A), were injected into the reconstructed C. elegans connectome (Varshney et al., 2011) (Section 4.6). microcircuit through virtual thalamo-cortical ﬁbers in which Again, we found many more high-dimensional simplices than spike trains were induced using patterns recorded in vivo (Bale expected from a random circuit with the same number of neurons et al., 2015; Figure S4; Section 4.7). These stimuli diﬀered (Figure S3). primarily in the degree of synchronous input received by the neurons. As expected, the neurons in the microcircuit responded 2.2.5. Simplicial Architecture of Neocortical to the inputs with various spiking patterns (Figures 4B1,B2,B4). Microcircuitry We then calculated for each connected pair of neurons the To understand the simplicial architecture of the microcircuit, we correlation of their spiking activity (Figure 4C) and found a began by analyzing the sub-graphs formed only by excitatory broad distribution of correlation coeﬃcients, with only∼12% of Frontiers in Computational Neuroscience | www.frontiersin.org 5 June 2017 | Volume 11 | Article 48 Reimann et al. Cliques, Cavities, Structure, and Function FIGURE 3 | (A1) Number of simplices in each dimension in the excitatory subgraph (shaded area: standard deviation across seven instantiations). (A2) Same, for the inhibitory subgraph. (A3) Same, for the subgraphs of individual layers. (B) Distribution across seven instantiations of the Bio-M graph of the number of 3- simplices an excitatory (red) or inhibitory (blue) neuron belongs to (simplices/neuron). (C) Mean over neurons in individual layers of the highest dimension of a simplex that they belong to. (D) Simplices/neuron by layer and dimension. (E) Correlation of 3-simplices/neuron and degree in the graph for all neurons. FIGURE 4 | (A) Patterns of thalamic innervation in the reconstruction. Each circle represents the center of innervation of a thalamic ﬁber. Each color represents a unique thalamic spike train assigned to that ﬁber. (B1) Exemplary directed simplex in a microcircuit. (B2) Connectivity and morphological types of neurons in the exemplary simplex. (B3) Raster plot and PSTH (1t =10 ms) of spiking response of neurons in (B1,B2) to stimulus S30b. (B4) Correlation coefﬁcients of all pairs of PSTHs in (B3). (C) Correlation coefﬁcients of PSTHs for all stimuli and all connected pairs of neurons in a microcircuit (1t = 25 ms). (D) Mean correlation coefﬁcients for connected pairs of neurons against the number of maximal simplices the edge between them belongs to, dimension by dimension. Means of fewer than 1,000 samples omitted. (E) Mean correlation coefﬁcient of pairs of neurons, given their position within a simplex and its dimension. Frontiers in Computational Neuroscience | www.frontiersin.org 6 June 2017 | Volume 11 | Article 48 Reimann et al. Cliques, Cavities, Structure, and Function connections where either the pre- or postsynaptic neuron failed the pair consisting of the ﬁrst (source) and last (sink) neurons to respond during all stimuli. (Figure 4E, purple), while the number of shared inputs is highest To avoid redundant sampling when testing the relationship for the last and second-to-last neurons (Figure 4E, red). The between simplex dimension and activity, we restricted our ﬁrst (source) and second neurons (Figure 4E, green) serve as a analysis to maximal simplices, i.e., directed simplices that are control because they have the smallest numbers of both indirect not part of any higher-dimensional simplex (Section 4.1.2). A connections and shared inputs in the simplex. connection can be part of many higher-dimensional maximal We found that correlations were signiﬁcantly higher for the simplices, unless it is itself a maximal 1-simplex. Despite the last two neurons in the simplex, suggesting that shared input restriction to maximal simplices, we retained all information generates more of the pairwise correlation in spiking than −6 about the structure of the microcircuit because the complete indirect connections in directed simplices (p < 8 · 10 , all structure is fully determined by its list of maximal simplices dimensions except 1D). Moreover, the spiking correlation of the (Section 4.1.2). Correlations were calculated from histograms source and sink neurons was similar to the correlation of the of the average spiking response (peri-stimulus time histogram, ﬁrst and second neurons (Figure 4E, purple and green), further PSTH; bin size, 25 ms) to ﬁve seconds of thalamo-cortical suggesting that spike correlations tend to increase as shared input over 30 repetitions of a given input pattern (Figure 4B3). input increases. These results hold for a range of histogram We then calculated the normalized cross-covariance of the time bin sizes (Figure S5). The speciﬁc positions of neurons in histograms for all connections (Figure 4C; Section 4.8) and local structures such as directed simplices therefore shape the compared it to the number of maximal simplices associated with emergence of correlated activity in response to stimuli. each connection in each dimension (see Figure 4D). The neurons forming maximal 1-simplices displayed 2.4. Cliques of Neurons Bound into Cavities a signiﬁcantly lower spiking correlation than the mean Simplices are the mathematical building blocks of the (Figure 4D), an indication of the fragility and lack of integration microcircuitry. To gain insight into how its global structure of the connection into the network. The mean correlation shapes activity, it is necessary to consider how simplices are initially decreased with the number of maximal 2-simplices a bound together. This can be achieved by analyzing the directed connection belongs to, and then increased slightly. We observed ﬂag complex, which is the set of all directed simplices together that the greater the number of maximal 2-simplices a connection with the set of all sub-simplices for each simplex (Figure belongs to, the less likely it is to belong to higher-dimensional S6, Section 4.1.2). The directed ﬂag complex is a complete maximal simplices, with the minimum correlation occurring representation of the graph, including in particular the cycles when the connection belongs to no simplices of dimension neglected when examining directed simplices in isolation. The higher than 3. In higher dimensions, the correlation increased relationship between any two directed simplices depends on with the number of maximal simplices to which a connection how they share sub-simplices. Just as any simplex can be realized belongs. While very high mean correlation can be attained for as a polyhedron, a directed ﬂag complex can be realized as a connections belonging to many maximal 3- or 4-simplices, the geometric object, built out of these polyhedra. If two simplices mean correlation of connections belonging to just one maximal share a sub-simplex, the corresponding polyhedra are glued 5- or 6-simplex was already considerably greater than the together along a common face (Figure 5A). The “shape” (or, mean. These ﬁndings reveal a strong relationship between the more precisely, the topology) of this geometric object fully structure of the network and its emergent activity and speciﬁcally describes the global structure of the network. that spike correlations depend on the level of participation of To analyze directed ﬂag complexes we computed two connections in high-dimensional simplices. descriptors, the Euler characteristic and Betti numbers (Section To determine the full extent to which the topological 4.1.5). The Euler characteristic of a ﬂag complex is given by the structure could organize activity of neurons, we examined alternating sum of the number of simplices in each dimension, spike correlations between pairs of neurons within individual from zero through the highest dimension (Figure 5A). The Betti simplices. These correlations increased with simplex dimension numbers together provide an indication of the number of cavities (Figure 4E, blue), again demonstrating that the degree (or more precisely, homology classes) fully enclosed by directed of organization in the activity increases with structural simplices in the geometric object realizing the directed ﬂag organization. Spike correlation between pairs of neurons is complex, where the dimension of a cavity is determined by the normally an ambiguous measurement of connection strength dimension of the enclosing simplices. The n-th Betti number, because it is inﬂuenced by the local structure, speciﬁcally by denoted β , indicates the number of n-dimensional cavities. For indirect connections and/or shared inputs (Palm et al., 1988; example, in Figure 5A, there is one 2-dimensional cavity (and Brody, 1999). However, since in our case the local structure is therefore β = 1) enclosed by the eight triangles; if an edge were known and described in terms of directed simplices, we could added between any two non-connected nodes, then the geometric infer how the local structural organization inﬂuences spike object realizing the corresponding ﬂag complex would be ﬁlled in correlations. We compared the impact of indirect connections with solid tetrahedra, and the cavity would disappear. In the ﬂag and of shared inputs on correlated activity by calculating the complexes of the reconstructions, it was not possible to compute average correlation of pairs of neurons at diﬀerent positions more than the zeroth and top nonzero Betti numbers, as lower in a simplex when ordered from source to sink (Figure 4E, dimensions were computationally too expensive (Section 4.2.2). right panel). The number of indirect connections is highest for We could easily compute all Betti numbers for the C. elegans Frontiers in Computational Neuroscience | www.frontiersin.org 7 June 2017 | Volume 11 | Article 48 Reimann et al. Cliques, Cavities, Structure, and Function FIGURE 5 | (A) Example of the calculation of the Euler characteristic of a directed ﬂag complex as an alternating sum of Betti numbers or simplex counts. (B) Euler characteristic against the highest non-zero Betti number (β ) for seven instances of reconstructed microcircuits based on ﬁve different biological datasets (Bio 1-5). (C) Top: The transmission-response (TR) graph of the activity of a microcircuit is a subgraph of its structural connectivity containing all nodes, but only a subset of the edges (connections). Bottom: An edge is contained if its presynaptic neuron spikes in a deﬁned time bin and its postsynaptic neurons spikes within 10 ms of the presynaptic spike. (D) Fraction of edges active against fraction of high-dimensional simplices active in TR graphs for various time bins of a simulation. Error bars indicate the standard deviation over 10 repetitions of the simulation. Blue triangles: 4-dimensional simplices, blue squares: 5-dimensional simplices. Red symbols and dashed lines indicate the results for choosing edges randomly from the structural graph and the number expected for random choice, respectively. connectome, however, as it has many fewer nodes and edges 2.5. Cliques and Cavities in Active (Figure S3). Sub-Graphs The Betti number computations showed that there are Thus far we have shown that the structural network guides cavities of dimension 5 (cavities completely enclosed by 5- the emergence of correlated activity. To determine whether simplices/6-neuron directed cliques) in all seven instances of this correlated activity is suﬃciently organized to bind neurons each of the reconstructions (Bio1-Bio5, Figure 5B; Bio-M not together to form active cliques and to bind cliques together to shown). In contrast, the ER- and PR-control models have form active cavities out of the structural graph, we represented no cavities of dimension higher than 3, and the GB-model the spiking activity during a simulation as a time series of sub- has no cavities of dimension higher than 4, demonstrating graphs for which we computed the corresponding directed ﬂag that there are not only non-random building blocks in complexes. Each sub-graph in this series comprises the same the reconstruction, but also non-random relationships among nodes (neurons) as the reconstruction, but only a subset of them. We found as well that the information encoded in the edges (synaptic connections), which are considered active, β and the Euler characteristic together captures enough i.e., the presynaptic neuron spikes in a time bin of size 1t of the structure of the ﬂag complex of a reconstruction and the postsynaptic neuron spikes within a time 1t after the to reveal subtle diﬀerences in their connectivity arising presynaptic spike (Figure 5C and Figure S7, Section 4.9). By from the underlying biological data (Figure 5B, diﬀerent considering subsequent, non-overlapping time bins of constant colors). size 1t , we obtain a time series of transmission-response (TR) Frontiers in Computational Neuroscience | www.frontiersin.org 8 June 2017 | Volume 11 | Article 48 Reimann et al. Cliques, Cavities, Structure, and Function graphs reﬂecting correlated activity in the microcircuitry. We The variation in Betti numbers and Euler characteristic over converted the time series of TR graphs in response to the time indicates that neurons become bound into cliques and diﬀerent patterns of thalamo-cortical inputs (see Figure 4A) into cavities by correlated activity (Figure 6A and Figure S8). time series of directed ﬂag complexes. We found signiﬁcantly When we plotted the number of cavities of dimension 1 more simplices in the TR graphs (1t = 5 ms, 1t = 10 (β ) against the number of those of dimension 3 (β ) (the 1 2 1 3 ms) than would be expected based on the number of edges highest dimension in which cavities consistently occur), the alone (Figure 5D), indicating that correlated activity becomes trajectory over the course of ∼100 ms (Figure 6B) began preferentially concentrated in directed simplices. ∼50 ms after stimulus onset with the formation of a large The nine stimuli generated diﬀerent spatio-temporal number of 1-dimensional cavities, followed by the emergence responses and diﬀerent numbers of active edges (Figure 6A). of 2-dimensional (not shown) and 3-dimensional cavities. The FIGURE 6 | (A) Number of edges, β , β , and Euler characteristic of the time series of TR graphs in response to the stimulus patterns shown in Figure 4 (mean and 1 3 SEM of 30 repetitions of each stimulus). (B) Trace of the time series of β against β for three of the stimuli. Shading of colors indicates Gaussian proﬁles at each time 1 3 step with means and standard deviations interpolated from 30 repetitions of each stimulus. (C) Trace for one of the stimuli in B, along with the mean ﬁring activity at different locations of the microcircuit during time steps of 2 ms. (D) Like (B), but for TR graphs of Bio 1-5, in response to stimulus S15b. Frontiers in Computational Neuroscience | www.frontiersin.org 9 June 2017 | Volume 11 | Article 48 Reimann et al. Cliques, Cavities, Structure, and Function decrease in β began while β was still increasing and continued 3. DISCUSSION 1 3 until β reached its peak, indicating that higher-dimensional This study provides a simple, powerful, parameter-free, and relationships between directed simplices continued to be formed unambiguous mathematical framework for relating the activity by correlated activity as the lower dimensional relationships of a neural network to its underlying structure, both locally (in subside. terms of simplices) and globally (in terms of cavities formed Diﬀerent stimuli led to Betti number trajectories of by these simplices). Using this framework revealed an intricate diﬀerent amplitudes, where higher degrees of synchrony in topology of synaptic connectivity containing an abundance of the thalamic input produced higher amplitudes. The trajectories cliques of neurons and of cavities binding the cliques together. all followed a similar progression of cavity formation toward The study also provides novel insight into how correlated activity a peak level of functional organization followed by relatively emerges in the network and how the network responds to rapid disintegration. The center of the projection of each stimuli. trajectory onto the β -axis (its β -center) was approximately 1 1 Such a vast number and variety of directed cliques and the same. Together, these characteristics of the trajectories cavities had not been observed before in any neural network. reveal a stereotypical evolution of cliques and cavities in The numbers of high-dimensional cliques and cavities found response to stimuli. These observations are consistent in the reconstruction are also far higher than in null models, with experimentally recorded in vivo responses to sensory even in those closely resembling the biology-based reconstructed stimuli in terms of onset delay, response duration, and the microcircuit, but with some of the biological constraints presence of distinct phases of the response (Luczak et al., released. We veriﬁed the existence of high-dimensional directed 2015). simplices in actual neocortical tissue. We further found To determine the neurons involved in this robust evolution of similar structures in a nervous system as phylogenetically functional organization, we recorded the mean levels of spiking diﬀerent as that of the worm C. elegans (Varshney et al., activity at diﬀerent spatial locations within the microcircuit 2011), suggesting that the presence of high-dimensional for one exemplary stimulus (Figure 6C). The activity started at topological structures is a general phenomenon across nervous depths that correspond to the locations of the thalamo-cortical systems. input (Meyer et al., 2010; Markram et al., 2015), increasing We showed that the spike correlation of a pair of neurons in layer 4 and at the top of layer 6, before propagating strongly increases with the number and dimension of the cliques downwards, reaching the top of layer 5 and the center of they belong to and that it even depends on their speciﬁc position layer 6 as β peaks, consistent with the ﬁnding that most in a directed clique. In particular, spike correlation increases with directed simplices are in these layers. The transition from proximity of the pair of neurons to the sink of a directed clique, as increasing β to increasing β coincided with the spread of 1 3 the degree of shared input increases. These observations indicate the upper activity zone deeper into layer 5 and the top of that the emergence of correlated activity mirrors the topological layer 6, consistent with the presence of the highest dimensional complexity of the network. While previous studies have found directed simplices in these layers. The bottom activity zone a similar link for motifs built from 2-dimensional simplices also continued moving deeper, until it eventually subsided. (Pajevic and Plenz, 2012; Chambers and MacLean, 2016), we As the top activity zone reached the bottom of layer 5, β generalize this to higher dimensions. The fact that each neuron attained its peak. The zones of activity at the peaks of β belongs to many directed cliques of various dimensions explains and β are highly complementary: zones active at the peak of in vivo observations that neurons can “ﬂexibly join multiple β were generally inactive at the peak of β and vice versa. 1 3 ensembles” (Miller et al., 2014). Braids of directed simplices The activity zone then remained in layer 5 until the cavities connected along their appropriate faces could possibly act as collapsed. synﬁre chains (Abeles, 1982), with a superposition of chains Finally, we applied the same stimulus to the reconstructions (Bienenstock, 1995) supported by the high number of cliques based on variations in the underlying biological data (see each neuron belongs to. Figure 5B, Bio-1 to 5) and found similar Betti number Topological metrics reﬂecting relationships among the trajectories, indicating that the general sequence of cavity cliques revealed biological diﬀerences in the connectivity of formation toward peak functional organization followed by reconstructed microcircuits. The same topological metrics disintegration is preserved across individuals. On the other applied to time-series of transmission-response sub-graphs hand, we observed markedly diﬀerent amplitudes, indicating that revealed a sequence of cavity formation and disintegration biological variability leads to variation in the number of high- in response to stimuli, consistent across diﬀerent stimuli dimensional cavities formed by correlated activity (Figure 6D). and individual microcircuits. The size of the trajectory was We also found that, unlike the case of diﬀerent stimuli determined by the degree of synchronous input and the applied to the same microcircuit (Figure 6B), trajectories arising biological parameters of the microcircuit, while its location from diﬀerent biological variations have diﬀerent β -centers. depended mainly on the biological parameters. Neuronal activity In some cases, we observed reverberant trajectories that also is therefore organized not only within and by directed cliques, but followed a similar sequence of cavity formation, though smaller also by highly structured relationships between directed cliques, in amplitude. The general sequence of cavity formation and consistent with a recent hypothesis concerning the relationship disintegration, however, appears to be stereotypic across stimuli between structure and function (Luczak et al., 2015). and individuals. Frontiers in Computational Neuroscience | www.frontiersin.org 10 June 2017 | Volume 11 | Article 48 Reimann et al. Cliques, Cavities, Structure, and Function Term Description Section The higher degree of topological complexity of the reconstruction compared to any of the null models was found Directed graph Network where each edge has a 4.1.1 to depend on the morphological detail of neurons, suggesting source and a target that the local statistics of branching of the dendrites and axons Simplex Clique of all-to-all connected nodes 4.1.2 is a crucial factor in forming directed cliques and cavities, Directed simplex Simplex in a directed graph, with a 4.1.3 though the exact mechanism by which this occurs remains to source and a sink be determined (but see Stepanyants and Chklovskii, 2005). The Source (of simplex) The node that is only a source of 4.1.3 number of directed 2-, 3-, and 4-simplices found per 12-patch edges in a directed simplex in vitro recording was higher than in the digital reconstruction, Sink (of simplex) The node that is only a target of 4.1.3 suggesting that the level of structural organization we found edges in a directed simplex is a conservative estimate of the actual complexity. Since the Face (of simplex) Obtained by leaving out one or more 4.1.2 reconstructions are stochastic instantiations at a speciﬁc age of nodes of a simplex the neocortex, they do not take into account rewiring driven by Simplicial complex A collection of simplices “glued” 4.1.2 plasticity during development and learning. Rewiring is readily together along common faces triggered by stimuli as well as spontaneous activity (Le Be and Maximal simplex Not a face of any larger simplex 4.1.2 Markram, 2006), which leads to a higher degree of organization Directionality Formalized, intuitive measure of 4.1.4 (Chklovskii et al., 2004; Holtmaat and Svoboda, 2009) that is directionality in a graph likely to increase the number of cliques. The diﬀerence may Betti numbers Description of a graph in terms of 4.1.5.1 also partly be due to incomplete axonal reconstructions that the number of cavities would lead to lower connectivity, but such an eﬀect would be Euler characteristic Alternating sum of number of 4.1.5.2 minor because the connection rate between the speciﬁc neurons simplices recorded for this comparison is reasonably well constrained (Reimann et al., 2015). The digital reconstruction does not take into account 4.1.1. Directed Graphs intracortical connections beyond the microcircuit. The increase A directed graph G consists of a pair of ﬁnite sets (V, E) and a in correlations between neurons with the number of cliques to function τ = (τ , τ ) : E → V × V. The elements of the set V 1 2 which they belong should be unaﬀected when these connections are the vertices of G, the elements of E are the edges of G, and the are taken into account because the overall correlation between function τ associates with each edge an ordered pair of vertices. neurons saturates already for a microcircuit of the size considered The direction of an edge e with τ(e) = (v , v ) is taken to be 1 2 in this study, as we have previously shown (Markram et al., 2015). from τ (e) = v , the source vertex, to τ (v) = v , the target However, the time course of responses to stimuli and hence the 1 1 2 2 vertex. The function τ is required to satisfy the following two speciﬁc shape of trajectories may be aﬀected by the neighboring conditions. tissue. In conclusion, this study suggests that neocortical 1. There are no (self-) loops in the graph (i.e., for each e ∈ E, if microcircuits process information through a stereotypical τ(e) = (v , v ), then v 6= v ). 1 2 1 2 progression of clique and cavity formation and disintegration, 2. For any pair of vertices (v , v ), there is at most one edge 1 2 consistent with a recent hypothesis of common strategies directed from v to v (i.e., the function τ is injective). 1 2 for information processing across the neocortex (Harris and Notice that a directed graph may contain pairs of vertices that Shepherd, 2015). We conjecture that a stimulus may be are reciprocally connected, i.e., there may exist edges e, e ∈ E processed by binding neurons into cliques of increasingly higher such that τ(e) = (v , v ) and τ(e ) = (v , v ) (Figure S6A1ii). dimension, as a speciﬁc class of cell assemblies, possibly to 1 2 2 1 represent features of the stimulus (Hebb, 1949; Braitenberg, A vertex v ∈ G is said to be a sink if there exists no 1978), and by binding these cliques into cavities of increasing e ∈ E such that v = τ (e), but there is at least one edge complexity, possibly to represent the associations between 1 ′ ′ e ∈ E such that τ (e ) = v. Similarly v is said to be a the features (Willshaw et al., 1969; Engel and Singer, 2001; 2 source is if there exists no e ∈ E such that v = τ (e), Knoblauch et al., 2009). 2 ′ ′ but there is at least one e ∈ E such that τ (e ) = v (Figures S6A1i,iii). A path in a directed graph G consists of 4. MATERIALS AND METHODS a sequence of edges (e , ..., e ) such that for all 1 ≤ k < 1 n 4.1. The Topological Toolbox n, the target of e is the source of e , i.e., τ (e ) = k k+1 2 k Specializing basic concepts of algebraic topology, we have τ (e ) (Figure S6A1iii). The length of the path (e , ..., e ) is 1 1 n k+1 formulated precise deﬁnitions of cliques (simplices) and cavities n. If, in addition, the target of e is the source of e , i.e., n 1 (as counted by Betti numbers) associated to directed networks. τ (e ) = τ (e ), then (e , ..., e ) is an oriented cycle. A graph 2 n 1 1 1 n What follows is a short introduction to directed graphs, simplicial that contains no oriented cycles is said to be acyclic (Figure complexes associated to directed graphs, and homology, as well S6A1i). as to the notion of directionality in directed graphs used in A directed graph is said to be fully connected if for every pair this study. We deﬁne, among others, the following terms and of distinct vertices, there exists an edge from one to the other, in concepts. at least one direction. Frontiers in Computational Neuroscience | www.frontiersin.org 11 June 2017 | Volume 11 | Article 48 Reimann et al. Cliques, Cavities, Structure, and Function 4.1.2. Simplices, Simplicial Complexes, and Flag then the directed ﬂag complex associated to G is the abstract directed simplicial complex S = S(G), with S = V and whose Complexes An abstract directed simplicial complex is a collection S of ﬁnite, directed n-simplices S for n ≥ 1 are (n+ 1)-tuples (v , . . . , v ), n 0 n of vertices such that for each 0 ≤ i < j ≤ n, there is an edge in ordered sets with the property that if σ ∈ S, then every subset τ of σ , with the natural ordering inherited from σ , is also a member G directed from v to v . The vertex v is called the source of the i j 0 simplex (v , . . . , v ), as there is an edge directed from v to v for of S. A subcomplex of an abstract directed simplicial complex is a 0 n 0 i all 0 < i ≤ n. Conversely, the vertex v is called the sink of the sub-collection S ⊆ S that is itself an abstract directed simplicial complex. Abstract directed simplicial complexes are a variation simplex (v , . . . , v ), as there is an edge directed from v to v for 0 n i n all 0 ≤ i < n. on the more common ordinary abstract simplicial complexes, where the sets forming the collection S are not assumed to be Notice that because of the assumptions on τ, an n-simplex in S is characterized by the (ordered) sequence (v , . . . , v ), but ordered. To be able to study directed graphs, we use this slightly 0 n more subtle concept. Henceforth, we always refer to abstract not by the underlying set of vertices. For instance (v , v , v ) and 1 2 3 (v , v , v ) are distinct 2-simplices with the same set of vertices. directed simplicial complexes as simplicial complexes. 