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Cited by 527Neural dynamics and brain functionTheoretical and Computational PhysicsFunctional Brain Connectivity Studies

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Broadband Criticality of Human Brain Network Synchronization

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Kitzbichler, Manfred G., Smith, Marie L., Christensen, Søren R., Bullmore, Ed·2009·PLoS Computational Biology
DOI 10.1371/journal.pcbi.1000314PMC PMC2647739
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Abstract

Self-organized criticality is an attractive model for human brain dynamics, but there has been little direct evidence for its existence in large-scale systems measured by neuroimaging. In general, critical systems are associated with fractal or power law scaling, long-range correlations in space and time, and rapid reconfiguration in response to external inputs. Here, we consider two measures of phase synchronization: the phase-lock interval, or duration of coupling between a pair of (neurophysiological) processes, and the lability of global synchronization of a (brain functional) network. Using computational simulations of two mechanistically distinct systems displaying complex dynamics, the Ising model and the Kuramoto model, we show that both synchronization metrics have power law probability distributions specifically when these systems are in a critical state. We then demonstrate power law scaling of both pairwise and global synchronization metrics in functional MRI and magnetoencephalographic data recorded from normal volunteers under resting conditions. These results strongly suggest that human brain functional systems exist in an endogenous state of dynamical criticality, characterized by a greater than random probability of both prolonged periods of phase-locking and occurrence of large rapid changes in the state of global synchronization, analogous to the neuronal "avalanches" previously described in cellular systems. Moreover, evidence for critical dynamics was identified consistently in neurophysiological systems operating at frequency intervals ranging from 0.05-0.11 to 62.5-125 Hz, confirming that criticality is a property of human brain functional network organization at all frequency intervals in the brain's physiological bandwidth.

