Focus on human brain mapping

Dynamic models of large-scale brain activity

Journal name:
Nature Neuroscience
Volume:
20,
Pages:
340–352
Year published:
DOI:
doi:10.1038/nn.4497
Received
Accepted
Published online

Abstract

Movement, cognition and perception arise from the collective activity of neurons within cortical circuits and across large-scale systems of the brain. While the causes of single neuron spikes have been understood for decades, the processes that support collective neural behavior in large-scale cortical systems are less clear and have been at times the subject of contention. Modeling large-scale brain activity with nonlinear dynamical systems theory allows the integration of experimental data from multiple modalities into a common framework that facilitates prediction, testing and possible refutation. This work reviews the core assumptions that underlie this computational approach, the methodological framework that fosters the translation of theory into the laboratory, and the emerging body of supporting evidence. While substantial challenges remain, evidence supports the view that collective, nonlinear dynamics are central to adaptive cortical activity. Likewise, aberrant dynamic processes appear to underlie a number of brain disorders.

At a glance

Figures

  1. A dynamical system is defined by a differential equation dX/dt = f(X).
    Figure 1: A dynamical system is defined by a differential equation dX/dt = f(X).

    Here X is composed of the two state variables x (the cell membrane potential) and y (the conductance of a fast-depolarizing ion channel). (a) The phase space is the geometric space spanned by the state variables: in this case, simply the Cartesian plane composed of axes for x and y. The dynamical system then defines a vector of length and direction given by f(x,y) at each point—that is, for each combination of membrane potential and ion channel conductance. (b) The flow (also called a vector field) is the set of all such vectors and shows how the dynamical system will flow through phase space: here, a distinctive clockwise flow is evident. (c) An orbit is a solution to the flow—a smooth line that is tangent to the flow. (d) Orbits converge onto the attractors, the long-term solutions of the system. Here there is just a single limit cycle attractor (red) reached from many different starting points (other colors). (eg) By adding a slow recovery variable z (middle), the system can show a simple limit cycle (e, top), corresponding to regular spiking (e, bottom); or a more complex limit cycle (f, top), yielding regular bursting (f, bottom); or a chaotic (strange) attractor (g, top) with irregular spiking (g, bottom) when the time scales of the spiking and recovery variable mix.

  2. Principles of the neuronal ensemble reduction.
    Figure 2: Principles of the neuronal ensemble reduction.

    (a) Complex spatial systems composed of interacting components, such as human cortical columns, are characterized by interactions that weaken with distance. (b) If the resulting correlations decay quickly compared to the size of the system, the statistics of the system converge toward a Gaussian distribution (inset), even if the statistics within the individual components are highly non-Gaussian. (c) The Fokker–Planck equation describes how the statistics passively change when the inputs and strength of stochastic fluctuations change. Here the inputs increase and the noise becomes less influential. The mean rate drifts up and the ensemble distribution becomes more precise as we move in the direction of the arrow. (d) If correlations within the system become stronger—for example, owing to synchrony—the correlation length diverges toward the size of the system in the direction of the arrow toward the blue curve. The assumptions underlying the diffusion approximation may not be met. (e) If strong ensemble correlations exist, the statistics may converge toward a non-Gaussian distribution (blue curve). Typical fluctuations shrink toward the mean (and hence the distribution becomes more tent-like), but the left and right tails (extremes) become fatter, corresponding to infrequent but high-amplitude, synchronous fluctuations.

