Metastable brain waves

Traveling patterns of neuronal activity—brain waves—have been observed across a breadth of neuronal recordings, states of awareness, and species, but their emergence in the human brain lacks a firm understanding. Here we analyze the complex nonlinear dynamics that emerge from modeling large-scale spontaneous neural activity on a whole-brain network derived from human tractography. We find a rich array of three-dimensional wave patterns, including traveling waves, spiral waves, sources, and sinks. These patterns are metastable, such that multiple spatiotemporal wave patterns are visited in sequence. Transitions between states correspond to reconfigurations of underlying phase flows, characterized by nonlinear instabilities. These metastable dynamics accord with empirical data from multiple imaging modalities, including electrical waves in cortical tissue, sequential spatiotemporal patterns in resting-state MEG data, and large-scale waves in human electrocorticography. By moving the study of functional networks from a spatially static to an inherently dynamic (wave-like) frame, our work unifies apparently diverse phenomena across functional neuroimaging modalities and makes specific predictions for further experimentation.


Coherence cross-correlation partitions
It turns out there is specific information in the interhemispheric partition (Supplementary Figure 3): front-back and top-bottom partitions are more similar to each other than they are to the left-right partition, possibly reflecting the unique 'gating role' of the corpus callosum. Thus some transitions occur across multiple partitions, while other partitions are sensitive to transitions with specific spatial (or topological) structure. This suggests a hierarchy of transitions: transitions for which the choice of spatial (or topological) partition matters, and more dramatic transitions that are evident in all partitions. Future work is required to catalog these possibilities.

Robustness across connectomes
We verified that the existence of wave patterns reproduces across connectomic data (Supplementary Movie 9). Waves similar to those in our fully-connected weighted connectome exist also in sparser networks (thresholded down to 10% density), and occur whether using traditional weight-based thresholding 1 or consistency-based thresholding 2 (mean nodal mean speeds 27 m s -1 and 29 m s -1 , respectively). Waves exist also on a 10%density binary network thresholded by weight (with all weights set to 1; mean nodal mean speed 33 m s -1 ). This is consistent with the lattice-like formation of binary networks, such that shorter connections are much more probable than longer connections. This is a natural substrate to support the increased local synchrony required for wave patterns. We also observed waves arising on two entirely independent connectomes: an elderly connectome derived using probabilistic tractography 3 , and the 998-node Hagmann et al. connectome derived using deterministic tractography from diffusion spectrum imaging 4 . Notably, waves arise on the connectomes from individual subjects, with similar speeds (subject mean±SD = 33±2 m s -1 ) and similar dwell times (subject mean±SD = 92±8 ms) across the cohort (Supplementary Figure 5a,b). These distributions are fairly narrow, but there is some variability due to the individual-subject connectomes. Future work will be needed to determine how the details of the patterns vary across subjects as well as the impact on wave properties of connectomic disturbances in brain disorders.

Robustness to initial conditions
We also verified robustness to initial conditions. Using an ensemble of 100 random initial conditions and the group average connectome with = 0.6, = 1 ms, we observed waves with little variability across the ensemble in the nodal mean speeds (ensemble mean±SD = 35.3±0.3 m s -1 ) and dwell times (ensemble mean±SD = 89±2 ms) (Supplementary Figure  5c,d). Thus the choice of initial conditions only makes a small contribution to the summary statistics of these waves.

Robustness to noise
We verified that wave patterns persist in the presence of weak noise (Supplementary Movie 10). These wave patterns have lower local synchrony than the noise-free case but also exhibit metastability (Supplementary Figure 6). Interestingly, weak noise also increases the global synchrony and dwell times, similar to the effects of stochastic resonance in other systems 5 . Further increases in noise decrease local synchrony and extinguish the waves (at fixed coupling).

Robustness to choice of model
We verified that emergent waves are not restricted to our particular choice of neural mass model. We tested two additional models: a network extension of the Wilson-Cowan model 6-8 and the Kuramoto model 9,10 ; see Methods for details. In both models we found large-scale waves (Supplementary Movie 11). In the Wilson-Cowan case, we observed complex dynamics with smooth waves in competition with partially-coherent activity that slowly intrudes onto the waves, leaving wave-like but less-coherent dynamics in its wake. This coexistence of coherent and incoherent states is an example of a chimera state 11 . In the Kuramoto case we observed waves when the model is in a regime of partial synchronization. Both of these models have complex dependencies on parameter values, similar to the neural mass model studied here.

Supplementary Figures
Supplementary Figure 1: Correlation between nodal mean speed and node degree. Degrees are calculated for the network thresholded to 30% density. Line is a least-squares fit. Source data are provided as a Source Data file.

Supplementary Tables
Supplementary Table 1 Statistics for comparisons in Fig. 7c,d, bolded entries correspond to the colored comparisons in the figure. Numbers are two-tailed p-values for all pair-wise t-tests between the numbers of visits to each functional network, corrected for multiple comparisons (Bonferroni). Upper triangle is for comparisons between sink visits and functional networks, lower triangle (shaded) is for comparisons between source visits and functional networks.