2 1 3 The elements σ of a simplicial complex S are called its 4.1.4. Directionality of Directed Graphs simplices. We deﬁne the dimension of σ (denoted dim(σ )) to We give a mathematical deﬁnition of the notion of directionality be the cardinality of the set σ minus one. If σ is a simplex of in directed graphs, and prove that directed simplices are fully dimension n, then we refer to σ as an n-simplex of S. The set of all connected directed graphs with maximal directionality. Let G = n-simplices of S is denoted S . A simplex τ is said to be a face of σ (V, E, τ) be a directed graph. For each vertex v ∈ G, deﬁne the if τ is a subset of σ of a strictly smaller cardinality. The front face signed degree of v to be of an n-simplex σ = (v , ..., v ) is a face τ = (v , ..., v ) for some 0 n 0 m m < n. Similarly, the back face of σ is a face τ = (v , . . . , v ) i n sd(v) = Indeg(v)− Outdeg(v). (1) for some 0 < i < n. If σ = (v , . . . , v ) ∈ S then, for each 0 n n th i 0 ≤ i ≤ n, the i face of σ is the (n − 1)-simplex σ obtained P Note that for any ﬁnite graph G, sd(v) = 0. We deﬁne the v∈G from σ by removing the vertex v . A simplex that is not a face n−i directionality of G, denoted Dr(G), to be the sum over all vertices of any other simplex is said to be maximal. The set of all maximal of the square of their signed degrees (Figure S1), simplices of a simplicial complex determines the entire simplicial complex, since every simplex is either maximal itself or a face of Dr(G) = sd(v) . (2) a maximal simplex. v∈V A simplicial complex gives rise to a topological space by geometric realization. A 0-simplex is realized by a single point, Let G denote a directed n-simplex, i.e., a fully connected directed a 1-simplex by a line segment, a 2-simplex by a (ﬁlled in) triangle, graph on n+ 1 vertices such that every complete subgraph has a and so on for higher dimensions. (see Munkres, 1984, Section unique source and a unique sink. Note that a directed n-simplex 1). To form the geometric realization of the simplicial complex, has no reciprocal connections. If G is any directed graph on one then glues the geometrically realized simplices together along n + 1 vertices, then Dr(G) ≤ Dr(G ). If additionally G is a common faces. The intersection of two simplices in S, neither of fully connected directed graph without reciprocal connections, which is a face of the other, is a proper subset, and hence a face, then equality holds if and only if G is isomorphic to G as a of both of them. In the geometric realization this means that the directed graph. A full proof of these statements is given in the geometric simplices that realize the abstract simplices intersect Supplementary Methods. on common faces, and hence give rise to a well-deﬁned geometric object. 4.1.5. Homology (n) If S is a simplicial complex, then the union S = S ∪···∪ Betti numbers and Euler characteristic are numerical quantities S , called the n-skeleton of S, is a subcomplex of S. We say that S associated to simplicial complexes that arise from an important (n) is n-dimensional if S = S , and n is minimal with this property. and very useful algebraic object one can associate with any If S is n-dimensional, and k ≤ n, then the collection S ∪. . .∪S k simplicial complex, called homology. Homology serves to is not a subcomplex of S because it is not closed under taking measure the “topological complexity” of simplicial complexes, subsets. However, if one adds to that collection all the faces of all leading us to refer to Betti numbers and Euler characteristic as simplices in S ∪ . . .∪ S , one obtains a subcomplex of S called k n topological metrics. In this study we use only mod 2 simplicial the k-coskeleton of S, which we will denote by S . Coskeleta are (k) homology, computationally the simplest variant of homology, important for computing homology (see Section 4.2.2). which is why it is very commonly used in applications (Bauer et al., 2017). What follows is an elementary description of 4.1.3. Simplicial Complexes of Directed Graphs homology and its basic properties. Directed graphs give rise to directed simplicial complexes in a natural way. The directed simplicial complex associated to 4.1.5.1. Betti numbers a directed graph G is called the directed ﬂag complex of G Let F denote the ﬁeld of two elements. Let S be a simplicial (Figure S6A2). This concept is a variation on the more common complex. Deﬁne the chain complex C (S, F ) to be the sequence ∗ 2 construction of a ﬂag complex associated with an undirected {C = C (S, F )} , such that C is the F -vector space whose n n 2 n≥0 n 2 graph (Aharoni et al., 2005). If G = (V, E, τ) is a directed graph, basis elements are the n-simplices σ ∈ S , for each n ≥ 0. In Frontiers in Computational Neuroscience | www.frontiersin.org 12 June 2017 | Volume 11 | Article 48 Reimann et al. Cliques, Cavities, Structure, and Function other words, the elements of C are formal sums of n-simplices 4.2. Computation of Simplices and in S. Homology For each n ≥ 1, there is a linear transformation called a 4.2.1. Generating Directed Flag Complexes with diﬀerential Hasse Diagrams To obtain the simplices, Betti numbers and Euler characteristic ∂ : C → C (3) n n n−1 of a directed graph, we ﬁrst generate the directed ﬂag complex associated to the graph. Our algorithm encodes a directed 0 1 n speciﬁed by ∂ (σ ) = σ + σ +···+ σ for every n-simplex σ , graph and its ﬂag complex as a Hasse diagram. The Hasse where σ is the i-th face of σ , as deﬁned above. Having deﬁned diagram then gives immediate access to all simplices and simplex ∂ on the basis, one then extends it linearly to the entire vector counts. The algorithm to generate the Hasse diagrams is fully space C . The n-th Betti number β (S) of a simplicial complex n n described in the Supplementary Methods Section 2.2, and the S is the F -vector space dimension of its n-th mod 2 homology C++ implementation of the code is publicly available at http:// group, which is deﬁned by neurotop.gforge.inria.fr/. H (S, F ) = Ker(∂ )/Im(∂ ) (4) n 2 n n+1 4.2.2. Homology Computations Betti numbers and Euler characteristic are computed from the directed ﬂag complexes. All homology computations carried out for n ≥ 1 and for this paper were made with F coeﬃcients, using the boundary matrix reduced by an algorithm from the PHAT library (Bauer H (S, F ) = C /Im(∂ ). (5) 0 2 0 1 et al., 2017). The complexity of computing the n-th Betti numbers scales For all n ≥ 1, there is an inclusion of vector subspaces with the number of simplices in dimensions n − 1, n, and Im(∂ ) ⊆ Ker(∂ ) ⊆ C , and thus the deﬁnition of homology n+1 n n n + 1. In particular, it requires the computation of rank and makes sense. nullity of matrices with shapes (n − 1) × n and n × (n + 1). Computing the Betti numbers of a simplicial complex is conceptually very easy. Let|S | denote the number of n-simplices Due to the millions of simplices in dimensions 2 and 3 in the reconstructed microcircuits (see Results), the calculation of Betti in the simplicial complex S. If one encodes the diﬀerential ∂ as a |S | × |S | -matrix D with entries in F , then one can numbers above 0 or below 5 was computationally not viable, n−1 n n 2 while the computation of the 5th Betti number was possible using easily compute its nullity, null(∂ ), and its rank, rk(∂ ), which n n are the F -dimensions of the null-space and the column space the 5-coskeleton for each of the complexes. Nevertheless, our Euler characteristic computations imply that at least one of β of D , respectively. The Betti numbers of S are then a sequence of n 2 natural numbers deﬁned by or β must be nonzero, and it is highly likely the β is nonzero 4 k for all k ≤ 5. β (S) = dim (C )− rk(∂ ), and 0 F 0 1 β (S) = null(∂ )− rk(∂ ). (6) n n n+1 4.3. Model of Neocortical Microcircuitry Analyses of connectivity and simulations of electrical activity are based on a previously published model of neocortical Since Im(∂ ) ⊆ Ker(∂ ) for all n ≥ 1, the Betti numbers n+1 n microcircuitry and related methods (Markram et al., 2015). are always non-negative. The n-th Betti number β gives an We analyzed microcircuits that were reconstructed with layer indication of the number of “n-dimensional cavities” in the height and cell density data from ﬁve diﬀerent animals (Bio-1- geometric realization of S. 5), with seven microcircuits per animal forming a mesocircuit (35 microcircuits in total). In addition, we analyzed microcircuits 4.1.5.2. Euler characteristic that were reconstructed using average data (Bio-M, seven If S is a simplicial complex, and |S | denotes the cardinality of microcircuits). Simulations were run on one microcircuit each of the set of n-simplices in S, then the Euler characteristic of S is Bio-1-5 and Bio-M. Each microcircuit contains∼31,000 neurons deﬁned to be and ∼8 million connections. Data about the microcircuit and the neuron models used in the simulations, as well as the χ(S) = (−1) |S |. (7) connection matrices, are available on https://bbp.epﬂ.ch/nmc- n≥0 portal/ (Ramaswamy et al., 2015). There is a well-known, close relationship between Euler characterstic and Betti numbers (Munkres, 1984, Theorem 22.2), 4.4. Control Networks which is expressed as follows. If β (S) is the sequence of n≥0 Additional control models of connectivity were constructed Betti numbers for S, then by removing diﬀerent biological constraints on connectivity. We created three types of random matrices of sizes and χ(S) = (−1) β (S). (8) connection probabilities identical to the connectivity matrices of n≥0 the reconstructed microcircuits. Frontiers in Computational Neuroscience | www.frontiersin.org 13 June 2017 | Volume 11 | Article 48 Reimann et al. Cliques, Cavities, Structure, and Function 4.4.1. ER-Model (Random-Independent Graph) limited number of patching pipettes available in vitro (12) and An empty square connection matrix of the same size as the the failure rate of the patching. This ﬁltering step was optimized connection matrix of the reconstruction was instantiated and to match the in silico and in vitro cluster size distributions. then randomly selected oﬀ-diagonal entries were activated. 4.6. C. elegans Connectome Speciﬁcally, entries were randomly selected with equal We analyzed part of the C. elegans connectome (Varshney et al., probabilities until the same number of entries as in the 2011), consisting of 6,393 directed chemical synapses, obtained reconstruction were active. The directed graph corresponding to from www.wormatlas.org/neuronalwiring.html. such a matrix is the directed analog of an Erdo˝s-Rényi random graph (Erdos and Rényi, 1960). 4.7. Simulation of Electrical Activity We performed simulations of neuronal electrical activity during 4.4.2. PR-Model (Morphology-Only, “Peters’ Rule”) stimulation with spatio-temporal patterns of thalamic input A square connection matrix was generated based on the existence at the in vivo-like state (as in Markram et al., 2015), in the of spatial appositions between neurons in the reconstruction, i.e., central microcircuit of Bio-M. Additionally, we repeated the instances where the axon of one neuron is within 1 μm of a same simulations in the central microcircuits of the Bio-1- dendrite of the other neuron. Appositions were then randomly 5 reconstructions. We ran simulations using nine diﬀerent removed from the matrix with equal probabilities until the same organizations of thalamic input spike trains (see below). number of connections as in the reconstruction remained. 4.7.1. Thalamic Stimulation 4.4.3. GB-Model (Shufﬂed, Preserving Distance We used spike trains of 42 VPM neurons extracted from Dependance) extracellular recordings of the response to texture-induced The connection matrix of a reconstructed microcircuit was split whisker motion in anesthetized rats, with up to nine cells in the into 55 submatrices based on the morphological types of pre- same barreloid recorded simultaneously (Bale et al., 2015). Each and postsynaptic neurons. Each submatrix was then randomized reconstructed microcircuit is innervated by 310 virtual thalamo- by shuﬄing its connections as follows. Connections in a sub- cortical ﬁbers (Markram et al., 2015). To generate sets of stimuli matrix were ﬁrst grouped into bins according to the distance with diﬀerent degrees of synchronous input, we assigned to between the somata of their pre- and postsynaptic cells. Next, each ﬁber one of 5 (SS5), 15 (SS15), or 30 (SS30) spike trains, for each connection a new postsynaptic target was randomly recorded from distinct VPM neurons. In addition, we used k- selected from the same distance bin. We selected a distance bin means clustering to form clusters of ﬁbers of size 1 (SSa), 5 (SSb), size of 75 μm, which was the largest bin size that preserved the and 10 (SSc) (scikit-learn, sklearn.cluster.KMeans, Pedregosa distribution of soma-distances of connected pairs of neurons in et al., 2011) that were assigned the same spike train. This all sub-matrices (no statistically signiﬁcant diﬀerence; p > 0.05, leads to diﬀerent spatial arrangements of the identical thalamic KL-test). inputs, and therefore to diﬀerent degrees of synchronous input to individual neurons in the microcircuit. 