  3. Models of large-scale brain dynamics.
    Figure 3: Models of large-scale brain dynamics.

    (a) Neural mass models (NMMs) are obtained by taking the average states in all neurons of each class (pyramidal, inhibitory) in a local population, here a cortical column (left). Traditional spiking neural models treat each neuron as an idealized individual unit (center). NMMs go further, reducing the entire population dynamics to a low-dimensional differential equation representing the average states of local interacting neurons (right). The variability in the cell threshold (inset, top right) smooths the all-or-nothing action potential of each neuron into a smooth sigmoidal map between average membrane potential and average firing rate. Integrating this equation yields an attractor for the local dynamics (bottom right). (b) Brain network models (BNMs) are composed by coupling an ensemble of NMMs into a large-scale system (top), with connections informed by the connectome (adapted from G. Roberts, A. Perry, A. Lord, A. Frankland, V. Leung et al., Mol. Psychiatry, in the press). Due to strong short-range connections, BNMs can yield wave patterns (bottom). This example shows a wave traveling diagonally to the right. (c) In a neural field model (NFM), the cortex is treated as a smooth sheet (top) that supports waves of propagating activity (top inset). Neural field models that include a neural mass in the thalamus (bottom inset) yield alpha oscillations with the same spectral properties as those observed in empirical data (bottom; adapted from ref. 11, Oxford University Press).

  4. Multistable large-scale brain rhythms.
    Figure 4: Multistable large-scale brain rhythms.

    (a) Empirical recordings of human resting-state (eyes closed) scalp EEG show the characteristic fluctuating 10 Hz alpha rhythm (middle). The instantaneous power (the square of the amplitude fluctuations) jumps erratically between a low-power (black) and a high-power (red) mode. Hence the histogram of the power fluctuations (right) is not unimodal (blue dotted line), but is composed of two distinct modes (colored red and black to match the time series)19. (b) An exemplar multistable dynamical system consists of a fixed-point attractor (red dot) and a limit cycle attractor (red circle), separated by a circular basin boundary (black circle). For weak noise, the system becomes trapped in the basin boundary of either the fixed point (left) or periodic attractor (middle). For strong noise, this multistable system jumps erratically between the attractor basins. Time series panels show corresponding fluctuations in a state variable. (c) When populated with physiologically realistic parameters, a corticothalamic NFM exhibits similar multistability. For realistic amplitude noise, the dynamics of this model show a strikingly close match to the higher order statistics of the empirical data19 (middle and right panels; compare to those of a). Middle and right panels of a,c adapted from ref. 31 under a Creative Commons CC BY 4.0 license.

  5. Technical and conceptual framework for empirical testing of NMMs and NFMs.
    Figure 5: Technical and conceptual framework for empirical testing of NMMs and NFMs.

    (a) Models of large-scale dynamics are derived from detailed neurophysiology through abstraction. A combination of mathematical analysis and numerical simulations can then be employed to understand the emergent dynamics supported by these models. This step can be constrained by ensuring that neurophysiological parameters are constrained to lie within realistic values. A forward model (biomagnetic or hemodynamic; the latter is illustrated) is then required to predict empirical data from these models97. Bottom right panel adapted from ref. 97 under a Creative Commons CC BY 4.0 license. (b) Empirical experiments using brain imaging technology yield empirical data across a range of spatial and temporal apertures. High-quality fMRI and EEG can be acquired simultaneously to test model predictions. Going from neural models to empirical data corresponds to model prediction. Using variational schemes and appropriate penalties for model complexity, the mismatch between prediction and observation can be used for model inversion and comparison.

  6. Application of neural field model to human epilepsy. (a) Human scalp EEG recording showing a characteristic 3 Hz absence seizure.
    Figure 6: Application of neural field model to human epilepsy11. (a) Human scalp EEG recording showing a characteristic 3 Hz absence seizure.

    Spectrogram shows fundamental 3 Hz frequency as well as higher order harmonics, reflective of nonlinear time series properties. Phase space reconstruction (right) shows rapid divergence of orbits from the resting-state (fixed point, blue arrow) attractor to a high-amplitude complex limit cycle (red arrow). (b) Left, corticothalamic neural field model perturbed through a 3 Hz Hopf bifurcation shows a striking match to empirical data in a (middle), including the overall symmetric seizure shape, the spike and wave waveforms at onset and offset, and the stippled spectrogram. Right, the onset of the seizure shows the divergence of the orbits from the fixed point (blue arrow) to the limit cycle (red arrow) upon introduction of nonlinear instabilities into the neural field model through the bifurcation. Right panels in a,b adapted from ref. 11, Oxford University Press.

