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Topological analysis of the connectome of digital reconstructions of neural microcircuits
Pawe Dotko, Kathryn Hess, Ran Levi, Max Nolte, Michael Reimann, Martina Scolamiero, Katharine Turner, Eilif Muller, Henry Markram
TL;DR
Traditional graph methods may not fully capture the complexity of a large biological connectome, motivating an algebraic-topological analysis of structural and functional organization. The study constructs directed flag complexes and transmission-response graphs, finding distinctive high-dimensional structure and effective stimulus classification. It presents topology as a useful tool for analyzing structural and functional neural connectomes, while leaving the biological significance of the discovered structures and homology classes for future work.
Problem
The study asks whether algebraic topology can provide a useful perspective on the structural and functional organization of a highly complex biological neural network.
Method
The authors analyze 42 reconstructed microcircuits with directed flag complexes and simulated-activity transmission-response graphs, measuring simplices, Betti numbers, Euler characteristic, and classification performance.
Results
The reconstructed microcircuits differed significantly from randomized graphs, contained many higher-dimensional simplices, and topological metrics effectively distinguished Point and Circle responses.
Takeaways & Limitations
Topology is supported as a useful mathematical tool for analyzing structural and functional connectomes of neural circuits.
Takeaways & Limitations
The biological significance of the discovered high-dimensional simplices and homology classes remains an open question for future studies.
Abstract
from arXiv · showhide
A recent publication provides the network graph for a neocortical microcircuit comprising 8 million connections between 31,000 neurons (H. Markram, et al., Reconstruction and simulation of neocortical microcircuitry, Cell, 163 (2015) no. 2, 456-492). Since traditional graph-theoretical methods may not be sufficient to understand the immense complexity of such a biological network, we explored whether methods from algebraic topology could provide a new perspective on its structural and functional organization. Structural topological analysis revealed that directed graphs representing connectivity among neurons in the microcircuit deviated significantly from different varieties of randomized graph. In particular, the directed graphs contained in the order of $10^7$ simplices Ð groups of neurons with all-to-all directed connectivity. Some of these simplices contained up to 8 neurons, making them the most extreme neuronal clustering motif ever reported. Functional topological analysis of simulated neuronal activity in the microcircuit revealed novel spatio-temporal metrics that provide an effective classification of functional responses to qualitatively different stimuli. This study represents the first algebraic topological analysis of structural connectomics and connectomics-based spatio-temporal activity in a biologically realistic neural microcircuit. The methods used in the study show promise for more general applications in network science.
1. Structural topology
The study represents reconstructed microcircuit connectivity as directed flag complexes and compares its simplex structure and homological organization with randomized controls. Biological complexes contain more and higher-dimensional simplices, while their highest nonzero Betti number occurs in dimension 5.
- Directed flag complexes: 42 reconstructed microcircuits were converted into directed flag complexes, with directed cliques represented as oriented simplices.Each vertex corresponds to a neuron, and each directed connection is encoded in the structural adjacency matrix.
- Randomized controls: 31,000 vertices and 0.8% connection probability defined the Erdős-Rényi comparison complexes.Additional controls preserved distance-dependent connection counts across cortical layers or morphological neuron types while randomizing otherwise.
- Simplex distributions: N-complexes contained many more simplices, including significantly higher-dimensional simplices, than the various random complexes.Their Euler characteristics were large and positive because even-dimensional simplices, particularly 2-dimensional ones, predominated.
- Homological organization: Betti numbers measure chains of intersecting directed cliques that form higher-dimensional holes, providing a finer complexity measure than simplex counts or Euler characteristic.The n-th Betti number counts chains of simplices intersecting along faces to create an n-dimensional hole.
- Homological organization: β5 was nonzero in every N-complex, varying from 1 to 80, whereas higher Betti numbers vanished above dimension 3 or 4 in the random complexes.Computing these Betti numbers required restricting attention to the 5th and 6th coskeleta because of the complexes’ size.
2. Functional topology
The functional analysis tested whether topology could distinguish Point and Circle stimuli in simulated microcircuit activity beyond mean firing rate. Transmission-response graphs and their time-varying metrics classified the stimuli most effectively in selected post-stimulation bins, especially using 2-dimensional simplex counts and β2.
- Stimulus experiment: Twenty trials each of Point and Circle stimuli were simulated with synchronous 20 Hz thalamo-cortical activation for one second.The point and circle patterns were chosen to produce essentially the same average firing rate.
- Classification results: Between 10 ms and 50 ms after initial stimulation, none of the metrics classified the stimuli very well because activity propagation was strongly similar.This period contrasted with selected early and post-second-stimulation bins where topological metrics performed well.
- Transmission-response graphs: Transmission-response edges required a physical connection, presynaptic firing in the time bin, and postsynaptic firing within 7.5 ms.The resulting matrices were interpreted as capturing likely influence of neuron j on neuron k under these conditions.
- Classification results: The 2-dimensional simplex count and β2 classified very well across the key bins and, in the 55–60 ms bin, were the only metrics to do so.In these cases, topological metrics far outperformed mean firing rate.
