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Cited by 264Neural dynamics and brain functionFunctional Brain Connectivity StudiesNonlinear Dynamics and Pattern Formation

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Metastable brain waves

Open access
Roberts, James A., Gollo, Leonardo L., Abeysuriya, Romesh G., Roberts, Gloria, Mitchell, Philip B., Woolrich, Mark W., Breakspear, Michael·2019·Nature Communications
DOI 10.1038/s41467-019-08999-0PMC PMC6401142

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Abstract

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.

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Figures

Fig. 1 Large-scale wave patterns in the model. Ten snapshots of the dynamics of the pyramidal mean membrane potential V at latencies indicated on the time axes, for a a traveling wave, b a rotating wave, and c a pattern with sinks (red areas shrinking for latencies 0–3 ms) and diffuse sources (broad red areas emerging for latencies 6–9 ms). These results are for strong coupling c = 0.6 and short delay τ = 1 ms
Fig. 1 Large-scale wave patterns in the model. Ten snapshots of the dynamics of the pyramidal mean membrane potential V at latencies indicated on the time axes, for a a traveling wave, b a rotating wave, and c a pattern with sinks (red areas shrinking for latencies 0–3 ms) and diffuse sources (broad red areas emerging for latencies 6–9 ms). These results are for strong coupling c = 0.6 and short delay τ = 1 msCC BY · PubMed Central
Fig. 2 Wave propagation speeds. a Histogram of log10(speed) across all nodes and times. b Histogram of average speed (in m s−1) at each node. Red line shows a kernel density estimate. c Mean speed in each region as a function of the lateral distance from the midline (x = 0). d Spatial distribution of nodal mean speeds as viewed (clockwise from left) from the top, right, and back. These results are for strong coupling c = 0.6 and short delay τ = 1 ms, as in Fig. 1. Source data are provided as a Source Data file
Fig. 2 Wave propagation speeds. a Histogram of log10(speed) across all nodes and times. b Histogram of average speed (in m s−1) at each node. Red line shows a kernel density estimate. c Mean speed in each region as a function of the lateral distance from the midline (x = 0). d Spatial distribution of nodal mean speeds as viewed (clockwise from left) from the top, right, and back. These results are for strong coupling c = 0.6 and short delay τ = 1 ms, as in Fig. 1. Source data are provided as a Source Data fileCC BY · PubMed Central
Fig. 3 Metastable transitions. a Each pattern has a signature in the interhemispheric cross-correlation, with transitions between different patterns revealing a brief period of desynchronization. b Vertical lines depict instances of low values of the interhemispheric cross-correlation function, corresponding to wave transitions. These results are for strong coupling c = 0.6 and short delay τ = 1 ms, as in Fig. 1
Fig. 3 Metastable transitions. a Each pattern has a signature in the interhemispheric cross-correlation, with transitions between different patterns revealing a brief period of desynchronization. b Vertical lines depict instances of low values of the interhemispheric cross-correlation function, corresponding to wave transitions. These results are for strong coupling c = 0.6 and short delay τ = 1 ms, as in Fig. 1CC BY · PubMed Central
Fig. 4 Dwell-time distributions. a Model for c = 0.6, d = 1 ms. b Resting-state MEG data from ref. 52. c, d Upper cumulative distributions (black circles) on double logarithmic axes for dwell times in c the model and d MEG data. Lines are maximum likelihood fits to upper tails for the power law (Pareto, red), exponentially truncated power law (green), lognormal (blue), exponential (cyan), and stretched exponential (Weibull, magenta) distributions. Tail cutoffs at dwell times of 100 ms. Source data are provided as a Source Data file
Fig. 4 Dwell-time distributions. a Model for c = 0.6, d = 1 ms. b Resting-state MEG data from ref. 52. c, d Upper cumulative distributions (black circles) on double logarithmic axes for dwell times in c the model and d MEG data. Lines are maximum likelihood fits to upper tails for the power law (Pareto, red), exponentially truncated power law (green), lognormal (blue), exponential (cyan), and stretched exponential (Weibull, magenta) distributions. Tail cutoffs at dwell times of 100 ms. Source data are provided as a Source Data fileCC BY · PubMed Central
