Topological analysis of the connectome of digital reconstructions of neural microcircuits

1. Structural topology

• Each microcircuit’s binary adjacency matrix was converted into a directed flag complex, encoding directed cliques as oriented simplices and their directed subcliques as faces.The 1 in structural-matrix entry (j, k) denotes a directed connection from neuron GID j to neuron GID k; the resulting complex is called an N-complex.Source: §1 · pp. 4–5
• Comparisons used five ER graphs and layer-, m-type-, and Peters’ Rule controls; layer and m-type randomizations preserved category-pair connectivity and 25 µm distance-bin counts.The ER graphs had 31,000 vertices and 0.8% connection probability; the biological controls included 20 partly randomized matrices and 10 Peters’ Rule matrices.Source: §1 · p. 5
• N-complexes had many more simplices and reached significantly higher dimensions than ER-, L-, M-, and P-complexes.Euler characteristics were large and positive in all cases, reflecting the predominance of even-dimensional simplices, particularly 2-dimensional ones.Source: §1 · p. 5
• β5 ranged from 1 to 80 in N-complexes, while βn=0 above dimension 3 in ER/P and above dimension 4 in L/M complexes.For L- and M-complexes, β4 ranged from 0 to 6 and was described as almost negligible; Betti computations were restricted to 5th and 6th coskeleta because full computation was practically infeasible.Source: §1 · p. 6

2. Functional topology

• The test compared point- and circle-shaped stimuli with essentially the same average firing rate, each delivered synchronously at 20 Hz for one second across 20 trials.The analysis aimed to determine whether topological metrics classified the stimuli better than firing rate.Source: §2 · pp. 6–7
• Activity was divided into 5 ms bins, with a transmission-response edge when a structural connection linked a neuron spiking in the bin to a neuron spiking within 7.5 ms afterward.The study classified trials per time bin using topological metrics and compared them with mean firing rates from three adjacent 5 ms intervals.Source: §2 · p. 7
• At 0–5 ms, 1D/2D simplex counts and β1/β2 classified well; at 5–10 ms, 2D count and β2 remained strong and Euler characteristic classified even better.These findings concern the first two time bins after the initial stimulation.Source: §2 · p. 7
• At 10–50 ms and 50–55 ms, metrics did not classify well; at 50–55 ms, 2D simplex count and β2 had the highest success rate, while at 55–60 ms they alone classified very well and topological metrics outperformed firing rate.Source: §2 · pp. 7–8

• Directed neural connectivity graphs differed significantly from both Erdős-Rényi and biologically constrained, distance-dependent random graphs, while containing numerous high-dimensional simplices of all-to-all connectivity.Some of these simplices represent the most extreme circuit motifs reported for brain tissue.Source: §3 · p. 8
• Topological methods effectively distinguished functional responses to distinct thalamic stimuli, introducing a measure of neural tissue’s spatio-temporal activity responses.Source: §3 · p. 8
• The biological significance of discovered simplices and homology classes remains an open question; the authors hypothesize that topological time series could capture a spatio-temporal code beyond rate or timing and reflect structural constraints on functional states.They suggest age-dependent reconstructions and synaptic-plasticity changes may reveal structures related to development, learning, and memory.Source: §3 · pp. 9–10
• The authors expect their directed-graph approach to apply beyond neuroscience, including gene and protein networks, VLSI circuits, and electrical grids, and note it may motivate random-graph results analogous to Kahle’s work on undirected flag complexes.Source: §3 · p. 10

4.1. Computation of flag complexes and their Betti numbers.

• Flag complexes were stored with a reference-based structure requiring space linear in vertices and edges, and homology used F2 coefficients with PHAT boundary-matrix reduction.The authors note that a public C++ implementation would be available.Source: §4.1 · p. 10
• Point and circle stimuli used synchronous spikes in 46 central or 56 peripheral VPM fibers, calibrated for similar mean firing rates and repeated every 50 ms.The microcircuit had 310 evenly distributed VPM fibers, enabling topographically distinct stimulation by selecting subsets.Source: §4.2 · p. 10

4.2. The Point vs. Circle experiment.

• A Gaussian naïve Bayes classifier was evaluated over 500 randomized trials, training on 15 trials and testing on five per stimulus, with firing-rate classification using three consecutive time bins.The three-bin comparison was used to account fairly for edge information spanning more than two time bins over a 12.5 ms range.Source: §4.2 · p. 10

4.3. Computation of transmission-response matrices.

• Transmission-response matrices were computed with a custom Python program combining the structural synaptic-connection matrix with simulated spiking output and time steps ∆t1 and ∆t2 of 5 and 7.5 ms.Further details are given in the supplementary methods.Source: §4.3 · pp. 10–11

4.4. Gaussian Bayes classifiers.

• The Gaussian Bayes classifier modeled dot and circle classes as Gaussian distributions from randomly split training data, assuming a uniform prior.The method minimizes misclassification probability under the Gaussian-distribution assumption.Source: §4.4 · p. 11

Supplementary Methods

• Each binary matrix marks structurally connected neuron pairs when a presynaptic spike in a ∆t1 bin is followed by a postsynaptic spike within ∆t2; 49 parameter pairs were tested on spontaneous activity.The study generated sequences of 20 matrices for each pair, with both intervals drawn from {1, 2, 5, 10, 20, 50, 100} ms.Source: Supplementary Methods · p. 12

SM.1.3. Null hypotheses: no neural processing.

