Emergence of metastability in frustrated oscillatory networks: the key role of hierarchical modularity

Abstract

Oscillatory complex networks in the metastable regime have been used to study the emergence of integrated and segregated activity in the brain, which are hypothesised to be fundamental for cognition. Yet, the parameters and the underlying mechanisms necessary to achieve the metastable regime are hard to identify, often relying on maximising the correlation with empirical functional connectivity dynamics. Here, we propose and show that the brain's hierarchically modular mesoscale structure alone can give rise to robust metastable dynamics and (metastable) chimera states in the presence of phase frustration. We construct unweighted 3-layer hierarchical networks of identical Kuramoto-Sakaguchi oscillators, parameterized by the average degree of the network and a structural parameter determining the ratio of connections between and within blocks in the upper two layers. Together, these parameters affect the characteristic timescales of the system. Away from the critical synchronization point, we detect the emergence of metastable states in the lowest hierarchical layer coexisting with chimera and metastable states in the upper layers. Using the Laplacian renormalization group flow approach, we uncover two distinct pathways towards achieving the metastable regimes detected in these distinct layers. In the upper layers, we show how the symmetry-breaking states depend on the slow eigenmodes of the system. In the lowest layer instead, metastable dynamics can be achieved as the separation of timescales between layers reaches a critical threshold. Our results show an explicit relationship between metastability, chimera states, and the eigenmodes of the system, bridging the gap between harmonic based studies of empirical data and oscillatory models.

Keywords: chimera states; criticality; hierarchical modularity; metastability; network physiology; oscillations; renormalisation; whole-brain.

FIGURE 1

Example of a 3-layer hierarchical network. (A) At each layer $l$ , we define a SBM with $B_l$ blocks containing $n_l$ blocks from the layer below (or nodes if $l=1$ ). The last layer is fixed to be a single block, $B_3=1$ , containing $n_3=2$ blocks from layer 2. In this example in layer 1 (blue blocks ${\mu }_i$ ) we have $B_1=8,n_1=2$ , in layer 2 (green blocks ${\rho }_i$ ) we have $B_2=2,n_2=4$ , in layer 3 (red block) we have $B_3=1,n_3=2$ . (B) Corresponding partition matrix. (C) Using our constraints, the 3-layer hierarchical network is fully parametrized by the connection probabilities $p_i,i=\mathrm{1,2,3}$ (blue, green, red).

FIGURE 2

(top) Example adjacency matrices for $H=0.0$ (A) $H=0.5$ (B) and $H=1$ (C). Each network was generated using $n_1=16,n_2=8$ and exemplifying average degree $k=51.2$ . Note how for $H=0.0$ the generated network is similar to a SBM while for $H=1.0$ the two populations are decoupled. (bottom) Modularity index $Q_l$ calculated over partitions $\{P_{l,i}|i\in \mathcal{V}\}$ for $l=1$ (D) and $l=2$ (E) as a function of $H$ and $k$ .

FIGURE 3

(A) Degree of synchronization, $R$ (blue), and index of metastability, ${\sigma }_{\text{met}}^3$ (purple), in the $l=3$ (whole-system) layer as a function of the structural parameter $H$ . (B) Degree of synchronization $R$ as a function of $H$ and the average degree of the network $k$ . (C) Index of metastability ${\sigma }_{\text{met}}^3$ as a function of $H$ and $k$ . Results were obtained using $n_1=16$ , $n_2=8$ , and $k=51.2$ , averaged over 100 seeds.

FIGURE 4

(A) Mean degree of synchronization $R_{{\rho }_i},i=\mathrm{1,2}$ for varying values of $H$ . (B) Populations’ metastability index, ${\sigma }_{\text{met}}^2$ , as a function of $H$ . (C) (Blue line) mean difference between the populations’ local KOP $\overline{d}$ and (orange line) standard deviation of the difference between local KOPs, $\sigma (d)$ as a function of $H$ . Red dashed lines separate the regions of the parameter space in which symmetry-breaking states are detected. $(i-iv)$ Examples of the evolution in time of the populations’ local KOPs and the whole-system KOP (black dashed lines) in distinct dynamical regimes. $(i)$ For low values of $H$ the system displays coherent dynamics and the two populations display the same degree of synchronization. $(ii,iii)$ For intermediate values of $H$ stable and breathing chimera states emerge, respectively. $(iv)$ Metastable chimera states emerge for $H=0.5$ . Results obtained using $n_1=16$ , $n_2=8$ , and $k=51.2$ , averaged over 100 seeds.