Fig. 5 Recurring flow patterns. a Recurrences between the mean alignment of velocity fields vi and vj at times ti and tj, respectively, in the model wave dynamics for c = 0.6, τ = 1 ms. b Recurrences for one instance of an amplitude-adjusted Fourier surrogate time series derived from the simulation used in a. c Histograms of recurrence values for the model (blue) and one surrogate (red). d Recurrence points where the model recurrence alignment was greater than (red) or less than (blue) all 100 surrogates. Red points correspond to flows aligned in the same direction, whereas all blue points correspond to flows aligned in opposite directions
Fig. 5 Recurring flow patterns. a Recurrences between the mean alignment of velocity fields vi and vj at times ti and tj, respectively, in the model wave dynamics for c = 0.6, τ = 1 ms. b Recurrences for one instance of an amplitude-adjusted Fourier surrogate time series derived from the simulation used in a. c Histograms of recurrence values for the model (blue) and one surrogate (red). d Recurrence points where the model recurrence alignment was greater than (red) or less than (blue) all 100 surrogates. Red points correspond to flows aligned in the same direction, whereas all blue points correspond to flows aligned in opposite directionsCC BY · PubMed Central
Fig. 6 Phase flow tracked across space and time. a Snapshots of waves (left, colored by voltage as in Fig. 1), the corresponding phase flow vectors (middle, colored by orientation in the 2D plane shown), and phase flow streamlines (right, blue and red denote forward and backward streamlines, respectively). b Exemplar streamlines near metastable transitions, colored as in a, viewed from the top, back, and right (rows 1–3, respectively). Shown are two sets of three snapshots, each surrounding a transition as indicated (gray) in the panel below. Highlighted points (filled circles) denote clusters that form sources (red) and sinks (blue). Black arrows denote the progression of time; colored arrows denote the features referred to in the text. c Lateral positions (displacement from the midline, x) of sinks (top) and sources (bottom) plotted across time, colored by vertical (dorsoventral) position z. Vertical black lines denote transition times calculated using interhemispheric cross-correlation
Fig. 6 Phase flow tracked across space and time. a Snapshots of waves (left, colored by voltage as in Fig. 1), the corresponding phase flow vectors (middle, colored by orientation in the 2D plane shown), and phase flow streamlines (right, blue and red denote forward and backward streamlines, respectively). b Exemplar streamlines near metastable transitions, colored as in a, viewed from the top, back, and right (rows 1–3, respectively). Shown are two sets of three snapshots, each surrounding a transition as indicated (gray) in the panel below. Highlighted points (filled circles) denote clusters that form sources (red) and sinks (blue). Black arrows denote the progression of time; colored arrows denote the features referred to in the text. c Lateral positions (displacement from the midline, x) of sinks (top) and sources (bottom) plotted across time, colored by vertical (dorsoventral) position z. Vertical black lines denote transition times calculated using interhemispheric cross-correlationCC BY · PubMed Central
Fig. 7 Sink and source properties. a Spatial distribution of sinks. b Spatial distribution of sources. c Overlap of sinks with functional networks. Blue denotes networks with fewer visits than red and gray denotes no significant difference from any other group. Networks are labeled as follows: AUD auditory, CO cingulo-opercular, DA dorsal attention, DM default mode, FP fronto-parietal, MEM memory, SAL salience, SH somatomotor hand, SM somatomotor mouth, SUB subcortical, UNC unclassified, VA ventral attention, VIS visual. d Overlap of sources with functional networks. Colors as per c. e Overlap of sinks with hubs (top 75 nodes by strength), feeders, and non-hubs (bottom 75 nodes by strength). Colors as per c. f Overlap of sources with hubs, feeders, and non-hubs. Colors as per c. White circles in violin plots denote group medians; violins are kernel density estimates. Statistics for c, d are given in Supplementary Table 1 and for e, f in Supplementary Table 2. Source data are provided as a Source Data file.
Fig. 7 Sink and source properties. a Spatial distribution of sinks. b Spatial distribution of sources. c Overlap of sinks with functional networks. Blue denotes networks with fewer visits than red and gray denotes no significant difference from any other group. Networks are labeled as follows: AUD auditory, CO cingulo-opercular, DA dorsal attention, DM default mode, FP fronto-parietal, MEM memory, SAL salience, SH somatomotor hand, SM somatomotor mouth, SUB subcortical, UNC unclassified, VA ventral attention, VIS visual. d Overlap of sources with functional networks. Colors as per c. e Overlap of sinks with hubs (top 75 nodes by strength), feeders, and non-hubs (bottom 75 nodes by strength). Colors as per c. f Overlap of sources with hubs, feeders, and non-hubs. Colors as per c. White circles in violin plots denote group medians; violins are kernel density estimates. Statistics for c, d are given in Supplementary Table 1 and for e, f in Supplementary Table 2. Source data are provided as a Source Data file.CC BY · PubMed Central