References

  1. Hodgkin, A.L. & Huxley, A.F. Propagation of electrical signals along giant nerve fibres. Philos. Trans. R. Soc. Lond. B Biol. Sci. 140, 177183 (1952).
  2. Kelso, J.S. Dynamic Patterns: The Self-Organization of Brain and Behavior (MIT Press, 1997).
  3. Hoel, E.P., Albantakis, L. & Tononi, G. Quantifying causal emergence shows that macro can beat micro. Proc. Natl. Acad. Sci. USA 110, 1979019795 (2013).
  4. Nunez, P.L. & Srinivasan, R. Electric Fields of the Brain: The Neurophysics of EEG (Oxford Univ. Press, 2006).
  5. Haken, H. Synergetik: Eine Einführung (Springer, 1982).
  6. Jirsa, V.K. & Haken, H. Field theory of electromagnetic brain activity. Phys. Rev. Lett. 77, 960963 (1996).
  7. Robinson, P.A., Rennie, C.J. & Wright, J.J. Propagation and stability of waves of electrical activity in the cerebral cortex. Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 56, 826 (1997).
  8. Coombes, S. et al. Modeling electrocortical activity through improved local approximations of integral neural field equations. Phys. Rev. E 76, 051901 (2007).
  9. Freeman, W.J. Nonlinear gain mediating cortical stimulus-response relations. Biol. Cybern. 33, 237247 (1979).
  10. Freeman, W.J. Mass Action in the Nervous System: Examination of the Neurophysiological Basis of Adaptive Behavior through the EEG (Academic Press, London, 1975).
  11. Breakspear, M. et al. A unifying explanation of primary generalized seizures through nonlinear brain modeling and bifurcation analysis. Cereb. Cortex 16, 12961313 (2006).
  12. Roberts, J.A., Iyer, K.K., Finnigan, S., Vanhatalo, S. & Breakspear, M. Scale-free bursting in human cortex following hypoxia at birth. J. Neurosci. 34, 65576572 (2014).
  13. Bojak, I., Stoyanov, Z.V. & Liley, D.T. Emergence of spatially heterogeneous burst suppression in a neural field model of electrocortical activity. Front. Syst. Neurosci. 9, 18 (2015).
  14. Phillips, A.J. & Robinson, P.A. A quantitative model of sleep-wake dynamics based on the physiology of the brainstem ascending arousal system. J. Biol. Rhythms 22, 167179 (2007).
  15. Bojak, I. & Liley, D.T. Modeling the effects of anesthesia on the electroencephalogram. Phys. Rev. E 71, 041902 (2005).
  16. Honey, C.J., Kötter, R., Breakspear, M. & Sporns, O. Network structure of cerebral cortex shapes functional connectivity on multiple time scales. Proc. Natl. Acad. Sci. USA 104, 1024010245 (2007).
  17. Deco, G., Jirsa, V., McIntosh, A.R., Sporns, O. & Kötter, R. Key role of coupling, delay, and noise in resting brain fluctuations. Proc. Natl. Acad. Sci. USA 106, 1030210307 (2009).
  18. Robinson, P.A., Rennie, C.J. & Rowe, D.L. Dynamics of large-scale brain activity in normal arousal states and epileptic seizures. Phys. Rev. E 65, 041924 (2002).
  19. Freyer, F. et al. Biophysical mechanisms of multistability in resting-state cortical rhythms. J. Neurosci. 31, 63536361 (2011).
  20. Valdes-Sosa, P.A. et al. Model driven EEG/fMRI fusion of brain oscillations. Hum. Brain Mapp. 30, 27012721 (2009).
  21. Daunizeau, J., Stephan, K.E. & Friston, K.J. Stochastic dynamic causal modelling of fMRI data: should we care about neural noise? Neuroimage 62, 464481 (2012).
  22. Wilson, H.R. & Cowan, J.D. Excitatory and inhibitory interactions in localized populations of model neurons. Biophys. J. 12, 124 (1972).
  23. Bruns, H. Über die Integrale des Vielkörper-problems. Acta Math. 11, 2596 (1887).