4.5. Patch Clamp Experiments 4.5.1. In vitro 4.8. Spike Train Correlations Connectivity between layer 5 thick-tufted pyramidal cells was We constructed postsynaptic time histograms (PSTHs) for each analyzed using multiple somatic whole-cell recordings (6–12 cells neuron for each stimulus, using the mean response to 30 trials of simultaneously) on 300 μm slices of primary somatosensory 5 s of thalamic stimulation (with bin size of 25 ms; for additional cortex of 14- to 16-day-old rats. Monosynaptic, direct excitatory control, bin sizes of 10, 50, 100, 250, and 500 ms were also connections were identiﬁed by stimulation of a presynaptic cell used). We then computed the normalized covariance matrix of with a 20-70 Hz train of 5-15 strong and brief current pulses the PSTHs of all neurons (1–2 nA, 2–4 ms). Experiments were carried out according to ij the Swiss national and institutional guidelines. Further details are R = p , (9) ij C C ii jj explained in the Supplementary Methods. where C is the covariance of the PSTHs of neurons i and ij 4.5.2. In silico j. PSTHs of simulations with diﬀerent thalamic stimuli were In order to obtain in silico cell groups comparable to their patched concatenated for each neuron to yield an average correlation in vitro counterparts, we designed a cell selection procedure coeﬃcient for all stimuli. In total, correlations are based on the approximating several of the experimental constraints of the in response of all neurons during 30 trials of nine stimuli for 5 s of vitro patch-clamp setup used in this study and explained above. activity (22.5 min). In brief, layer 5 thick-tufted pyramidal cells were selected from a volume with dimensions of 200× 200× 20 μm. The size of the 4.9. Transmission-Response Matrices volume was chosen to match the ﬁeld of view usually available in The temporal sequence of transmission-response matrices the in vitro patch-clamp setup and to account for the tendency associated to a simulation of neuronal activity of duration T is to patch nearby cells, which increases the probability of ﬁnding deﬁned as connected cells. The total number of cells was then reduced by randomly discarding a fraction of them, approximating the TR(1t , 1t ) := {A(n) = A(n, 1t , 1t )} , (10) 1 2 1 2 n=1 Frontiers in Computational Neuroscience | www.frontiersin.org 14 June 2017 | Volume 11 | Article 48 Reimann et al. Cliques, Cavities, Structure, and Function where the n-th matrix, A(n), is a binary matrix describing spiking to simulate the activity. MN performed the simulations. MN, activity in the time interval [n·1t , (n+1)·1t +1t ], and where MR, and PD generated the directed ﬂag complexes from the 1 1 2 N = T/1t . The (j, k)-coeﬃcient of A(n) corresponding to the connection matrices for analysis. KH and RL developed the n-th time bin is 1 if and only if the following three conditions are theory for directed cliques and directed simplicial complexes. MR and RL developed the deﬁnition of directionality within satisﬁed, where s denotes the time of the i-th spike of neuron j. motifs and directed cliques. MR developed the deﬁnition for (1) The (j, k)-coeﬃcient of the structural matrix is 1, i.e., there transmission response matrices. PD developed the code to is a structural connection from neuron j to neuron k, so that isolate simplices and directed simplices and performed initial they form a pre-post synaptic pair. computations. MS performed topological and statistical analyses (2) There is some i such that n1t ≤ s < (n+1)1t , i.e., neuron 1 1 on the ﬂag complexes and on the C. elegans connectome. KT j spikes in the n-th time bin. helped with initial statistical analysis of network responses to (3) There is some l such that 0 < s − s < 1t , i.e., neuron k stimuli. MR and MN analyzed the simulation data, mapped l i spikes after neuron j, within a 1t interval. it onto the topological data and generated the ﬁgures. RP performed the patch-clamp experiments. GC and MR performed In other words, a non-zero entry in a transmission-response the corresponding in silico experiments. HM, KH, RL, MR, and matrix denotes a presynaptic spike, closely followed by a MN wrote the paper. postsynaptic spike, maximizing the possibility of a causal relationship between the spikes. Based on ﬁring data from ACKNOWLEDGMENTS spontaneous activity in the reconstructed microcircuit, we optimized 1t such that the resulting transmission-response This work was supported by funding from the ETH Domain matrices best reﬂect the actual sucessful transmission of signals for the Blue Brain Project and the Laboratory of Neural between the neurons in the microcircuit (see Supplementary Microcircuitry. The Blue Brain Project’s IBM BlueGene/Q Methods). Unless noted otherwise, 1t = 5 and 1t = 10 ms 1 2 system, BlueBrain IV, is funded by the ETH Board and hosted were used throughout the study. at the Swiss National Supercomputing Center (CSCS). MS was supported by the NCCR Synapsy grant of the Swiss National 4.10. Data Analysis and Statistical Tests Science Foundation. Partial support for PD was provided by the Analysis of the model and simulations was performed on a Linux GUDHI project, supported by an Advanced Investigator Grant computing-cluster using Python 2.7, including the numpy and of the European Research Council and hosted by INRIA. We scipy libraries (Jones et al., 2001), and custom Python scripts. We thank Eilif Muller for providing input on the analysis, Magdalena calculated p-values using Welch’s t-test (scipy.stats), unless noted Kedziorek for help with proving maximality in directed cliques, otherwise. Gard Spreemann for help with the analysis of the C. elegans connectome, and Taylor H. Newton for helpful discussions about AUTHOR CONTRIBUTIONS statistical methods. HM and RL developed and initially conceived the study over 10 years of discussions. HM, RL, and KH conceived and directed the SUPPLEMENTARY MATERIAL ﬁnal study. 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Publisher: Pubmed Central
Copyright: Copyright © 2017 Reimann, Nolte, Scolamiero, Turner, Perin, Chindemi, Dłotko, Levi, Hess and Markram.
ISSN: 1662-5188
eISSN: 1662-5188
DOI: 10.3389/fncom.2017.00048
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Abstract

ORIGINAL RESEARCH published: 12 June 2017 doi: 10.3389/fncom.2017.00048 Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function 1 † 1 † 2 2 Michael W. Reimann , Max Nolte , Martina Scolamiero , Katharine Turner , 3 1 4‡ 5‡ 2 ‡ Rodrigo Perin , Giuseppe Chindemi , Paweł Dłotko , Ran Levi , Kathryn Hess * and 1, 3 ‡ Henry Markram * 1 2 Blue Brain Project, École Polytechnique Fédérale de Lausanne, Geneva, Switzerland, Laboratory for Topology and Neuroscience, Brain Mind Institute, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, Laboratory of Neural Microcircuitry, Brain Mind Institute, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, DataShape, INRIA Saclay, Palaiseau, France, Institute of Mathematics, University of Aberdeen, Aberdeen, United Kingdom The lack of a formal link between neural network structure and its emergent function has hampered our understanding of how the brain processes information. We have now come closer to describing such a link by taking the direction of synaptic transmission into account, constructing graphs of a network that reﬂect the direction of information ﬂow, and analyzing these directed graphs using algebraic topology. Applying this approach to a local network of neurons in the neocortex revealed a remarkably intricate and previously Edited by: unseen topology of synaptic connectivity. The synaptic network contains an abundance Paul Miller, Brandeis University, United States of cliques of neurons bound into cavities that guide the emergence of correlated activity. Reviewed by: In response to stimuli, correlated activity binds synaptically connected neurons into Cees van Leeuwen, functional cliques and cavities that evolve in a stereotypical sequence toward peak KU Leuven, Belgium complexity. We propose that the brain processes stimuli by forming increasingly complex Andreas Knoblauch, Hochschule Albstadt-Sigmaringen, functional cliques and cavities. Germany Keywords: connectomics, topology, directed networks, structure-function, correlations, Betti numbers *Correspondence: Henry Markram henry.markram@epﬂ.ch 1. INTRODUCTION Kathryn Hess kathryn.hess@epﬂ.ch † How the structure of a network determines its function is not well understood. For neural networks These authors have contributed speciﬁcally, we lack a unifying mathematical framework to unambiguously describe the emergent equally to this work. behavior of the network in terms of its underlying structure (Bassett and Sporns, 2017). While Co-senior author. graph theory has been used to analyze network topology with some success (Bullmore and Sporns, 2009), current methods are usually constrained to analyzing how local connectivity inﬂuences local Received: 11 March 2017 Accepted: 18 May 2017 activity (Pajevic and Plenz, 2012; Chambers and MacLean, 2016) or global network dynamics (Hu Published: 12 June 2017 et al., 2014), or how global network properties like connectivity and balance of excitatory and Citation: inhibitory neurons inﬂuence network dynamics (Renart et al., 2010; Rosenbaum et al., 2017). One Reimann MW, Nolte M, Scolamiero M, such global network property is small-worldness. While it has been shown that small-worldness Turner K, Perin R, Chindemi G, optimizes information exchange (Latora and Marchiori, 2001), and that adaptive rewiring during Dłotko P, Levi R, Hess K and chaotic activity leads to small world networks (Gong and Leeuwen, 2004), the degree of small- Markram H (2017) Cliques of Neurons worldness cannot describe most local network properties, such as the diﬀerent roles of individual Bound into Cavities Provide a Missing neurons. Link between Structure and Function. Algebraic topology (Munkres, 1984) oﬀers the unique advantage of providing methods to Front. Comput. Neurosci. 11:48. doi: 10.3389/fncom.2017.00048 describe quantitatively both local network properties and the global network properties that Frontiers in Computational Neuroscience | www.frontiersin.org 1 June 2017 | Volume 11 | Article 48 Reimann et al. Cliques, Cavities, Structure, and Function emerge from local structure, thus unifying both levels. More reconstruction provide a statistical and biological range of recently, algebraic topology has been applied to functional microcircuits for analysis. networks between brain regions using fMRI (Petri et al., 2014) We found a remarkably high number and variety of high- and between neurons using neural activity (Giusti et al., 2015), dimensional directed cliques and cavities, which had not been but the underlying synaptic connections (structural network) seen before in neural networks, either biological or artiﬁcial, were unknown. Furthermore, all formal topological analyses have and in far greater numbers than those found in various null overlooked the direction of information ﬂow, since they analyzed models of directed networks. Topological metrics reﬂecting the only undirected graphs. number of directed cliques and cavities not only distinguished We developed a mathematical framework to analyze both the the reconstructions from all null models, they also revealed subtle structural and the functional topology of the network, integrating diﬀerences between reconstructions based on biological datasets local and global descriptions, enabling us to establish a clear from diﬀerent animals, suggesting that individual variations in relationship between them. We represent a network as a directed biological detail of neocortical microcircuits are reﬂected in the graph, with neurons as the vertices and the synaptic connections repertoire of directed cliques and cavities. When we simulated directed from pre- to postsynaptic neurons as the edges, which microcircuit activity in response to sensory stimuli, we observed can be analyzed using elementary tools from algebraic topology that pairwise correlations in neuronal activity increased with the (Munkres, 1984). The structural graph contains all synaptic number and dimension of the directed cliques to which a pair connections, while a functional graph is a sub-graph of the of neurons belongs, indicating that the hierarchical structure of structural graph containing only those connections that are active the network shapes a hierarchy of correlated activity. In fact, we within a speciﬁc time bin (i.e., in which a postsynaptic neuron found a hierarchy of correlated activity between neurons even ﬁres within a short time of a presynaptic spike). The response to within a single directed clique. During activity, many more high- a stimulus can then be represented and studied as a time series of dimensional directed cliques formed than would be expected functional graphs. from the number of active connections, further suggesting that Networks are often analyzed in terms of groups of nodes correlated activity tends to bind neurons into high-dimensional that are all-to-all connected, known as cliques. The number of active cliques. neurons in a clique determines its size, or more formally, its Following a spatio-temporal stimulus to the network, we dimension. In directed graphs it is natural to consider directed found that during correlated activity, active cliques form cliques, which are cliques containing a single source neuron and a increasingly high-dimensional cavities (i.e., cavities formed by single sink neuron and reﬂecting a speciﬁc motif of connectivity increasingly larger cliques). Moreover, we discovered that while diﬀerent spatio-temporal stimuli applied to the same circuit (Song et al., 2005; Perin et al., 2011), wherein the ﬂow of information through a group of neurons has an unambiguous and the same stimulus applied to diﬀerent circuits produced direction. The manner in which directed cliques bind together diﬀerent activity patterns, they all exhibited the same general can be represented geometrically. When directed cliques bind evolution, where functional relationships among increasingly appropriately by sharing neurons, and without forming a larger higher-dimensional cliques form and then disintegrate. clique due to missing connections, they form cavities (“holes,” “voids”) in this geometric representation, with high-dimensional cavities forming when high-dimensional (large) cliques bind 2. RESULTS together. Directed cliques describe the ﬂow of information in the network at the local level, while cavities provide a global measure 2.1. The Case for Directed Simplices of information ﬂow in the whole network. Using these naturally Networks of neurons connected by electrical synapses (gap arising structures, we established a direct relationship between junctions) can be represented as undirected graphs, where the structural graph and the emergent ﬂow of information in information can ﬂow in both directions. Networks with response to stimuli, as captured through time series of functional chemical synapses, which impose a single direction of synaptic graphs. communication from the pre- to the postsynaptic neuron We applied this framework to digital reconstructions of rat (Figures 1B2,B3), are more accurately represented as directed neocortical microcircuitry that closely resemble the biological graphs. Sub-sampling networks of neurons experimentally has tissue in terms of the numbers, types, and densities of neurons revealed small motifs of synaptic connectivity, but not large and their synaptic connectivity (a “microconnectome” model cliques of neurons (Song et al., 2005; Perin et al., 2011). Knowing for a cortical microcircuit, Figures 1A,B; see Markram et al., the complete directed network of neurons, as we do in the case 2015; Reimann et al., 2015). Simulations of the reconstructed of the reconstructed microcircuit, enables us to detect all cliques, microcircuitry reproduce multiple emergent electrical behaviors directed, and otherwise (Figure 1). found experimentally in the neocortex (Markram et al., When the direction of connections is not taken into account, 2015). The microcircuit, formed by ∼8 million connections a great deal of information is lost. For example, in the undirected (edges) between ∼31,000 neurons (vertices), was reconstructed case, there is only one possible conﬁguration for a clique of from experimental data, guided by biological principles of four fully connected neurons (Figure 2A1, left). However, in the organization, and iteratively reﬁned until validated against directed case, there are 3 = 729 possible conﬁgurations, as each a battery of independent anatomical and physiological data of the six connections can be in one of three states (i→ j, j← i, obtained from experiments. Multiple instantiations of the or i↔ j connection types; Figure 2A1 right). Frontiers in Computational Neuroscience | www.frontiersin.org 2 June 2017 | Volume 11 | Article 48 Reimann et al. Cliques, Cavities, Structure, and Function FIGURE 1 | (A) Thin (10 μm) slice of in silico reconstructed tissue. Red: A clique formed by ﬁve pyramidal cells in layer 5. (B1) Full connection matrix of a reconstructed microcircuit with 31,146 neurons. Neurons are sorted by cortical layer and morphological type within each layer. Pre-/postsynaptic neurons along the vertical/horizontal axis. Each grayscale pixel indicates the connections between two groups of 62 neurons each, ranging from white (no connections) to black (≥8% connected pairs). (B2) Zoom into the connectivity between two groups of 434 neurons each in layer 5, i.e., 7 by 7 pixels in (A), followed by a further zoom into the clique of 5 neurons shown in (A). Black indicates presence, and white absence of a connection. (B3) Zoom into the somata of the clique in (A) and representation of their connectivity as a directed graph. A clique with reciprocal connections contains two or Directed cliques have the highest net directionality among all more cliques consisting only of uni-directional connections cliques (Figure S1; Section 4.1.4). A clique that contains cycles (Figure 2A2). When only uni-directional connections are always decomposes into directed cliques with the same number considered, there are 2 possible conﬁgurations of four fully of neurons or fewer, at the very least any single connection connected neurons, which are of two types: those that contain between two neurons forms a 2-clique. A cyclical clique of three cycles (40 conﬁgurations; Figure 2A3 left; Section 4.1.3) and neurons therefore decomposes into three 2-cliques. Following the those that do not (24 conﬁgurations; Figure 2A3 right). Directed conventions in algebraic topology, we refer to directed cliques of cliques are exactly the acyclic cliques. The net directionality n neurons as directed simplices of dimension n-1 or directed (n-1)- of information ﬂow through any motif can be deﬁned as the simplices (which reﬂects their natural geometric representation sum over all neurons of the squares of the diﬀerences between as (n-1)-dimensional polyhedra) (see Figure S2; Section 4.1.3). their in-degree and their out-degree (see Equation 2, Figure S1). Correspondingly, their sub-cliques are called sub-simplices. Frontiers in Computational Neuroscience | www.frontiersin.org 3 June 2017 | Volume 11 | Article 48 Reimann et al. Cliques, Cavities, Structure, and Function FIGURE 2 | (A1) A 4-clique in the undirected connectivity graph has one of 729 conﬁgurations in the directed graph. (A2) Conﬁgurations containing bidirectional connections are resolved by considering all sub-graphs without bidirectional connections. (A3) Without bidirectional connections, 64 possible conﬁgurations remain, 24 of which are acyclic, with a clear sink-source structure (directed simplices, in this case of dimension 3). (B) Number of simplices in each dimension in the Bio-M reconstruction (shaded area: standard deviation of seven statistical instantiations) and in three types of random control networks. (C) Examples of neurons forming high-dimensional simplices in the reconstruction. Bottom: Their representation as directed graphs. (D) (Left) Number of directed simplices of various dimensions found in 55 in vitro patch-clamp experiments sampling groups of pyramidal cells in layer 5. (Right) Number of simplices of various dimensions found in 100,000 in silico experiments mimicking the patch-clamp procedure of (B). 2.2. An Abundance of Directed Simplices many as 80 million directed 3-simplices (Figure 2B; blue). This is 2.2.1. Reconstructed Neocortical Microcircuitry the ﬁrst indication of the existence of such a vast number of high- dimensional directed simplices in neocortical microcircuitry, or We analyzed 42 variants of the reconstructed microconnectome, grouped into six sets, each comprised of seven statistically in any neural network. varying instantiations (Markram et al., 2015; Section 4.3). The ﬁrst ﬁve sets were based on speciﬁc heights of the six layers 2.2.2. Control Models of the neocortex, cell densities, and distributions of diﬀerent To compare these results with null models, we examined how the cell types experimentally measured in ﬁve diﬀerent rats (Bio1- numbers of directed simplices in these reconstructions diﬀered from those of artiﬁcial circuits and from circuits in which some 5), while the sixth represents the mean of these measurements (Bio-M). Individual instantiations within a set varied with the of the biological rules of connectivity were omitted (see Section outcome of the stochastic portions of the reconstruction process. 4.4). For one control, we generated ﬁve Erdo˝s-Rényi random Surprisingly, we found that the reconstructions consistently graphs (ER) of equal size (∼31,000 vertices) and the same average contained directed simplices of dimensions up to 6 or 7, with as connection probability as the Bio-M circuit (∼0.8%; ∼8 million Frontiers in Computational Neuroscience | www.frontiersin.org 4 June 2017 | Volume 11 | Article 48 Reimann et al. Cliques, Cavities, Structure, and Function edges) (Figure 2B; dark green). For another, we constructed a neurons, only by inhibitory neurons, and only in individual circuit with the same 3D model neurons as the Bio-M circuit, but layers by both excitatory and inhibitory neurons. Restricting to connected the neurons using a random connectivity rule [“Peters’ only excitatory neurons barely reduces the number of simplices Rule” (Peters and Feldman, 1976), PR; Figure 2B, red]. For the in each dimension (Figure 3A1), while simplex counts in last control we connected the neurons in the Bio-M circuit inhibitory sub-graphs are multiple orders of magnitude smaller according to the distance-dependent connection probabilities (Figure 3A2), consistent with the fact that most neurons in the between the diﬀerent morphological types of neurons. Since microcircuitry are excitatory. Analyzing the sub-graphs of the this control is similar to deriving connectivity from the average layers in isolation shows that layers 5 and 6, where most of overlap of neuronal arbors (Shepherd et al., 2005), it retains the excitatory neurons reside (Markram et al., 2015), contain the general biological (GB) features of connectivity between the most simplices and the largest number of high-dimensional diﬀerent types of neurons (Reimann et al., 2015), excluding simplices (Figure 3A3). only explicit pairwise connectivity between individual neurons, The large number of simplices relative to the number of which is determined by the overlap of their speciﬁc arbors neurons in the microcircuit implies that each neuron belongs (Figure 2B, yellow). In all cases, the number of directed simplices to many directed simplices. Indeed, when we counted the of dimensions larger than 1 was far smaller than in the Bio-M number of simplices to which each neuron belongs across circuit. In addition, the relative diﬀerences between the Bio-M dimensions, we observed a long-tailed distribution such that and the null models increased markedly with dimension. a neuron belongs on average to thousands of simplices (Figure 3B). Both the mean maximal dimension and the number 2.2.3. In vitro of simplices a neuron belongs to are highest in the deeper Simplices of high dimensions (such as those depicted in cortical layers (Figure 3C). Neurons in layer 5 belong to the Figure 2C) have not yet been observed experimentally, as largest number of simplices, many spanning multiple layers doing so would require simultaneous intracellular recording (Figure 3D), consistent with the abundance of neurons with of large numbers of neurons. To obtain an indication of the the largest morphologies, which are connected to all layers. On presence of many high-dimensional directed simplices in the the other hand, layer 6 has the largest number of simplices actual neocortical tissue, we performed multi-neuron patch- that are fully contained in the layer (Figure 3A3), consistent clamp experiments with up to 12 neurons at a time in in vitro with the fact that layer 6 contains the most neurons. While the slices of the neocortex of the same age and brain region as the number of simplices that can form in the microcircuitry depends digitally reconstructed tissue (Section 4.5.1). Although limited by essentially on the number of neurons, the number of simplices the number of neurons we could simultaneously record from, we to which a single neuron belongs depends fundamentally found a substantial number of directed simplices up to dimension on its number of incoming and outgoing connections (its 3, and even one 4-dimensional simplex, in just 55 multi-neuron degree), which in turn depends on its morphological size recording experiments (Figure 2D, left). We then mimicked (Figure 3E). these experiments on the reconstructed microcircuit by repeating the same multi-neuron patch-clamp recordings in silico (Section 4.5.2) and found a similar shape of the distribution of 4-, 3-, and 2.3. Topology Organizes Spike Correlations 2-simplices, though in lower frequencies than in the actual tissue The presence of vast numbers of directed cliques across a (Figure 2D, right). These ﬁndings not only conﬁrm that high- range of dimensions in the neocortex, far more than in dimensional directed simplices are prevalent in the neocortical null models, demonstrates that connectivity between these tissue, they also suggest that the degree of organization in the neurons is highly organized into fundamental building blocks neocortex is even greater than that in the reconstruction, which of increasing complexity. Since the structural topology of the is already highly signiﬁcant (see Section 3). neural network takes into account the direction of information ﬂow, we hypothesized that emergent electrical activity of the 2.2.4. C. elegans microcircuitry mirrors its hierarchical structural organization. To test whether the presence of large numbers of high- To test this hypothesis, we simulated the electrical activity of dimensional directed simplices is a general phenomenon of the microcircuit under in vivo-like conditions (Markram et al., neural networks rather than a speciﬁc phenomenon found in this 2015). part of the brain of this particular animal and at this particular Stimuli, conﬁgured as nine diﬀerent spatio-temporal input age, we computed the numbers of directed simplices in the patterns (Figure 4A), were injected into the reconstructed C. elegans connectome (Varshney et al., 2011) (Section 4.6). microcircuit through virtual thalamo-cortical ﬁbers in which Again, we found many more high-dimensional simplices than spike trains were induced using patterns recorded in vivo (Bale expected from a random circuit with the same number of neurons et al., 2015; Figure S4; Section 4.7). These stimuli diﬀered (Figure S3). primarily in the degree of synchronous input received by the neurons. As expected, the neurons in the microcircuit responded 2.2.5. Simplicial Architecture of Neocortical to the inputs with various spiking patterns (Figures 4B1,B2,B4). Microcircuitry We then calculated for each connected pair of neurons the To understand the simplicial architecture of the microcircuit, we correlation of their spiking activity (Figure 4C) and found a began by analyzing the sub-graphs formed only by excitatory broad distribution of correlation coeﬃcients, with only∼12% of Frontiers in Computational Neuroscience | www.frontiersin.org 5 June 2017 | Volume 11 | Article 48 Reimann et al. Cliques, Cavities, Structure, and Function FIGURE 3 | (A1) Number of simplices in each dimension in the excitatory subgraph (shaded area: standard deviation across seven instantiations). (A2) Same, for the inhibitory subgraph. (A3) Same, for the subgraphs of individual layers. (B) Distribution across seven instantiations of the Bio-M graph of the number of 3- simplices an excitatory (red) or inhibitory (blue) neuron belongs to (simplices/neuron). (C) Mean over neurons in individual layers of the highest dimension of a simplex that they belong to. (D) Simplices/neuron by layer and dimension. (E) Correlation of 3-simplices/neuron and degree in the graph for all neurons. FIGURE 4 | (A) Patterns of thalamic innervation in the reconstruction. Each circle represents the center of innervation of a thalamic ﬁber. Each color represents a unique thalamic spike train assigned to that ﬁber. (B1) Exemplary directed simplex in a microcircuit. (B2) Connectivity and morphological types of neurons in the exemplary simplex. (B3) Raster plot and PSTH (1t =10 ms) of spiking response of neurons in (B1,B2) to stimulus S30b. (B4) Correlation coefﬁcients of all pairs of PSTHs in (B3). (C) Correlation coefﬁcients of PSTHs for all stimuli and all connected pairs of neurons in a microcircuit (1t = 25 ms). (D) Mean correlation coefﬁcients for connected pairs of neurons against the number of maximal simplices the edge between them belongs to, dimension by dimension. Means of fewer than 1,000 samples omitted. (E) Mean correlation coefﬁcient of pairs of neurons, given their position within a simplex and its dimension. Frontiers in Computational Neuroscience | www.frontiersin.org 6 June 2017 | Volume 11 | Article 48 Reimann et al. Cliques, Cavities, Structure, and Function connections where either the pre- or postsynaptic neuron failed the pair consisting of the ﬁrst (source) and last (sink) neurons to respond during all stimuli. (Figure 4E, purple), while the number of shared inputs is highest To avoid redundant sampling when testing the relationship for the last and second-to-last neurons (Figure 4E, red). The between simplex dimension and activity, we restricted our ﬁrst (source) and second neurons (Figure 4E, green) serve as a analysis to maximal simplices, i.e., directed simplices that are control because they have the smallest numbers of both indirect not part of any higher-dimensional simplex (Section 4.1.2). A connections and shared inputs in the simplex. connection can be part of many higher-dimensional maximal We found that correlations were signiﬁcantly higher for the simplices, unless it is itself a maximal 1-simplex. Despite the last two neurons in the simplex, suggesting that shared input restriction to maximal simplices, we retained all information generates more of the pairwise correlation in spiking than −6 about the structure of the microcircuit because the complete indirect connections in directed simplices (p < 8 · 10 , all structure is fully determined by its list of maximal simplices dimensions except 1D). Moreover, the spiking correlation of the (Section 4.1.2). Correlations were calculated from histograms source and sink neurons was similar to the correlation of the of the average spiking response (peri-stimulus time histogram, ﬁrst and second neurons (Figure 4E, purple and green), further PSTH; bin size, 25 ms) to ﬁve seconds of thalamo-cortical suggesting that spike correlations tend to increase as