Fig. 8 Dynamics as a function of coupling strength c and delay τ. a Functional connectivity matrices calculated directly from the neuronal time series (corresponding results after convolution of the neuronal time series with a hemodynamic response function are provided in Supplementary Fig. 4). Each tile shows one FC matrix with axes indexing the nodes 1–513. b Interhemispheric cross-correlation functions, showing a 1 s segment (time is on the horizontal axis in each tile) for lags between − 30 ms and 30 ms (lag is on the vertical axis in each tile). Exemplars in the text are highlighted here with colored outlines and shown in corresponding panels: c weakly coherent waves; d interhemispheric cross-correlation dynamics despite negligible average FC; and e lurching waves
Fig. 8 Dynamics as a function of coupling strength c and delay τ. a Functional connectivity matrices calculated directly from the neuronal time series (corresponding results after convolution of the neuronal time series with a hemodynamic response function are provided in Supplementary Fig. 4). Each tile shows one FC matrix with axes indexing the nodes 1–513. b Interhemispheric cross-correlation functions, showing a 1 s segment (time is on the horizontal axis in each tile) for lags between − 30 ms and 30 ms (lag is on the vertical axis in each tile). Exemplars in the text are highlighted here with colored outlines and shown in corresponding panels: c weakly coherent waves; d interhemispheric cross-correlation dynamics despite negligible average FC; and e lurching wavesCC BY · PubMed Central
Fig. 9 Correlation between modeled and empirical functional connectivity as a function of coupling and delay. a Pearson’s correlation between empirical and modeled FC values for each pair of regions, without GSR. Numbers 1–3 indicate the top three highest correlations. b Same as a but with GSR
Fig. 9 Correlation between modeled and empirical functional connectivity as a function of coupling and delay. a Pearson’s correlation between empirical and modeled FC values for each pair of regions, without GSR. Numbers 1–3 indicate the top three highest correlations. b Same as a but with GSRCC BY · PubMed Central
Fig. 10 Local and global synchrony for surrogate networks. a Local and global synchrony in fully randomized networks as a function of coupling strength c. Lines show synchronization order parameters Rlocal (solid) and Rglobal (dashed). Insets show the interhemispheric cross-correlations at representative values of c, denoted by the arrows. b Loss of waves with progressive structural network randomization, parameterized by the proportion of randomized edges. Insets show the (log-)weight-vs.-fiber length relationships (blue point clouds) and the interhemispheric cross-correlations for the proportion of randomized edges denoted by the arrows. Lines are an average over an ensemble of ten random surrogates at each point. c Waves in synthetic networks with pure exponential weight–distance relationship, parameterized by the slope of the linear log10 (weight)-vs.-fiber length relationship. Insets show the (log-)weight-vs.-fiber length relationships for the synthetic exponential networks (blue) and the original network (gray), and the interhemispheric cross-correlations for the slope values denoted by the arrows. Source data are provided as a Source Data file
Fig. 10 Local and global synchrony for surrogate networks. a Local and global synchrony in fully randomized networks as a function of coupling strength c. Lines show synchronization order parameters Rlocal (solid) and Rglobal (dashed). Insets show the interhemispheric cross-correlations at representative values of c, denoted by the arrows. b Loss of waves with progressive structural network randomization, parameterized by the proportion of randomized edges. Insets show the (log-)weight-vs.-fiber length relationships (blue point clouds) and the interhemispheric cross-correlations for the proportion of randomized edges denoted by the arrows. Lines are an average over an ensemble of ten random surrogates at each point. c Waves in synthetic networks with pure exponential weight–distance relationship, parameterized by the slope of the linear log10 (weight)-vs.-fiber length relationship. Insets show the (log-)weight-vs.-fiber length relationships for the synthetic exponential networks (blue) and the original network (gray), and the interhemispheric cross-correlations for the slope values denoted by the arrows. Source data are provided as a Source Data fileCC BY · PubMed Central

