Communities, hubs and rich-club organisation

Background & rationale

Modules in neuronal co-activity graphs correspond to cell assemblies, groups of neurons that fire together (Hebb 1949; Carrillo-Reid et al. 2016). Within modular networks, hub nodes can be classified by their within-module degree z and participation coefficient P into provincial and connector hubs (Guimerà & Amaral 2005). Rich-club organisation, densely interconnected high-degree nodes, is a recurrent feature of neural networks (van den Heuvel & Sporns 2011) and has been reported in cortical microcircuits (Nigam et al. 2016). Stress-related hippocampal network reorganisation is driven disproportionately by highly active neurons (Transl Psychiatry 2022), which motivates asking whether hubs change under defeat. Day-to-day stability of modules connects to work on representational drift (Ziv et al. 2013).

Methodology

Graphs. 10%-density proportional graphs per session (as in S04).

Modules. Louvain community detection (resolution 1, fixed seed). Modularity was compared with circular-shift surrogates at equal density (Q − null Q, from S04).

Cartography. For every neuron: within-module degree z and participation coefficient P (Guimerà & Amaral 2005). Hubs are neurons in the top 10% of degree. Of these, connector hubs have P > 0.3 and provincial hubs P ≤ 0.3. Session summaries: mean P of hubs, fraction of hubs that are connectors.

Rich club. Rich-club coefficient φ(k) was normalised by the mean of 10 degree-preserving rewired graphs (double-edge swaps). The session index is the mean normalised φ over k in the 70th–90th percentile of the degree distribution (> 1 indicates a rich club).

Stability. Because the same neurons are tracked across sessions, partitions are directly comparable. - Module stability: adjusted Rand index (ARI) between Louvain partitions of consecutive same-condition sessions (e.g. SD day 1 vs SD day 2), compared with ARI after permuting neuron labels (200 permutations). - Hub persistence: Jaccard overlap of hub sets between consecutive same-condition sessions vs the expectation for random sets of equal size.

Statistics. Per-animal means over days that contain all of BL, SD and SS (days 1–9 without day 7) were compared with Friedman and exact Wilcoxon tests, FDR-corrected across metrics × contrasts. Module and hub stability against their nulls were tested on per-animal means with one-sided exact Wilcoxon tests.

Parameters

Results

Modular organisation. At 10% density, networks split into 7 non-trivial modules per session (median). Modularity exceeded circular-shift surrogates by 0.049 on average, indicating co-activity modules beyond what each neuron's own temporal statistics produce.

Hubs and rich club. Hubs (top 10% degree) had mean participation P = 0.38, and 70% of them were connector hubs. The normalised rich-club index averaged 1.33. Condition contrasts are in Table 1 and Figure 2. 0 of 12 tests survived FDR correction.

Stability across days. Partitions of consecutive same-condition sessions agreed above chance (mean ARI 0.337 vs -0.000; p = 0.016). Hub sets overlapped more than random sets (Jaccard 0.20 vs 0.06; p = 0.016). Module stability differed between conditions with Friedman p = 0.223 (Figure 4).

Limitations

• Louvain is stochastic. A single seed was used per session; consensus clustering would reduce variability.
• BL, SD and SS are recorded in a fixed order within each day, so condition contrasts are confounded with position in the recording.
• The hub definition (top 10% degree, P > 0.3) follows common practice but is a choice. The direction of effects should be checked at other cut-offs.
• Rich-club normalisation with 10 rewired graphs per session gives coarse estimates.

References

1. Guimerà R. & Amaral L.A.N. (2005) Functional cartography of complex metabolic networks. Nature 433:895–900.
2. van den Heuvel M.P. & Sporns O. (2011) Rich-club organization of the human connectome. J Neurosci 31:15775–15786.
3. Colizza V. et al. (2006) Detecting rich-club ordering in complex networks. Nat Phys 2:110–115.
4. Nigam S. et al. (2016) Rich-club organization in effective connectivity among cortical neurons. J Neurosci 36:670–684.
5. Carrillo-Reid L. et al. (2016) Imprinting and recalling cortical ensembles. Science 353:691–694.
6. Ziv Y. et al. (2013) Long-term dynamics of CA1 hippocampal place codes. Nat Neurosci 16:264–266.
7. Hubert L. & Arabie P. (1985) Comparing partitions. J Classif 2:193–218.