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 unseen topology of synaptic connectivity. The synaptic network contains an abundance of cliques of neurons bound into cavities that guide the emergence of correlated activity. In response to stimuli, correlated activity binds synaptically connected neurons into functional cliques and cavities that evolve in a stereotypical sequence toward peak complexity. We propose that the brain processes stimuli by forming increasingly complex functional cliques and cavities.

Introduction

How the structure of a network determines its function is not well understood. For neural networks specifically, we lack a unifying mathematical framework to unambiguously describe the emergent behavior of the network in terms of its underlying structure. While graph theory has been used to analyze network topology with some success, current methods are usually constrained to analyzing how local connectivity influences local activity or global network dynamics, or how global network properties like connectivity and balance of excitatory and inhibitory neurons influence network dynamics. One such global network property is small-worldness. While it has been shown that small-worldness optimizes information exchange, and that adaptive rewiring during chaotic activity leads to small world networks, the degree of small-worldness cannot describe most local network properties, such as the different roles of individual neurons.

Algebraic topology offers the unique advantage of providing methods to describe quantitatively both local network properties and the global network properties that emerge from local structure, thus unifying both levels. More recently, algebraic topology has been applied to functional networks between brain regions using fMRI and between neurons using neural activity, but the underlying synaptic connections (structural network) were unknown. Furthermore, all formal topological analyses have overlooked the direction of information flow, since they analyzed only undirected graphs.

We developed a mathematical framework to analyze both the structural and the functional topology of the network, integrating local and global descriptions, enabling us to establish a clear relationship between them. We represent a network as a directed graph, with neurons as the vertices and the synaptic connections directed from pre- to postsynaptic neurons as the edges, which can be analyzed using elementary tools from algebraic topology. The structural graph contains all synaptic connections, while a functional graph is a sub-graph of the structural graph containing only those connections that are active within a specific time bin (i.e., in which a postsynaptic neuron fires within a short time of a presynaptic spike). The response to a stimulus can then be represented and studied as a time series of functional graphs.

Networks are often analyzed in terms of groups of nodes that are all-to-all connected, known as cliques. The number of neurons in a clique determines its size, or more formally, its dimension. In directed graphs it is natural to consider directed cliques, which are cliques containing a single source neuron and a single sink neuron and reflecting a specific motif of connectivity, wherein the flow of information through a group of neurons has an unambiguous direction. The manner in which directed cliques bind together can be represented geometrically. When directed cliques bind appropriately by sharing neurons, and without forming a larger 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 together. Directed cliques describe the flow of information in the network at the local level, while cavities provide a global measure of information flow in the whole network. Using these naturally arising structures, we established a direct relationship between the structural graph and the emergent flow of information in response to stimuli, as captured through time series of functional graphs.

We applied this framework to digital reconstructions of rat neocortical microcircuitry that closely resemble the biological tissue in terms of the numbers, types, and densities of neurons and their synaptic connectivity (a “microconnectome” model for a cortical microcircuit, Figures 1A,B; see Markram et al., 2015; Reimann et al., 2015). Simulations of the reconstructed microcircuitry reproduce multiple emergent electrical behaviors found experimentally in the neocortex. The microcircuit, formed by ~8 million connections (edges) between ~31,000 neurons (vertices), was reconstructed from experimental data, guided by biological principles of organization, and iteratively refined until validated against a battery of independent anatomical and physiological data obtained from experiments. Multiple instantiations of the reconstruction provide a statistical and biological range of microcircuits for analysis.

We found a remarkably high number and variety of high-dimensional directed cliques and cavities, which had not been seen before in neural networks, either biological or artificial, and in far greater numbers than those found in various null models of directed networks. Topological metrics reflecting the number of directed cliques and cavities not only distinguished the reconstructions from all null models, they also revealed subtle differences between reconstructions based on biological datasets from different animals, suggesting that individual variations in biological detail of neocortical microcircuits are reflected in the repertoire of directed cliques and cavities. When we simulated microcircuit activity in response to sensory stimuli, we observed that pairwise correlations in neuronal activity increased with the number and dimension of the directed cliques to which a pair of neurons belongs, indicating that the hierarchical structure of the network shapes a hierarchy of correlated activity. In fact, we found a hierarchy of correlated activity between neurons even within a single directed clique. During activity, many more high-dimensional directed cliques formed than would be expected from the number of active connections, further suggesting that correlated activity tends to bind neurons into high-dimensional active cliques.

Following a spatio-temporal stimulus to the network, we found that during correlated activity, active cliques form increasingly high-dimensional cavities (i.e., cavities formed by increasingly larger cliques). Moreover, we discovered that while different spatio-temporal stimuli applied to the same circuit and the same stimulus applied to different circuits produced different activity patterns, they all exhibited the same general evolution, where functional relationships among increasingly higher-dimensional cliques form and then disintegrate.

Results

Networks of neurons connected by electrical synapses (gap junctions) can be represented as undirected graphs, where information can flow in both directions. Networks with chemical synapses, which impose a single direction of synaptic communication from the pre- to the postsynaptic neuron (Figures 1B2,B3), are more accurately represented as directed graphs. Sub-sampling networks of neurons experimentally has revealed small motifs of synaptic connectivity, but not large cliques of neurons (Song et al., 2005; Perin et al., 2011). Knowing the complete directed network of neurons, as we do in the case of the reconstructed microcircuit, enables us to detect all cliques, directed, and otherwise (Figure 1).

When the direction of connections is not taken into account, a great deal of information is lost. For example, in the undirected case, there is only one possible configuration for a clique of four fully connected neurons (Figure 2A1, left). However, in the directed case, there are 36 = 729 possible configurations, as each of the six connections can be in one of three states (i → j, j ← i, or i ↔ j connection types; Figure 2A1 right).

A clique with reciprocal connections contains two or more cliques consisting only of uni-directional connections (Figure 2A2). When only uni-directional connections are considered, there are 26 possible configurations of four fully connected neurons, which are of two types: those that contain cycles (40 configurations; Figure 2A3 left; Section 4.1.3) and those that do not (24 configurations; Figure 2A3 right). Directed cliques are exactly the acyclic cliques. The net directionality of information flow through any motif can be defined as the sum over all neurons of the squares of the differences between their in-degree and their out-degree (see Equation 2, Figure S1). Directed cliques have the highest net directionality among all cliques (Figure S1; Section 4.1.4). A clique that contains cycles always decomposes into directed cliques with the same number of neurons or fewer, at the very least any single connection between two neurons forms a 2-clique. A cyclical clique of three neurons therefore decomposes into three 2-cliques. Following the conventions in algebraic topology, we refer to directed cliques of n neurons as directed simplices of dimension n-1 or directed (n-1)-simplices (which reflects their natural geometric representation as (n-1)-dimensional polyhedra) (see Figure S2; Section 4.1.3). Correspondingly, their sub-cliques are called sub-simplices.

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https://www.frontiersin.org/articles/10.3389/fncom.2017.00048/full 

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