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Figure 1 Ising model simulations of a dynamic system at critical and non-critical temperatures. (A) Binary 128×128 lattices showing the configuration of spins after 2,000 timesteps at low temperature, (left); critical temperature, (middle); and high temperature, (right). At hot temperature the spins are randomly configured, at low temperature they are close to an entirely ordered state, and at critical temperature they have a fractal configuration. (B) Probability distribution of phase lock interval (PLI) between pairs of processes at critical (black line) and at hot temperature (red line) plotted on a log-log scale. The black dashed line represents a power law with slope . (C) Probability distribution of lability of global synchronization () at critical temperature (black line) and at hot temperature (red dotted line); the black dashed line represents a power law with slope . For the cold Ising model the equilibrium state of the system is a monolithic lattice with either all spins up or down, resulting in an entirely static system for which the PLI distribution is a Dirac Delta peak at the duration of the time series. The key point is that the probability distributions of both duration of pairwise synchronization, indexed by the phase lock interval, and lability of global synchronization, show power law behaviour for the 2D Ising model specifically at critical temperature.
Figure 1 Ising model simulations of a dynamic system at critical and non-critical temperatures. (A) Binary 128×128 lattices showing the configuration of spins after 2,000 timesteps at low temperature, (left); critical temperature, (middle); and high temperature, (right). At hot temperature the spins are randomly configured, at low temperature they are close to an entirely ordered state, and at critical temperature they have a fractal configuration. (B) Probability distribution of phase lock interval (PLI) between pairs of processes at critical (black line) and at hot temperature (red line) plotted on a log-log scale. The black dashed line represents a power law with slope . (C) Probability distribution of lability of global synchronization () at critical temperature (black line) and at hot temperature (red dotted line); the black dashed line represents a power law with slope . For the cold Ising model the equilibrium state of the system is a monolithic lattice with either all spins up or down, resulting in an entirely static system for which the PLI distribution is a Dirac Delta peak at the duration of the time series. The key point is that the probability distributions of both duration of pairwise synchronization, indexed by the phase lock interval, and lability of global synchronization, show power law behaviour for the 2D Ising model specifically at critical temperature.CC BY · PubMed Central
Figure 2 Kuramoto model simulations of dynamic system states as functions of coupling strength between oscillators. When the coupling strength has critical value , the system is metastable and demonstrates the greatest fluctuations in the mean field and in the number of synchronized pairs. Top panel: change of effective frequencies of oscillators (black lines) with coupling strength (equivalent to natural frequencies for low ). Vertically symmetric red bands indicate range over time of effective mean-field coupling strength . Natural frequencies lower than synchronize with the mean frequency , leading to variations in fraction of synchronized pairs subject to fluctuating mean-field strength . Bottom panel: dependence on coupling strength of time averaged order parameter (black circles), and of fraction of synchronized oscillators (red circles) with standard deviation indicated by error bars in gray. The open black symbols and green curve show fluctuation amplitudes of and , respectively.
Figure 2 Kuramoto model simulations of dynamic system states as functions of coupling strength between oscillators. When the coupling strength has critical value , the system is metastable and demonstrates the greatest fluctuations in the mean field and in the number of synchronized pairs. Top panel: change of effective frequencies of oscillators (black lines) with coupling strength (equivalent to natural frequencies for low ). Vertically symmetric red bands indicate range over time of effective mean-field coupling strength . Natural frequencies lower than synchronize with the mean frequency , leading to variations in fraction of synchronized pairs subject to fluctuating mean-field strength . Bottom panel: dependence on coupling strength of time averaged order parameter (black circles), and of fraction of synchronized oscillators (red circles) with standard deviation indicated by error bars in gray. The open black symbols and green curve show fluctuation amplitudes of and , respectively.CC BY · PubMed Central
Figure 3 Simulated Kuramoto model data. Top row: results from system at critical coupling strength , bottom row: no coupling, i.e. free running oscillators. In all panels simulation data is denoted by solid lines (filled symbols) and the corresponding surrogate data by dotted lines. The colors encode wavelet scales 3–11. Left column: Power spectrum of simulated Kuramoto model time series plotted on logarithmic axes. In the critical state the spectrum shows clustering of the effective frequencies forming a common broad peak and follows a power law with exponent −2 on the low-frequency end. The spectrum of the uncoupled model is a simple superposition of the natural oscillator frequencies. The colored vertical lines represent the frequency intervals corresponding to wavelet scales 3–11 (scales 1 and 2 indicated by dotted lines not used). Middle column: Probability distributions for phase-lock interval PLI. Only the critical system produces a power law, clearly distinct from the surrogate data showing an exponential fall-off. The black dashed line represents a power law with . Right column: Probability distribution for lability of global synchronization is plotted on logarithmic axes for each wavelet scale. Again a power law is only seen in the critical model, whereas surrogate data and uncoupled model produce exponential distributions. The straight dashed line represents a power law with .
Figure 3 Simulated Kuramoto model data. Top row: results from system at critical coupling strength , bottom row: no coupling, i.e. free running oscillators. In all panels simulation data is denoted by solid lines (filled symbols) and the corresponding surrogate data by dotted lines. The colors encode wavelet scales 3–11. Left column: Power spectrum of simulated Kuramoto model time series plotted on logarithmic axes. In the critical state the spectrum shows clustering of the effective frequencies forming a common broad peak and follows a power law with exponent −2 on the low-frequency end. The spectrum of the uncoupled model is a simple superposition of the natural oscillator frequencies. The colored vertical lines represent the frequency intervals corresponding to wavelet scales 3–11 (scales 1 and 2 indicated by dotted lines not used). Middle column: Probability distributions for phase-lock interval PLI. Only the critical system produces a power law, clearly distinct from the surrogate data showing an exponential fall-off. The black dashed line represents a power law with . Right column: Probability distribution for lability of global synchronization is plotted on logarithmic axes for each wavelet scale. Again a power law is only seen in the critical model, whereas surrogate data and uncoupled model produce exponential distributions. The straight dashed line represents a power law with .CC BY · PubMed Central
Figure 4 Illustration of phase synchronization between pairs of neurophysiological processes in low and high frequency intervals. (A) Top panel: Amplitude of functional MRI signals in the frequency interval 0.05–0.1 Hz (corresponding to wavelet scale 3) is shown for three brain regions in a single subject: left precentral gyrus (black), right precentral gyrus (red), and left olfactory cortex (green). Bottom panel: Phase difference between two pairs of fMRI processes is shown for right and left precentral gyrus (black), and left precentral gyrus and olfactory cortex (red). The shaded area represents phase differences less than ; while phase difference is in this regime the pair of processes is said to be phase-locked. (B) Top panel: Amplitude of MEG signals in the frequency interval 31–63 Hz (approximately equivalent to the classical band) is shown for three sensors in a single subject: two left temporal sources (black and green), and a left frontal source (red). Bottom panel: Phase difference between two pairs of MEG processes is shown for 2 left temporal sensors (black) and for left frontal and temporal sensors (red). The horizontal bars in the top-left corner of each panel denote the temporal extent of , respectively, corresponding to about 8 wavelet cycles. Only a short section of the actual time-series is shown.