shared input over 30 repetitions of a given input pattern (Figure 4B3). input increases. These results hold for a range of histogram We then calculated the normalized cross-covariance of the time bin sizes (Figure S5). The speciﬁc positions of neurons in histograms for all connections (Figure 4C; Section 4.8) and local structures such as directed simplices therefore shape the compared it to the number of maximal simplices associated with emergence of correlated activity in response to stimuli. each connection in each dimension (see Figure 4D). The neurons forming maximal 1-simplices displayed 2.4. Cliques of Neurons Bound into Cavities a signiﬁcantly lower spiking correlation than the mean Simplices are the mathematical building blocks of the (Figure 4D), an indication of the fragility and lack of integration microcircuitry. To gain insight into how its global structure of the connection into the network. The mean correlation shapes activity, it is necessary to consider how simplices are initially decreased with the number of maximal 2-simplices a bound together. This can be achieved by analyzing the directed connection belongs to, and then increased slightly. We observed ﬂag complex, which is the set of all directed simplices together that the greater the number of maximal 2-simplices a connection with the set of all sub-simplices for each simplex (Figure belongs to, the less likely it is to belong to higher-dimensional S6, Section 4.1.2). The directed ﬂag complex is a complete maximal simplices, with the minimum correlation occurring representation of the graph, including in particular the cycles when the connection belongs to no simplices of dimension neglected when examining directed simplices in isolation. The higher than 3. In higher dimensions, the correlation increased relationship between any two directed simplices depends on with the number of maximal simplices to which a connection how they share sub-simplices. Just as any simplex can be realized belongs. While very high mean correlation can be attained for as a polyhedron, a directed ﬂag complex can be realized as a connections belonging to many maximal 3- or 4-simplices, the geometric object, built out of these polyhedra. If two simplices mean correlation of connections belonging to just one maximal share a sub-simplex, the corresponding polyhedra are glued 5- or 6-simplex was already considerably greater than the together along a common face (Figure 5A). The “shape” (or, mean. These ﬁndings reveal a strong relationship between the more precisely, the topology) of this geometric object fully structure of the network and its emergent activity and speciﬁcally describes the global structure of the network. that spike correlations depend on the level of participation of To analyze directed ﬂag complexes we computed two connections in high-dimensional simplices. descriptors, the Euler characteristic and Betti numbers (Section To determine the full extent to which the topological 4.1.5). The Euler characteristic of a ﬂag complex is given by the structure could organize activity of neurons, we examined alternating sum of the number of simplices in each dimension, spike correlations between pairs of neurons within individual from zero through the highest dimension (Figure 5A). The Betti simplices. These correlations increased with simplex dimension numbers together provide an indication of the number of cavities (Figure 4E, blue), again demonstrating that the degree (or more precisely, homology classes) fully enclosed by directed of organization in the activity increases with structural simplices in the geometric object realizing the directed ﬂag organization. Spike correlation between pairs of neurons is complex, where the dimension of a cavity is determined by the normally an ambiguous measurement of connection strength dimension of the enclosing simplices. The n-th Betti number, because it is inﬂuenced by the local structure, speciﬁcally by denoted β , indicates the number of n-dimensional cavities. For indirect connections and/or shared inputs (Palm et al., 1988; example, in Figure 5A, there is one 2-dimensional cavity (and Brody, 1999). However, since in our case the local structure is therefore β = 1) enclosed by the eight triangles; if an edge were known and described in terms of directed simplices, we could added between any two non-connected nodes, then the geometric infer how the local structural organization inﬂuences spike object realizing the corresponding ﬂag complex would be ﬁlled in correlations. We compared the impact of indirect connections with solid tetrahedra, and the cavity would disappear. In the ﬂag and of shared inputs on correlated activity by calculating the complexes of the reconstructions, it was not possible to compute average correlation of pairs of neurons at diﬀerent positions more than the zeroth and top nonzero Betti numbers, as lower in a simplex when ordered from source to sink (Figure 4E, dimensions were computationally too expensive (Section 4.2.2). right panel). The number of indirect connections is highest for We could easily compute all Betti numbers for the C. elegans Frontiers in Computational Neuroscience | www.frontiersin.org 7 June 2017 | Volume 11 | Article 48 Reimann et al. Cliques, Cavities, Structure, and Function FIGURE 5 | (A) Example of the calculation of the Euler characteristic of a directed ﬂag complex as an alternating sum of Betti numbers or simplex counts. (B) Euler characteristic against the highest non-zero Betti number (β ) for seven instances of reconstructed microcircuits based on ﬁve different biological datasets (Bio 1-5). (C) Top: The transmission-response (TR) graph of the activity of a microcircuit is a subgraph of its structural connectivity containing all nodes, but only a subset of the edges (connections). Bottom: An edge is contained if its presynaptic neuron spikes in a deﬁned time bin and its postsynaptic neurons spikes within 10 ms of the presynaptic spike. (D) Fraction of edges active against fraction of high-dimensional simplices active in TR graphs for various time bins of a simulation. Error bars indicate the standard deviation over 10 repetitions of the simulation. Blue triangles: 4-dimensional simplices, blue squares: 5-dimensional simplices. Red symbols and dashed lines indicate the results for choosing edges randomly from the structural graph and the number expected for random choice, respectively. connectome, however, as it has many fewer nodes and edges 2.5. Cliques and Cavities in Active (Figure S3). Sub-Graphs The Betti number computations showed that there are Thus far we have shown that the structural network guides cavities of dimension 5 (cavities completely enclosed by 5- the emergence of correlated activity. To determine whether simplices/6-neuron directed cliques) in all seven instances of this correlated activity is suﬃciently organized to bind neurons each of the reconstructions (Bio1-Bio5, Figure 5B; Bio-M not together to form active cliques and to bind cliques together to shown). In contrast, the ER- and PR-control models have form active cavities out of the structural graph, we represented no cavities of dimension higher than 3, and the GB-model the spiking activity during a simulation as a time series of sub- has no cavities of dimension higher than 4, demonstrating graphs for which we computed the corresponding directed ﬂag that there are not only non-random building blocks in complexes. Each sub-graph in this series comprises the same the reconstruction, but also non-random relationships among nodes (neurons) as the reconstruction, but only a subset of them. We found as well that the information encoded in the edges (synaptic connections), which are considered active, β and the Euler characteristic together captures enough i.e., the presynaptic neuron spikes in a time bin of size 1t of the structure of the ﬂag complex of a reconstruction and the postsynaptic neuron spikes within a time 1t after the to reveal subtle diﬀerences in their connectivity arising presynaptic spike (Figure 5C and Figure S7, Section 4.9). By from the underlying biological data (Figure 5B, diﬀerent considering subsequent, non-overlapping time bins of constant colors). size 1t , we obtain a time series of transmission-response (TR) Frontiers in Computational Neuroscience | www.frontiersin.org 8 June 2017 | Volume 11 | Article 48 Reimann et al. Cliques, Cavities, Structure, and Function graphs reﬂecting correlated activity in the microcircuitry. We The variation in Betti numbers and Euler characteristic over converted the time series of TR graphs in response to the time indicates that neurons become bound into cliques and diﬀerent patterns of thalamo-cortical inputs (see Figure 4A) into cavities by correlated activity (Figure 6A and Figure S8). time series of directed ﬂag complexes. We found signiﬁcantly When we plotted the number of cavities of dimension 1 more simplices in the TR graphs (1t = 5 ms, 1t = 10 (β ) against the number of those of dimension 3 (β ) (the 1 2 1 3 ms) than would be expected based on the number of edges highest dimension in which cavities consistently occur), the alone (Figure 5D), indicating that correlated activity becomes trajectory over the course of ∼100 ms (Figure 6B) began preferentially concentrated in directed simplices. ∼50 ms after stimulus onset with the formation of a large The nine stimuli generated diﬀerent spatio-temporal number of 1-dimensional cavities, followed by the emergence responses and diﬀerent numbers of active edges (Figure 6A). of 2-dimensional (not shown) and 3-dimensional cavities. The FIGURE 6 | (A) Number of edges, β , β , and Euler characteristic of the time series of TR graphs in response to the stimulus patterns shown in Figure 4 (mean and 1 3 SEM of 30 repetitions of each stimulus). (B) Trace of the time series of β against β for three of the stimuli. Shading of colors indicates Gaussian proﬁles at each time 1 3 step with means and standard deviations interpolated from 30 repetitions of each stimulus. (C) Trace for one of the stimuli in B, along with the mean ﬁring activity at different locations of the microcircuit during time steps of 2 ms. (D) Like (B), but for TR graphs of Bio 1-5, in response to stimulus S15b. Frontiers in Computational Neuroscience | www.frontiersin.org 9 June 2017 | Volume 11 | Article 48 Reimann et al. Cliques, Cavities, Structure, and Function decrease in β began while β was still increasing and continued 3. DISCUSSION 1 3 until β reached its peak, indicating that higher-dimensional This study provides a simple, powerful, parameter-free, and relationships between directed simplices continued to be formed unambiguous mathematical framework for relating the activity by correlated activity as the lower dimensional relationships of a neural network to its underlying structure, both locally (in subside. terms of simplices) and globally (in terms of cavities formed Diﬀerent stimuli led to Betti number trajectories of by these simplices). Using this framework revealed an intricate diﬀerent amplitudes, where higher degrees of synchrony in topology of synaptic connectivity containing an abundance of the thalamic input produced higher amplitudes. The trajectories cliques of neurons and of cavities binding the cliques together. all followed a similar progression of cavity formation toward The study also provides novel insight into how correlated activity a peak level of functional organization followed by relatively emerges in the network and how the network responds to rapid disintegration. The center of the projection of each stimuli. trajectory onto the β -axis (its β -center) was approximately 1 1 Such a vast number and variety of directed cliques and the same. Together, these characteristics of the trajectories cavities had not been observed before in any neural network. reveal a stereotypical evolution of cliques and cavities in The numbers of high-dimensional cliques and cavities found response to stimuli. These observations are consistent in the reconstruction are also far higher than in null models, with experimentally recorded in vivo responses to sensory even in those closely resembling the biology-based reconstructed stimuli in terms of onset delay, response duration, and the microcircuit, but with some of the biological constraints presence of distinct phases of the response (Luczak et al., released. We veriﬁed the existence of high-dimensional directed 2015). simplices in actual neocortical tissue. We further found To determine the neurons involved in this robust evolution of similar structures in a nervous system as phylogenetically functional organization, we recorded the mean levels of spiking diﬀerent as that of the worm C. elegans (Varshney et al., activity at diﬀerent spatial locations within the microcircuit 2011), suggesting that the presence of high-dimensional for one exemplary stimulus (Figure 6C). The activity started at topological structures is a general phenomenon across nervous depths that correspond to the locations of the thalamo-cortical systems. input (Meyer et al., 2010; Markram et al., 2015), increasing We showed that the spike correlation of a pair of neurons in layer 4 and at the top of layer 6, before propagating strongly increases with the number and dimension of the cliques downwards, reaching the top of layer 5 and the center of they belong to and that it even depends on their speciﬁc position layer 6 as β peaks, consistent with the ﬁnding that most in a directed clique. In particular, spike correlation increases with directed simplices are in these layers. The transition from proximity of the pair of neurons to the sink of a directed clique, as increasing β to increasing β coincided with the spread of 1 3 the degree of shared input increases. These observations indicate the upper activity zone deeper into layer 5 and the top of that the emergence of correlated activity mirrors the topological layer 6, consistent with the presence of the highest dimensional complexity of the network. While previous studies have found directed simplices in these layers. The bottom activity zone a similar link for motifs built from 2-dimensional simplices also continued moving deeper, until it eventually subsided. (Pajevic and Plenz, 2012; Chambers and MacLean, 2016), we As the top activity zone reached the bottom of layer 5, β generalize this to higher dimensions. The fact that each neuron attained its peak. The zones of activity at the peaks of β belongs to many directed cliques of various dimensions explains and β are highly complementary: zones active at the peak of in vivo observations that neurons can “ﬂexibly join multiple β were generally inactive at the peak of β and vice versa. 