3. Discussion
The study argues that topology can analyze both structural and functional connectomes, revealing abundant high-dimensional connectivity motifs and distinguishing responses to distinct stimuli. It also identifies open questions about the biological significance of these structures and anticipates applications beyond neuroscience.
- 3. Discussion: Topological methods revealed high-dimensional simplices representing all-to-all neuronal connectivity, described as the most extreme circuit motifs reported for brain tissue.The analysis also found a surprisingly large number of these structures.
- 3. Discussion: Topological metrics effectively distinguished functional responses to distinct thalamic stimuli and introduced a spatio-temporal measure of neural activity responses.The authors present this as a new measure based on functional connectivity dynamics.
- 3. Discussion: The results provide a convincing argument for using topology to analyze structural and functional connectomes of neural circuits.
- 3. Discussion: The biological significance of the discovered high-dimensional simplices and homology classes remains an open question for future studies.
- 3. Discussion: The directed-graph topological approach may also apply to networks with intricate directed connectivity, including gene and protein networks, VLSI circuits, and electrical grids.
4. Materials and methods
The methods combine a reference-based representation of directed flag complexes with simulated stimulation experiments and Gaussian naïve Bayes classification. Structural matrices are combined with simulated spike outputs to construct transmission-response analyses.
- 4. Materials and methods: The directed flag complex uses vectors storing references to simplices, with storage growing linearly in the numbers of vertices and edges.The paper reports a publicly available C++ implementation and homology computations using F2 coefficients.
- 4. Materials and methods: The microcircuit receives input through 310 VPM fibers, enabling topographically distinct point and circle stimuli.The stimuli were calibrated by adjusting fiber counts to evoke similar overall mean firing rates.
- 4. Materials and methods: Gaussian naïve Bayes classification used 500 random train-test resampling trials, with 15 trials per stimulus for training and five for testing.Firing-rate classification used three consecutive time bins for a fairer comparison.
- 4. Materials and methods: Transmission-response matrices combined the BBP structural synaptic-connectivity matrix with spiking simulation outputs and user-defined time steps.The analyses used Δt1 and Δt2 values of 5 and 7.5 ms.
- 4. Materials and methods: The Gaussian Bayes classifier models the dot and circle classes as Gaussian distributions and minimizes misclassification probability under that assumption.The procedure uses randomly split training and testing data and assumes a uniform prior.
Supplementary Methods
The supplementary methods define transmission-response matrices from structural connectivity and spike timing, then optimize temporal parameters using a stochastic firing model. They also describe directed flag complexes and mod-2 homology as the topological framework for analyzing these matrices and reconstructed microcircuits.
- Transmission-response matrices: A transmission-response matrix marks a structural connection when the presynaptic neuron spikes in a time bin and the postsynaptic neuron spikes within Δt2.The construction uses three conditions: a structural edge, presynaptic firing during the Δt1 bin, and postsynaptic firing within the following Δt2 interval.
- Transmission-response matrices: 20 transmission-response matrix sequences were generated for each of 49 pairs of Δt1 and Δt2 values, with each interval selected from {1, 2, 5, 10, 20, 50, 100} ms.The sequences were generated from spontaneous activity in the reconstructed microcircuit.
- Transmission-response matrices: The true transmission-response matrices vary across repeated trials because the in silico microcircuit is partly stochastic, even when the same stimulus conditions are repeated.Different random-number-generator seeds can produce different transmission-response matrices.
- Parameter optimization: The stochastic firing model represents each neuron's spiking with a time-dependent instantaneous firing rate and incorporates neural processing into the estimated transmission probability.A simplified model that ignores neural processing is used for comparison after shuffling firing data.
- Parameter optimization: The selected parameters were Δt2 = 7.5 ms and Δt1 = 5 ms, although the ratio maximum occurred at Δt1 = 50 ms and between Δt2 = 5 ms and Δt2 = 10 ms.The shorter Δt1 was chosen because its ratio was only slightly lower and avoids more than one spike per neuron per bin.
6. Supplementary Figures
The supplementary figures illustrate transmission-response parameter sweeps and basic directed-graph and Hasse-diagram concepts used in the topological analysis.
- Transmission-response matrices: Figure S1 varies ∆t1 and ∆t2 to show the average number of ones in true transmission-response matrices during spontaneous activity.The simulation uses in-vivo-like spontaneous activity (Ca 1.2).
- Transmission-response matrices: Figure S2 compares true and randomized transmission-response matrices using the ratio of their average numbers of ones across ∆t1 and ∆t2.The analysis averages matrices over 20 time steps, while shuffling firing-neuron identities preserves spike counts.
- Directed graphs: Figures S3A–C distinguish directed-graph isomorphism using sources and sinks: A and B are isomorphic, whereas C has neither source nor sink.The displayed isomorphism maps a, b, c, d to 1, 2, 3, 4, respectively.
- Hasse diagrams: Figure S4 contrasts a non-stratified Hasse diagram with a stratified admissible one whose levels are 0, 1, and 2.The example labels front, neither, and back faces relative to vertex 1.