FIGURE 5

(A) Left: Metastability index ${\sigma }_{\text{met}}^1$ calculated over all modules ${\mu }_i$ as a function of $H$ and $k$ . Right: example of $R_{{\mu }_i}$ dynamics in time for a single simulation with average degree $k=51.2$ (top) and $k=21$ (bottom). (B) Average local KOP within each population $j=\mathrm{1,2}$ , i.e., $⟨R_{{\mu }_i}⟩$ for ${\mu }_i\subset {\rho }_j$ , for fixed $k=51.2$ and varying values of $H$ . (C) Metastability index within each population $j=\mathrm{1,2}$ , i.e., ${\sigma }_{\text{met}}(R_{{\mu }_i})$ , for ${\mu }_i\subset {\rho }_j$ , for fixed $k=51.2$ and varying values of $H$ , and overall metastability of the first layer blocks ${\sigma }_{\text{met}}^1$ (black dashed line). Red dashed lines separate the regions of the parameter space in which symmetry-breaking states are detected in layer 2.

FIGURE 6

Gaps between consecutive eigenvalues ${\lambda }_2,{\lambda }_3,\dots {\lambda }_N$ are affected by the structural parameter $H$ and the average degree $k$ . In this model, increasing $k$ reduces the size of the second spectral gap, between ${\lambda }_{B_1}$ and ${\lambda }_{B_1+1}$ (where $B_1=16$ is the number of layer 1 blocks). On the other hand, increasing $H$ increases the size of the first spectral gap (inset figure), between ${\lambda }_{B_2}$ and ${\lambda }_{B_2+1}$ (where $B_2=2$ is the number of layer 2 blocks). Each eigenvalue was obtained by averaging over 10 randomly generated 3-layer networks with $n_1=16,n_2=8$ .

FIGURE 7

(A) Left: example of populations’ dynamics in a stable chimera state in layer 2, where $A$ was obtained using the 3-layer variation of the nSBM with $H=0.2$ , $n_1=16,n_2=8$ , and $k=21$ . Right: the weighted coarse-grained adjacency matrix $A^{\prime }$ is obtained by time-rescaling the Laplacian associated with the original adjacency matrix $A$ . The symmetry breaking parameter $a^{\prime }=0.23$ , obtained by computing the normalized difference between intra-population $K_1$ (red squares) and inter-population $K_2$ couplings of the coarse-grained adjacency matrix $A^{\prime }$ , is also associated with stable chimera states in the two-population model (see (Abrams et al., 2008) and results therein). (B) The symmetry breaking parameter $a^{\prime }$ for varying $H$ and six exemplifying average degrees $k$ . (C) Log-log plot displaying the cutoff point when metastability in layer 1, ${\sigma }_{\text{met}}^1$ , starts to sharply decrease as the size of the second spectral gap, ${\lambda }_{B_1+1}-{\lambda }_{B_1}$ , increases.

FIGURE 8

Structural perturbation analysis for fixed $H=0.5$ and for $k=51.2$ (top row) and $k=21$ (bottom row). In each layer $l$ , we compute the average degree of synchronization $R_l$ of all blocks in the layer (left column) and the metastability index ${\sigma }_{\text{met}}^l$ (middle column). The modularity index $Q_l$ is computed with respect to the partitions in layer 1 and 2 (right column). Results were obtained for systems of size $n_1=16,n_2=8$ and averaged over 100 seeds.

FIGURE 9

Impact of heterogeneous frequencies with increasing $\delta \omega$ for $H=0.5$ and for $k=51.2$ (top row) and $k=21$ (bottom row). In each layer $l$ , we compute the average degree of synchronization $R_l$ of all blocks in layer $l$ (left column) and the metastability index ${\sigma }_{\text{met}}^l$ (right column). Results were obtained for systems of size $n_1=16,n_2=8$ , using normalized coupling $K/k$ with $K=50$ , and averaged over 100 seeds.