1 3 ensembles” (Miller et al., 2014). Braids of directed simplices The activity zone then remained in layer 5 until the cavities connected along their appropriate faces could possibly act as collapsed. synﬁre chains (Abeles, 1982), with a superposition of chains Finally, we applied the same stimulus to the reconstructions (Bienenstock, 1995) supported by the high number of cliques based on variations in the underlying biological data (see each neuron belongs to. Figure 5B, Bio-1 to 5) and found similar Betti number Topological metrics reﬂecting relationships among the trajectories, indicating that the general sequence of cavity cliques revealed biological diﬀerences in the connectivity of formation toward peak functional organization followed by reconstructed microcircuits. The same topological metrics disintegration is preserved across individuals. On the other applied to time-series of transmission-response sub-graphs hand, we observed markedly diﬀerent amplitudes, indicating that revealed a sequence of cavity formation and disintegration biological variability leads to variation in the number of high- in response to stimuli, consistent across diﬀerent stimuli dimensional cavities formed by correlated activity (Figure 6D). and individual microcircuits. The size of the trajectory was We also found that, unlike the case of diﬀerent stimuli determined by the degree of synchronous input and the applied to the same microcircuit (Figure 6B), trajectories arising biological parameters of the microcircuit, while its location from diﬀerent biological variations have diﬀerent β -centers. depended mainly on the biological parameters. Neuronal activity In some cases, we observed reverberant trajectories that also is therefore organized not only within and by directed cliques, but followed a similar sequence of cavity formation, though smaller also by highly structured relationships between directed cliques, in amplitude. The general sequence of cavity formation and consistent with a recent hypothesis concerning the relationship disintegration, however, appears to be stereotypic across stimuli between structure and function (Luczak et al., 2015). and individuals. Frontiers in Computational Neuroscience | www.frontiersin.org 10 June 2017 | Volume 11 | Article 48 Reimann et al. Cliques, Cavities, Structure, and Function Term Description Section The higher degree of topological complexity of the reconstruction compared to any of the null models was found Directed graph Network where each edge has a 4.1.1 to depend on the morphological detail of neurons, suggesting source and a target that the local statistics of branching of the dendrites and axons Simplex Clique of all-to-all connected nodes 4.1.2 is a crucial factor in forming directed cliques and cavities, Directed simplex Simplex in a directed graph, with a 4.1.3 though the exact mechanism by which this occurs remains to source and a sink be determined (but see Stepanyants and Chklovskii, 2005). The Source (of simplex) The node that is only a source of 4.1.3 number of directed 2-, 3-, and 4-simplices found per 12-patch edges in a directed simplex in vitro recording was higher than in the digital reconstruction, Sink (of simplex) The node that is only a target of 4.1.3 suggesting that the level of structural organization we found edges in a directed simplex is a conservative estimate of the actual complexity. Since the Face (of simplex) Obtained by leaving out one or more 4.1.2 reconstructions are stochastic instantiations at a speciﬁc age of nodes of a simplex the neocortex, they do not take into account rewiring driven by Simplicial complex A collection of simplices “glued” 4.1.2 plasticity during development and learning. Rewiring is readily together along common faces triggered by stimuli as well as spontaneous activity (Le Be and Maximal simplex Not a face of any larger simplex 4.1.2 Markram, 2006), which leads to a higher degree of organization Directionality Formalized, intuitive measure of 4.1.4 (Chklovskii et al., 2004; Holtmaat and Svoboda, 2009) that is directionality in a graph likely to increase the number of cliques. The diﬀerence may Betti numbers Description of a graph in terms of 4.1.5.1 also partly be due to incomplete axonal reconstructions that the number of cavities would lead to lower connectivity, but such an eﬀect would be Euler characteristic Alternating sum of number of 4.1.5.2 minor because the connection rate between the speciﬁc neurons simplices recorded for this comparison is reasonably well constrained (Reimann et al., 2015). The digital reconstruction does not take into account 4.1.1. Directed Graphs intracortical connections beyond the microcircuit. The increase A directed graph G consists of a pair of ﬁnite sets (V, E) and a in correlations between neurons with the number of cliques to function τ = (τ , τ ) : E → V × V. The elements of the set V 1 2 which they belong should be unaﬀected when these connections are the vertices of G, the elements of E are the edges of G, and the are taken into account because the overall correlation between function τ associates with each edge an ordered pair of vertices. neurons saturates already for a microcircuit of the size considered The direction of an edge e with τ(e) = (v , v ) is taken to be 1 2 in this study, as we have previously shown (Markram et al., 2015). from τ (e) = v , the source vertex, to τ (v) = v , the target However, the time course of responses to stimuli and hence the 1 1 2 2 vertex. The function τ is required to satisfy the following two speciﬁc shape of trajectories may be aﬀected by the neighboring conditions. tissue. In conclusion, this study suggests that neocortical 1. There are no (self-) loops in the graph (i.e., for each e ∈ E, if microcircuits process information through a stereotypical τ(e) = (v , v ), then v 6= v ). 1 2 1 2 progression of clique and cavity formation and disintegration, 2. For any pair of vertices (v , v ), there is at most one edge 1 2 consistent with a recent hypothesis of common strategies directed from v to v (i.e., the function τ is injective). 1 2 for information processing across the neocortex (Harris and Notice that a directed graph may contain pairs of vertices that Shepherd, 2015). We conjecture that a stimulus may be are reciprocally connected, i.e., there may exist edges e, e ∈ E processed by binding neurons into cliques of increasingly higher such that τ(e) = (v , v ) and τ(e ) = (v , v ) (Figure S6A1ii). dimension, as a speciﬁc class of cell assemblies, possibly to 1 2 2 1 represent features of the stimulus (Hebb, 1949; Braitenberg, A vertex v ∈ G is said to be a sink if there exists no 1978), and by binding these cliques into cavities of increasing e ∈ E such that v = τ (e), but there is at least one edge complexity, possibly to represent the associations between 1 ′ ′ e ∈ E such that τ (e ) = v. Similarly v is said to be a the features (Willshaw et al., 1969; Engel and Singer, 2001; 2 source is if there exists no e ∈ E such that v = τ (e), Knoblauch et al., 2009). 2 ′ ′ but there is at least one e ∈ E such that τ (e ) = v (Figures S6A1i,iii). A path in a directed graph G consists of 4. MATERIALS AND METHODS a sequence of edges (e , ..., e ) such that for all 1 ≤ k < 1 n 4.1. The Topological Toolbox n, the target of e is the source of e , i.e., τ (e ) = k k+1 2 k Specializing basic concepts of algebraic topology, we have τ (e ) (Figure S6A1iii). The length of the path (e , ..., e ) is 1 1 n k+1 formulated precise deﬁnitions of cliques (simplices) and cavities n. If, in addition, the target of e is the source of e , i.e., n 1 (as counted by Betti numbers) associated to directed networks. τ (e ) = τ (e ), then (e , ..., e ) is an oriented cycle. A graph 2 n 1 1 1 n What follows is a short introduction to directed graphs, simplicial that contains no oriented cycles is said to be acyclic (Figure complexes associated to directed graphs, and homology, as well S6A1i). as to the notion of directionality in directed graphs used in A directed graph is said to be fully connected if for every pair this study. We deﬁne, among others, the following terms and of distinct vertices, there exists an edge from one to the other, in concepts. at least one direction. Frontiers in Computational Neuroscience | www.frontiersin.org 11 June 2017 | Volume 11 | Article 48 Reimann et al. Cliques, Cavities, Structure, and Function 4.1.2. Simplices, Simplicial Complexes, and Flag then the directed ﬂag complex associated to G is the abstract directed simplicial complex S = S(G), with S = V and whose Complexes An abstract directed simplicial complex is a collection S of ﬁnite, directed n-simplices S for n ≥ 1 are (n+ 1)-tuples (v , . . . , v ), n 0 n of vertices such that for each 0 ≤ i < j ≤ n, there is an edge in ordered sets with the property that if σ ∈ S, then every subset τ of σ , with the natural ordering inherited from σ , is also a member G directed from v to v . The vertex v is called the source of the i j 0 simplex (v , . . . , v ), as there is an edge directed from v to v for of S. A subcomplex of an abstract directed simplicial complex is a 0 n 0 i all 0 < i ≤ n. Conversely, the vertex v is called the sink of the sub-collection S ⊆ S that is itself an abstract directed simplicial complex. Abstract directed simplicial complexes are a variation simplex (v , . . . , v ), as there is an edge directed from v to v for 0 n i n all 0 ≤ i < n. on the more common ordinary abstract simplicial complexes, where the sets forming the collection S are not assumed to be Notice that because of the assumptions on τ, an n-simplex in S is characterized by the (ordered) sequence (v , . . . , v ), but ordered. To be able to study directed graphs, we use this slightly 0 n more subtle concept. Henceforth, we always refer to abstract not by the underlying set of vertices. For instance (v , v , v ) and 1 2 3 (v , v , v ) are distinct 2-simplices with the same set of vertices. directed simplicial complexes as simplicial complexes. 2 1 3 The elements σ of a simplicial complex S are called its 4.1.4. Directionality of Directed Graphs simplices. We deﬁne the dimension of σ (denoted dim(σ )) to We give a mathematical deﬁnition of the notion of directionality be the cardinality of the set σ minus one. If σ is a simplex of in directed graphs, and prove that directed simplices are fully dimension n, then we refer to σ as an n-simplex of S. The set of all connected directed graphs with maximal directionality. Let G = n-simplices of S is denoted S . A simplex τ is said to be a face of σ (V, E, τ) be a directed graph. For each vertex v ∈ G, deﬁne the if τ is a subset of σ of a strictly smaller cardinality. The front face signed degree of v to be of an n-simplex σ = (v , ..., v ) is a face τ = (v , ..., v ) for some 0 n 0 m m < n. Similarly, the back face of σ is a face τ = (v , . . . , v ) i n sd(v) = Indeg(v)− Outdeg(v). (1) for some 0 < i < n. If σ = (v , . . . , v ) ∈ S then, for each 0 n n th i 0 ≤ i ≤ n, the i face of σ is the (n − 1)-simplex σ obtained P Note that for any ﬁnite graph G, sd(v) = 0. We deﬁne the v∈G from σ by removing the vertex v . A simplex that is not a face n−i directionality of G, denoted Dr(G), to be the sum over all vertices of any other simplex is said to be maximal. The set of all maximal of the square of their signed degrees (Figure S1), simplices of a simplicial complex determines the entire simplicial complex, since every simplex is either maximal itself or a face of Dr(G) = sd(v) . (2) a maximal simplex. v∈V A simplicial complex gives rise to a topological space by geometric realization. A 0-simplex is realized by a single point, Let G denote a directed n-simplex, i.e., a fully connected directed a 1-simplex by a line segment, a 2-simplex by a (ﬁlled in) triangle, graph on n+ 1 vertices such that every complete subgraph has a and so on for higher dimensions. (see Munkres, 1984, Section unique source and a unique sink. Note that a directed n-simplex 1). To form the geometric realization of the simplicial complex, has no reciprocal connections. If G is any directed graph on one then glues the geometrically realized simplices together along n + 1 vertices, then Dr(G) ≤ Dr(G ). If additionally G is a common faces. The intersection of two simplices in S, neither of fully connected directed graph without reciprocal connections, which is a face of the other, is a proper subset, and hence a face, then equality holds if and only if G is isomorphic to G as a of both of them. In the geometric realization this means that the directed graph. A full proof of these statements is given in the geometric simplices that realize the abstract simplices intersect Supplementary Methods. on common faces, and hence give rise to a well-deﬁned geometric object. 4.1.5. Homology (n) If S is a simplicial complex, then the union S = S ∪···∪ Betti numbers and Euler characteristic are numerical quantities S , called the n-skeleton of S, is a subcomplex of S. We say that S associated to simplicial complexes that arise from an important (n) is n-dimensional if S = S , and n is minimal with this property. and very useful algebraic object one can associate with any If S is n-dimensional, and k ≤ n, then the collection S ∪. . .∪S k simplicial complex, called homology. Homology serves to is not a subcomplex of S because it is not closed under taking measure the “topological complexity” of simplicial complexes, subsets. However, if one adds to that collection all the faces of all leading us to refer to Betti numbers and Euler characteristic as simplices in S ∪ . . .