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  1. Ester, M., Kriegel, H. -P., Sander, J. & Xu, X. in Proc. Second International Conference on Knowledge Discovery and Data Mining 226–231 (AAAI Press).

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Metadata

Authors
Roberts, J. A., Gollo, L. L., Abeysuriya, R. G., Roberts, G., Mitchell, P. B., Woolrich, M. W., & Breakspear, M.
Affiliations
  • QIMR Berghofer Medical Research Institute (AU)
  • University of Oxford (GB)
  • Black Dog Institute (AU)
Corresponding author
Roberts, James A.
ORCID
Venue
Nature Communications
Publisher
Springer Science and Business Media LLC
Published
2019-03-05
Volume
10
Issue
1
Pages
1056-1056
ISSN
2041-1723 (ISSN-L)
DOI
10.1038/s41467-019-08999-0
PMC
PMC6401142
OpenAlex
W2950869179
Cited by
264
References
94
Citations by year
2026: 32 · 2025: 42 · 2024: 36 · 2023: 41 · 2022: 37 · 2021: 26 · 2020: 30 · 2019: 19
FWCI
16.27
Type
journal article
Language
en
Open access
Licence
Locations
  • Nature Communications (publishedVersion, cc-by)
  • bioRxiv (Cold Spring Harbor Laboratory) (acceptedVersion, cc-by-nc-nd)
  • Europe PMC (PubMed Central) (publishedVersion, cc-by)
  • PubMed Central (publishedVersion, cc-by)
  • PubMed (publishedVersion)
  • RePEc: Research Papers in Economics (submittedVersion)
  • DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion)
  • DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
  • Figshare (submittedVersion, cc-by)
  • NOVA (University of Newcastle Australia) (submittedVersion, cc-by)
Funders
  • Wellcome Trust (MR/K005464/1, 106183/Z/14/Z, 203139/Z/16/Z, 106183, 203139, 203139/Z/16)
  • University of Nottingham
  • Medical Research Council (203139/Z/16/Z, MR/K005464/1)
  • Australian Research Council (1118153, 1144936, CE140100007)
  • National Health and Medical Research Council (1145168, CE140100007, 1144936, 1110975, 1118153, 1037196)
  • Centre of Excellence for Integrative Brain Function, Australian Research Council (CE140100007)
Topics
  • Life Sciences › Neuroscience › Cognitive Neuroscience › Neural dynamics and brain function
  • Life Sciences › Neuroscience › Cognitive Neuroscience › Functional Brain Connectivity Studies
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Nonlinear Dynamics and Pattern Formation
Keywords
neural oscillations, traveling waves, spiral waves, metastable dynamics, nonlinear dynamics, whole-brain networks, phase flow, nonlinear instability, spontaneous neuronal activity, resting-state MEG, electrocorticography, functional networks
MeSH terms
Electrocorticography, Adolescent, Adult, Brain, Computer Simulation, Female, Humans, Male, Models, Neurological, Nerve Net, Neurons, Nonlinear Dynamics, Young Adult, Diffusion Tensor Imaging, Brain Waves, Healthy Volunteers

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