Figure 4 Illustration of phase synchronization between pairs of neurophysiological processes in low and high frequency intervals. (A) Top panel: Amplitude of functional MRI signals in the frequency interval 0.05–0.1 Hz (corresponding to wavelet scale 3) is shown for three brain regions in a single subject: left precentral gyrus (black), right precentral gyrus (red), and left olfactory cortex (green). Bottom panel: Phase difference between two pairs of fMRI processes is shown for right and left precentral gyrus (black), and left precentral gyrus and olfactory cortex (red). The shaded area represents phase differences less than ; while phase difference is in this regime the pair of processes is said to be phase-locked. (B) Top panel: Amplitude of MEG signals in the frequency interval 31–63 Hz (approximately equivalent to the classical band) is shown for three sensors in a single subject: two left temporal sources (black and green), and a left frontal source (red). Bottom panel: Phase difference between two pairs of MEG processes is shown for 2 left temporal sensors (black) and for left frontal and temporal sensors (red). The horizontal bars in the top-left corner of each panel denote the temporal extent of , respectively, corresponding to about 8 wavelet cycles. Only a short section of the actual time-series is shown.CC BY · PubMed Central
Figure 5 Phase-locking and global synchronization in a low frequency network measured using functional MRI. Colors denote wavelet scales: black = scale 1 (0.45−0.22 Hz); red = scale 2 (0.22−0.11 Hz); green = scale 3 (0.11−0.05 Hz). (A) Probability distributions of phase-lock interval (PLI, s) are plotted on logarithmic axes for all pairs of processes (filled symbols) and for all (intra-modular) pairs of processes within the same functional module (open symbols). The corresponding distributions for phase-scrambled surrogate data are shown by the dotted lines, and the straight dashed line indicates a power-law with . (B) Cumulative probability distributions of phase-lock intervals are shown on logarithmic axes for all pairs of processes (solid lines) and surrogate data (dotted lines). Inset shows the power law scaling exponent as a function of wavelet scale (larger scales represent lower frequencies). (C) Weighted cumulative probability distribution of phase-lock intervals are shown on linear axes for all pairs of processes (solid lines), intra-modular pairs of processes (dashed lines) and surrogate data (thin dotted lines). The negative range on the x-axis stands for intervals without phase-lock. (D) Probability distributions for lability of global synchronization () are shown on logarithmic axes for fMRI data (filled symbols and solid lines) and surrogate data (dotted lines). The dashed straight line indicates a power law with to guide the eye.
Figure 5 Phase-locking and global synchronization in a low frequency network measured using functional MRI. Colors denote wavelet scales: black = scale 1 (0.45−0.22 Hz); red = scale 2 (0.22−0.11 Hz); green = scale 3 (0.11−0.05 Hz). (A) Probability distributions of phase-lock interval (PLI, s) are plotted on logarithmic axes for all pairs of processes (filled symbols) and for all (intra-modular) pairs of processes within the same functional module (open symbols). The corresponding distributions for phase-scrambled surrogate data are shown by the dotted lines, and the straight dashed line indicates a power-law with . (B) Cumulative probability distributions of phase-lock intervals are shown on logarithmic axes for all pairs of processes (solid lines) and surrogate data (dotted lines). Inset shows the power law scaling exponent as a function of wavelet scale (larger scales represent lower frequencies). (C) Weighted cumulative probability distribution of phase-lock intervals are shown on linear axes for all pairs of processes (solid lines), intra-modular pairs of processes (dashed lines) and surrogate data (thin dotted lines). The negative range on the x-axis stands for intervals without phase-lock. (D) Probability distributions for lability of global synchronization () are shown on logarithmic axes for fMRI data (filled symbols and solid lines) and surrogate data (dotted lines). The dashed straight line indicates a power law with to guide the eye.CC BY · PubMed Central
Figure 6 Effects of spatial proximity and modularity on scaling of phase locking between fMRI time series. Left: Dependence of phase-lock index on log of physical distance between a pair of brain regions (dots). The number density is indicated by contours, where the change in abundance with distance has been normalized out. The red line has a slope of 0.25 and serves to guide the eye. Right: Matrix representing the relative prevalence of long-term phase lock versus short-term phase lock intervals for all pairs of brain regions in resting-state fMRI data. The color of each element indicates the value of for a specific pair of processes (see text for exact definition of ). Intra-modular pairs of regions are located close to the diagonal and are segregated and identified by black rectangles (grey rectangles denote coarser anatomical separation of brain regions as labeled; see text). All graphs shown are for wavelet scale 1 in the fMRI data.
Figure 6 Effects of spatial proximity and modularity on scaling of phase locking between fMRI time series. Left: Dependence of phase-lock index on log of physical distance between a pair of brain regions (dots). The number density is indicated by contours, where the change in abundance with distance has been normalized out. The red line has a slope of 0.25 and serves to guide the eye. Right: Matrix representing the relative prevalence of long-term phase lock versus short-term phase lock intervals for all pairs of brain regions in resting-state fMRI data. The color of each element indicates the value of for a specific pair of processes (see text for exact definition of ). Intra-modular pairs of regions are located close to the diagonal and are segregated and identified by black rectangles (grey rectangles denote coarser anatomical separation of brain regions as labeled; see text). All graphs shown are for wavelet scale 1 in the fMRI data.CC BY · PubMed Central
Figure 7 Probability distributions for phase-lock interval and lability of global synchronization in MEG data. In all panels MEG data is denoted by solid lines (filled symbols) and the corresponding surrogate data by dotted lines. The colors encode wavelet scales as follows: black = scale 3 (125−62.5 Hz); red = 4 (62.5−31 Hz); green = 5 (31−15.5 Hz); blue = 6 (15.5−8 Hz); light blue = 7 (8−4 Hz); pink = 8 (4−2 Hz); yellow = 9 (2−1 Hz); gray = 10 (1−0.5 Hz); black = 11 (0.5−0 Hz). (A) Probability distribution of phase-lock intervals is plotted on logarithmic axes for each wavelet scale. The black dashed line represents a power law with . (B) Cumulative probability distribution of phase-lock intervals is plotted on logarithmic axes for each wavelet scale. The exponents of the power law distributions tend to increase as a function of increasing scale (inset). (C) Weighted cumulative distribution of phase-lock intervals on linear axes for each wavelet scale indicating the fraction of time spent locked for a time longer than the PLI indicated. (D) Probability distribution for lability of global synchronization is plotted on logarithmic axes for each wavelet scale. The straight dashed line represents a power law with .
Figure 7 Probability distributions for phase-lock interval and lability of global synchronization in MEG data. In all panels MEG data is denoted by solid lines (filled symbols) and the corresponding surrogate data by dotted lines. The colors encode wavelet scales as follows: black = scale 3 (125−62.5 Hz); red = 4 (62.5−31 Hz); green = 5 (31−15.5 Hz); blue = 6 (15.5−8 Hz); light blue = 7 (8−4 Hz); pink = 8 (4−2 Hz); yellow = 9 (2−1 Hz); gray = 10 (1−0.5 Hz); black = 11 (0.5−0 Hz). (A) Probability distribution of phase-lock intervals is plotted on logarithmic axes for each wavelet scale. The black dashed line represents a power law with . (B) Cumulative probability distribution of phase-lock intervals is plotted on logarithmic axes for each wavelet scale. The exponents of the power law distributions tend to increase as a function of increasing scale (inset). (C) Weighted cumulative distribution of phase-lock intervals on linear axes for each wavelet scale indicating the fraction of time spent locked for a time longer than the PLI indicated. (D) Probability distribution for lability of global synchronization is plotted on logarithmic axes for each wavelet scale. The straight dashed line represents a power law with .CC BY · PubMed Central

