  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 {\DH} 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.