• The null model ignores pre-postsynaptic interaction; shuffled spikes preserve each neuron’s and time bin’s spike counts, and the true and simplified matrices are compared by their ratio of ones.Maximizing the difference from this control is intended to identify intervals that best capture pre-postsynaptic interaction.Source: SM1.1. Properties of the… · p. 13, SM.1.3. Null hypotheses: no neural… · p. 15, SM.1.4. Optimization of parameters · p. 15

SM1.2. Stochastic model with neural processing.

• The stochastic model treats each neuron’s spiking as an inhomogeneous Poisson process and models transmission probability as depending on firing rates, presynaptic spiking history, and synaptic interaction.The authors assume true transmission-response matrices capture spike transmission according to this model.Source: SM1.2. Stochastic model with neural… · pp. 13–14

SM.1.4. Optimization of parameters.

• The selected intervals were ∆t2 = 7.5 ms and ∆t1 = 5 ms, despite a ∆t1 maximum at 50 ms, to avoid multiple spikes per neuron per bin.Across tested cases, the maximum lay between ∆t2 = 5 ms and ∆t2 = 10 ms; the ratio maximum at ∆t1 = 5 ms was only slightly lower than at 50 ms.Source: SM.1.4. Optimization of parameters · p. 16

SM2. Gaussian Bayes classifiers

• The Gaussian Bayes classifier estimates each class’s Gaussian distribution from sample means and variances and selects the most likely class, assuming a uniform prior in the described analysis.The Gaussian class-distribution assumption is the model used to estimate the otherwise unknown distribution from sample data.Source: SM2. Gaussian Bayes classifiers · p. 16

SM3. Randomization of connection matrices and other control cases

• Connectivity controls included random binary matrices at the reconstruction’s 0.8% connection probability and randomizations preserving layer-to-layer and m-type connectivity patterns.The layer-based method preserves connection counts and distance-dependent probabilities between layer pairs within 25 µm; the analogous m-type method partitions neurons into 55 morphological groups.Source: SM3. Randomization of connection… · p. 17, SM3.1. Generation of Erd˝os-R´enyi… · p. 17

SM3.4. Generation of connection matrices according to Peters’ Rule.

• The Peters’ Rule control began with connections between neurons whose arbors came within 3 µm, then randomly pruned them to the structural matrix’s connection count.The initial matrix had approximately 16 times more connections than the structural matrix.Source: SM3.4. Generation of connection… · p. 17

ST1. The topological toolbox

• The topological toolbox associates directed graphs with directed flag complexes and computes Euler characteristics and Betti numbers using implemented data structures and algorithms.The account introduces these methods for non-experts and notes their directed variants.Source: ST1. The topological toolbox · p. 18

ST1.2. Simplicial complexes.

• An abstract simplicial complex is closed under taking subsets; a directed flag complex consists of ordered vertex tuples with all forward edges present, realized geometrically as cliques.Its n-skeleton contains simplices through dimension n, while its k-coskeleton includes higher-dimensional simplices and their faces.Source: ST1.2. Simplicial complexes · p. 19

ST1.3. Homology, Betti numbers, and Euler characteristic.

• Mod-2 Betti numbers count n-dimensional holes and, for directed flag complexes, graph metamotifs; coskeleton invariance lets top-dimensional computations omit lower simplices that are not faces of higher ones.The paper uses F2 homology for computational simplicity, while noting that other homology variants may provide deeper information.Source: ST1.3. Homology, Betti numbers, and… · p. 20
• A simplicial complex is encoded by a stratified, oriented Hasse diagram whose levels correspond to simplex dimensions and whose edges link each simplex to its codimension-one faces.The Euler characteristic is preserved under isomorphisms and agrees between a simplicial complex and its associated stratified Hasse diagram.Source: ST1.3. Homology, Betti numbers, and… · pp. 21–22

ST2. Data structures and algorithms

• The Hasse-diagram data structure requires O((Sint·d)|V| + |E|·Sint) space, but the directed flag complex can grow exponentially, so construction may be stopped at a fixed dimension.Algorithm time is proportional to output-complex size times its maximal level.Source: ST2. Data structures and algorithms · pp. 22, 24
• Algorithm 1 iteratively extends simplices with vertices connected from every existing vertex, ensuring the generated complex has the required directed orientations.The method assumes the input graph is represented as an admissible Hasse diagram; homology computations use F2 coefficients and a boundary-matrix reduction algorithm from PHAT.Source: ST2. Data structures and algorithms · pp. 23–24, ST2.4. Homology and Betti numbers · p. 24