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

Abstract

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.

Keywords: Betti numbers; connectomics; correlations; directed networks; structure-function; topology.

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.

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).

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 (Δt = 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 (Δt = 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.

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 (β₅) 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.

Figure 6

(A) Number of edges, β₁, β₃, and Euler characteristic of the time series of TR graphs in response to the stimulus patterns shown in Figure 4 (mean and SEM of 30 repetitions of each stimulus). (B) Trace of the time series of β₁ against β₃ for three of the stimuli. Shading of colors indicates Gaussian profiles at each time 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.