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Metadata

Authors
Kitzbichler, M. G., Smith, M. L., Christensen, S. R., & Bullmore, E.
Affiliations
  • University of Cambridge (GB)
  • MRC Cognition and Brain Sciences Unit (GB)
  • GlaxoSmithKline (United Kingdom) (GB)
ORCID
Venue
PLoS Computational Biology
Publisher
Public Library of Science (PLoS)
Published
2009-03-20
Volume
5
Issue
3
Pages
e1000314
ISSN
  • 1553-734X (ISSN-L)
  • 1553-7358
DOI
10.1371/journal.pcbi.1000314
PMC
PMC2647739
OpenAlex
W2115990221
Cited by
527
References
52
Citations by year
2026: 10 · 2025: 16 · 2024: 14 · 2023: 28 · 2022: 28 · 2021: 31 · 2020: 25 · 2019: 38 · 2018: 37 · 2017: 37 · 2016: 24 · 2015: 24 · 2014: 39 · 2013: 44 · 2012: 54
FWCI
21.21
Type
journal article
Language
en
Open access
Licence
Locations
  • PLoS Computational Biology (publishedVersion, cc-by)
  • PubMed Central (publishedVersion, cc-by)
  • PubMed (publishedVersion)
  • BIROn (Birkbeck, University of London) (acceptedVersion)
  • RePEc: Research Papers in Economics (submittedVersion)
  • DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
  • Apollo (University of Cambridge) (submittedVersion, cc-by)
  • Apollo (cc-by)
Funders
  • GlaxoSmithKline
  • Wellcome Trust
  • Medical Research Council
  • National Institute of Mental Health
Topics
  • Life Sciences › Neuroscience › Cognitive Neuroscience › Neural dynamics and brain function
  • Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Theoretical and Computational Physics
  • Life Sciences › Neuroscience › Cognitive Neuroscience › Functional Brain Connectivity Studies
Keywords
self-organized criticality, functional brain networks, phase synchronization, power-law scaling, neuronal avalanches, Ising model, Kuramoto model, resting-state brain dynamics, functional MRI, magnetoencephalography
MeSH terms
Action Potentials, Animals, Biological Clocks, Brain, Computer Simulation, Cortical Synchronization, Humans, Models, Neurological, Nerve Net, Synaptic Transmission, Neurons

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