∪ S , one obtains a subcomplex of S called k n topological metrics. In this study we use only mod 2 simplicial the k-coskeleton of S, which we will denote by S . Coskeleta are (k) homology, computationally the simplest variant of homology, important for computing homology (see Section 4.2.2). which is why it is very commonly used in applications (Bauer et al., 2017). What follows is an elementary description of 4.1.3. Simplicial Complexes of Directed Graphs homology and its basic properties. Directed graphs give rise to directed simplicial complexes in a natural way. The directed simplicial complex associated to 4.1.5.1. Betti numbers a directed graph G is called the directed ﬂag complex of G Let F denote the ﬁeld of two elements. Let S be a simplicial (Figure S6A2). This concept is a variation on the more common complex. Deﬁne the chain complex C (S, F ) to be the sequence ∗ 2 construction of a ﬂag complex associated with an undirected {C = C (S, F )} , such that C is the F -vector space whose n n 2 n≥0 n 2 graph (Aharoni et al., 2005). If G = (V, E, τ) is a directed graph, basis elements are the n-simplices σ ∈ S , for each n ≥ 0. In Frontiers in Computational Neuroscience | www.frontiersin.org 12 June 2017 | Volume 11 | Article 48 Reimann et al. Cliques, Cavities, Structure, and Function other words, the elements of C are formal sums of n-simplices 4.2. Computation of Simplices and in S. Homology For each n ≥ 1, there is a linear transformation called a 4.2.1. Generating Directed Flag Complexes with diﬀerential Hasse Diagrams To obtain the simplices, Betti numbers and Euler characteristic ∂ : C → C (3) n n n−1 of a directed graph, we ﬁrst generate the directed ﬂag complex associated to the graph. Our algorithm encodes a directed 0 1 n speciﬁed by ∂ (σ ) = σ + σ +···+ σ for every n-simplex σ , graph and its ﬂag complex as a Hasse diagram. The Hasse where σ is the i-th face of σ , as deﬁned above. Having deﬁned diagram then gives immediate access to all simplices and simplex ∂ on the basis, one then extends it linearly to the entire vector counts. The algorithm to generate the Hasse diagrams is fully space C . The n-th Betti number β (S) of a simplicial complex n n described in the Supplementary Methods Section 2.2, and the S is the F -vector space dimension of its n-th mod 2 homology C++ implementation of the code is publicly available at http:// group, which is deﬁned by neurotop.gforge.inria.fr/. H (S, F ) = Ker(∂ )/Im(∂ ) (4) n 2 n n+1 4.2.2. Homology Computations Betti numbers and Euler characteristic are computed from the directed ﬂag complexes. All homology computations carried out for n ≥ 1 and for this paper were made with F coeﬃcients, using the boundary matrix reduced by an algorithm from the PHAT library (Bauer H (S, F ) = C /Im(∂ ). (5) 0 2 0 1 et al., 2017). The complexity of computing the n-th Betti numbers scales For all n ≥ 1, there is an inclusion of vector subspaces with the number of simplices in dimensions n − 1, n, and Im(∂ ) ⊆ Ker(∂ ) ⊆ C , and thus the deﬁnition of homology n+1 n n n + 1. In particular, it requires the computation of rank and makes sense. nullity of matrices with shapes (n − 1) × n and n × (n + 1). Computing the Betti numbers of a simplicial complex is conceptually very easy. Let|S | denote the number of n-simplices Due to the millions of simplices in dimensions 2 and 3 in the reconstructed microcircuits (see Results), the calculation of Betti in the simplicial complex S. If one encodes the diﬀerential ∂ as a |S | × |S | -matrix D with entries in F , then one can numbers above 0 or below 5 was computationally not viable, n−1 n n 2 while the computation of the 5th Betti number was possible using easily compute its nullity, null(∂ ), and its rank, rk(∂ ), which n n are the F -dimensions of the null-space and the column space the 5-coskeleton for each of the complexes. Nevertheless, our Euler characteristic computations imply that at least one of β of D , respectively. The Betti numbers of S are then a sequence of n 2 natural numbers deﬁned by or β must be nonzero, and it is highly likely the β is nonzero 4 k for all k ≤ 5. β (S) = dim (C )− rk(∂ ), and 0 F 0 1 β (S) = null(∂ )− rk(∂ ). (6) n n n+1 4.3. Model of Neocortical Microcircuitry Analyses of connectivity and simulations of electrical activity are based on a previously published model of neocortical Since Im(∂ ) ⊆ Ker(∂ ) for all n ≥ 1, the Betti numbers n+1 n microcircuitry and related methods (Markram et al., 2015). are always non-negative. The n-th Betti number β gives an We analyzed microcircuits that were reconstructed with layer indication of the number of “n-dimensional cavities” in the height and cell density data from ﬁve diﬀerent animals (Bio-1- geometric realization of S. 5), with seven microcircuits per animal forming a mesocircuit (35 microcircuits in total). In addition, we analyzed microcircuits 4.1.5.2. Euler characteristic that were reconstructed using average data (Bio-M, seven If S is a simplicial complex, and |S | denotes the cardinality of microcircuits). Simulations were run on one microcircuit each of the set of n-simplices in S, then the Euler characteristic of S is Bio-1-5 and Bio-M. Each microcircuit contains∼31,000 neurons deﬁned to be and ∼8 million connections. Data about the microcircuit and the neuron models used in the simulations, as well as the χ(S) = (−1) |S |. (7) connection matrices, are available on https://bbp.epﬂ.ch/nmc- n≥0 portal/ (Ramaswamy et al., 2015). There is a well-known, close relationship between Euler characterstic and Betti numbers (Munkres, 1984, Theorem 22.2), 4.4. Control Networks which is expressed as follows. If β (S) is the sequence of n≥0 Additional control models of connectivity were constructed Betti numbers for S, then by removing diﬀerent biological constraints on connectivity. We created three types of random matrices of sizes and χ(S) = (−1) β (S). (8) connection probabilities identical to the connectivity matrices of n≥0 the reconstructed microcircuits. Frontiers in Computational Neuroscience | www.frontiersin.org 13 June 2017 | Volume 11 | Article 48 Reimann et al. Cliques, Cavities, Structure, and Function 4.4.1. ER-Model (Random-Independent Graph) limited number of patching pipettes available in vitro (12) and An empty square connection matrix of the same size as the the failure rate of the patching. This ﬁltering step was optimized connection matrix of the reconstruction was instantiated and to match the in silico and in vitro cluster size distributions. then randomly selected oﬀ-diagonal entries were activated. 4.6. C. elegans Connectome Speciﬁcally, entries were randomly selected with equal We analyzed part of the C. elegans connectome (Varshney et al., probabilities until the same number of entries as in the 2011), consisting of 6,393 directed chemical synapses, obtained reconstruction were active. The directed graph corresponding to from www.wormatlas.org/neuronalwiring.html. such a matrix is the directed analog of an Erdo˝s-Rényi random graph (Erdos and Rényi, 1960). 4.7. Simulation of Electrical Activity We performed simulations of neuronal electrical activity during 4.4.2. PR-Model (Morphology-Only, “Peters’ Rule”) stimulation with spatio-temporal patterns of thalamic input A square connection matrix was generated based on the existence at the in vivo-like state (as in Markram et al., 2015), in the of spatial appositions between neurons in the reconstruction, i.e., central microcircuit of Bio-M. Additionally, we repeated the instances where the axon of one neuron is within 1 μm of a same simulations in the central microcircuits of the Bio-1- dendrite of the other neuron. Appositions were then randomly 5 reconstructions. We ran simulations using nine diﬀerent removed from the matrix with equal probabilities until the same organizations of thalamic input spike trains (see below). number of connections as in the reconstruction remained. 4.7.1. Thalamic Stimulation 4.4.3. GB-Model (Shufﬂed, Preserving Distance We used spike trains of 42 VPM neurons extracted from Dependance) extracellular recordings of the response to texture-induced The connection matrix of a reconstructed microcircuit was split whisker motion in anesthetized rats, with up to nine cells in the into 55 submatrices based on the morphological types of pre- same barreloid recorded simultaneously (Bale et al., 2015). Each and postsynaptic neurons. Each submatrix was then randomized reconstructed microcircuit is innervated by 310 virtual thalamo- by shuﬄing its connections as follows. Connections in a sub- cortical ﬁbers (Markram et al., 2015). To generate sets of stimuli matrix were ﬁrst grouped into bins according to the distance with diﬀerent degrees of synchronous input, we assigned to between the somata of their pre- and postsynaptic cells. Next, each ﬁber one of 5 (SS5), 15 (SS15), or 30 (SS30) spike trains, for each connection a new postsynaptic target was randomly recorded from distinct VPM neurons. In addition, we used k- selected from the same distance bin. We selected a distance bin means clustering to form clusters of ﬁbers of size 1 (SSa), 5 (SSb), size of 75 μm, which was the largest bin size that preserved the and 10 (SSc) (scikit-learn, sklearn.cluster.KMeans, Pedregosa distribution of soma-distances of connected pairs of neurons in et al., 2011) that were assigned the same spike train. This all sub-matrices (no statistically signiﬁcant diﬀerence; p > 0.05, leads to diﬀerent spatial arrangements of the identical thalamic KL-test). inputs, and therefore to diﬀerent degrees of synchronous input to individual neurons in the microcircuit. 4.5. Patch Clamp Experiments 4.5.1. In vitro 4.8. Spike Train Correlations Connectivity between layer 5 thick-tufted pyramidal cells was We constructed postsynaptic time histograms (PSTHs) for each analyzed using multiple somatic whole-cell recordings (6–12 cells neuron for each stimulus, using the mean response to 30 trials of simultaneously) on 300 μm slices of primary somatosensory 5 s of thalamic stimulation (with bin size of 25 ms; for additional cortex of 14- to 16-day-old rats. Monosynaptic, direct excitatory control, bin sizes of 10, 50, 100, 250, and 500 ms were also connections were identiﬁed by stimulation of a presynaptic cell used). We then computed the normalized covariance matrix of with a 20-70 Hz train of 5-15 strong and brief current pulses the PSTHs of all neurons (1–2 nA, 2–4 ms). Experiments were carried out according to ij the Swiss national and institutional guidelines. Further details are R = p , (9) ij C C ii jj explained in the Supplementary Methods. where C is the covariance of the PSTHs of neurons i and ij 4.5.2. In silico j. PSTHs of simulations with diﬀerent thalamic stimuli were In order to obtain in silico cell groups comparable to their patched concatenated for each neuron to yield an average correlation in vitro counterparts, we designed a cell selection procedure coeﬃcient for all stimuli. In total, correlations are based on the approximating several of the experimental constraints of the in response of all neurons during 30 trials of nine stimuli for 5 s of vitro patch-clamp setup used in this study and explained above. activity (22.5 min). In brief, layer 5 thick-tufted pyramidal cells were selected from a volume with dimensions of 200× 200× 20 μm. The size of the 4.9. Transmission-Response Matrices volume was chosen to match the ﬁeld of view usually available in The temporal sequence of transmission-response matrices the in vitro patch-clamp setup and to account for the tendency associated to a simulation of neuronal activity of duration T is to patch nearby cells, which increases the probability of ﬁnding deﬁned as connected cells. The total number of cells was then reduced by randomly discarding a fraction of them, approximating the TR(1t , 1t ) := {A(n) = A(n, 1t , 1t )} , (10) 1 2 1 2 n=1 Frontiers in Computational Neuroscience | www.frontiersin.org 14 June 2017 | Volume 11 | Article 48 Reimann et al. Cliques, Cavities, Structure, and Function where the n-th matrix, A(n), is a binary matrix describing spiking to simulate the activity. MN performed the simulations. MN, activity in the time interval [n·1t , (n+1)·1t +1t ], and where MR, and PD generated the directed ﬂag complexes from the 1 1 2 N = T/1t . The (j, k)-coeﬃcient of A(n) corresponding to the connection matrices for analysis. KH and RL developed the n-th time bin is 1 if and only if the following three conditions are theory for directed cliques and directed simplicial complexes. MR and RL developed the deﬁnition of directionality within satisﬁed, where s denotes the time of the i-th spike of neuron j. motifs and directed cliques. MR developed the deﬁnition for (1) The (j, k)-coeﬃcient of the structural matrix is 1, i.e., there transmission response matrices. PD developed the code to is a structural connection from neuron j to neuron k, so that isolate simplices and directed simplices and performed initial they form a pre-post synaptic pair. computations. MS performed topological and statistical analyses (2) There is some i such that n1t ≤ s < (n+1)1t , i.e., neuron 1 1 on the ﬂag complexes and on the C. elegans connectome. KT j spikes in the n-th time bin. helped with initial statistical analysis of network responses to (3) There is some l such that 0 < s − s < 1t , i.e., neuron k stimuli. MR and MN analyzed the simulation data, mapped l i spikes after neuron j, within a 1t interval. it onto the topological data and generated the ﬁgures. RP performed the patch-clamp experiments. GC and MR performed In other words, a non-zero entry in a transmission-response the corresponding in silico experiments. HM, KH, RL, MR, and matrix denotes a presynaptic spike, closely followed by a MN wrote the paper. postsynaptic spike, maximizing the possibility of a causal relationship between the spikes. Based on ﬁring data from ACKNOWLEDGMENTS spontaneous activity in the reconstructed microcircuit, we optimized 1t such that the resulting transmission-response This work was supported by funding from the ETH Domain matrices best reﬂect the actual sucessful transmission of signals for the Blue Brain Project and the Laboratory of Neural between the neurons in the microcircuit (see Supplementary Microcircuitry. The Blue Brain Project’s IBM BlueGene/Q Methods). Unless noted otherwise, 1t = 5 and 1t = 10 ms 1 2 system, BlueBrain IV, is funded by the ETH Board and hosted were used throughout the study. at the Swiss National Supercomputing Center (CSCS). MS was supported by the NCCR Synapsy grant of the Swiss National 4.10. Data Analysis and Statistical Tests Science Foundation. Partial support for PD was provided by the Analysis of the model and simulations was performed on a Linux GUDHI project, supported by an Advanced Investigator Grant computing-cluster using Python 2.7, including the numpy and of the European Research Council and hosted by INRIA. We scipy libraries (Jones et al., 2001), and custom Python scripts. We thank Eilif Muller for providing input on the analysis, Magdalena calculated p-values using Welch’s t-test (scipy.stats), unless noted Kedziorek for help with proving maximality in directed cliques, otherwise. Gard Spreemann for help with the analysis of the C. elegans connectome, and Taylor H. Newton for helpful discussions about AUTHOR CONTRIBUTIONS statistical methods. HM and RL developed and initially conceived the study over 10 years of discussions. HM, RL, and KH conceived and directed the SUPPLEMENTARY MATERIAL ﬁnal study. 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Available online at: http://www.jmlr.org/papers/v12/pedregosa11a. with accepted academic practice. No use, distribution or reproduction is permitted html which does not comply with these terms. Frontiers in Computational Neuroscience | www.frontiersin.org 16 June 2017 | Volume 11 | Article 48

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Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function

Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function

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Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function

Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function

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