Dynamical Reconnection and Stability Constraints on Cortical Network Architecture

Stability under dynamical changes to network connectivity is invoked alongside previous criteria to constrain brain network architecture. A new hierarchical network is introduced that satisfies all these constraints, unlike more commonly studied regular, random, and small-world networks. It is shown that hierarchical networks can simultaneously have high clustering, short path lengths, and low wiring costs, while being robustly stable under large scale reconnection of substructures.

Figure 1

Schematic connection matrices (CMs) of networks, with neural populations labeling rows and columns, and white entries for a connection between a given row and column. (a) Regular network. (b) Random network. (c) Small-world network. (d) Hierarchical network (HN) shaded to show levels. (e) Example HN. (f) Cat cortex CM adapted from [5].

Figure 2

(a) Clustering coefficient $C$ (shading) and path length $L$ (contours) vs $p_c$ and $q$ for HN with $n=64$, $m=16$, $s=4$, $p_m=1$, $N_c=⟨k⟩n$ [cf. Eq. (3)]. The network becomes disconnected ($L\to \infty$) slightly below the $L=2.5$ contour. (b) As for (a) but for $m=4$ and $s=16$.

Figure 3

HN properties for $n=512$, $m=128$, $s=4$, $p_m=1$, $p_c=1$, $q=0.22$, $N_c=⟨k⟩n$ from Eq. (3). (a) $\lambda$ spectrum. (b) Distribution of $\mathrm{Re}\lambda$ from an ensemble of HNs. (c) $P_s$ vs $g$.

Figure 4

Principal connection matrix eigenvalue ${\lambda }_1$ vs network size $n$ for RN (▵) $p=0.34$, $\mathrm{SW}(+)$ $p=0.25$ off diagonal with a regular spine $k=4$, regular (○) $k=11$ and HN (·) with $p_m=1$, $p_c=1$, $q=0.22$, $s=4$, $N_c=⟨k⟩n$.