  24. Poincaré, H. & Magini, R. Les méthodes nouvelles de la mécanique céleste. Nuovo Cimento 10, 128130 (1899).
  25. Abraham, R.H. & Shaw, C.D. Dynamics: The Geometry of Behavior (Aerial Press, 1983).
  26. Jirsa, V.K., Sporns, O., Breakspear, M., Deco, G. & McIntosh, A.R. Towards the virtual brain: network modeling of the intact and the damaged brain. Arch. Ital. Biol. 148, 189205 (2010).
  27. Haken, H., Kelso, J.A. & Bunz, H. A theoretical model of phase transitions in human hand movements. Biol. Cybern. 51, 347356 (1985).
  28. Faisal, A.A., Selen, L.P. & Wolpert, D.M. Noise in the nervous system. Nat. Rev. Neurosci. 9, 292303 (2008).
  29. Misic´, B., Mills, T., Taylor, M.J. & McIntosh, A.R. Brain noise is task dependent and region specific. J. Neurophysiol. 104, 26672676 (2010).
  30. Laing, C. & Lord, G.J. Stochastic Methods in Neuroscience (Oxford Univ. Press, 2010).
  31. Freyer, F., Roberts, J.A., Ritter, P. & Breakspear, M. A canonical model of multistability and scale-invariance in biological systems. PLoS Comput. Biol. 8, e1002634 (2012).
  32. Anderson, P.W. More is different. Science 177, 393396 (1972).
  33. Lopes da Silva, F. Neural mechanisms underlying brain waves: from neural membranes to networks. Electroencephalogr. Clin. Neurophysiol. 79, 8193 (1991).
  34. Omurtag, A., Knight, B.W. & Sirovich, L. On the simulation of large populations of neurons. J. Comput. Neurosci. 8, 5163 (2000).
  35. Fourcaud, N. & Brunel, N. Dynamics of the firing probability of noisy integrate-and-fire neurons. Neural Comput. 14, 20572110 (2002).
  36. El Boustani, S. & Destexhe, A. A master equation formalism for macroscopic modeling of asynchronous irregular activity states. Neural Comput. 21, 46100 (2009).
  37. Deco, G., Jirsa, V.K., Robinson, P.A., Breakspear, M. & Friston, K. The dynamic brain: from spiking neurons to neural masses and cortical fields. PLoS Comput. Biol. 4, e1000092 (2008).
  38. Harrison, L.M., David, O. & Friston, K.J. Stochastic models of neuronal dynamics. Phil. Trans. R. Soc. Lond. B 360, 10751091 (2005).
  39. Ma, W.J., Beck, J.M., Latham, P.E. & Pouget, A. Bayesian inference with probabilistic population codes. Nat. Neurosci. 9, 14321438 (2006).
  40. Friston, K. The free-energy principle: a unified brain theory? Nat. Rev. Neurosci. 11, 127138 (2010).
  41. Huys, Q.J., Maia, T.V. & Frank, M.J. Computational psychiatry as a bridge from neuroscience to clinical applications. Nat. Neurosci. 19, 404413 (2016).
  42. Beggs, J.M. & Plenz, D. Neuronal avalanches in neocortical circuits. J. Neurosci. 23, 1116711177 (2003).
  43. Friedman, N. et al. Universal critical dynamics in high resolution neuronal avalanche data. Phys. Rev. Lett. 108, 208102 (2012).
  44. Roberts, J.A., Boonstra, T.W. & Breakspear, M. The heavy tail of the human brain. Curr. Opin. Neurobiol. 31, 164172 (2015).
  45. Jansen, B.H. & Rit, V.G. Electroencephalogram and visual evoked potential generation in a mathematical model of coupled cortical columns. Biol. Cybern. 73, 357366 (1995).
  46. Lopes da Silva, F.H., Hoeks, A., Smits, H. & Zetterberg, L.H. Model of brain rhythmic activity. The alpha-rhythm of the thalamus. Kybernetik 15, 2737 (1974).
  47. Marreiros, A.C., Daunizeau, J., Kiebel, S.J. & Friston, K.J. Population dynamics: variance and the sigmoid activation function. Neuroimage 42, 147157 (2008).
  48. Larter, R., Speelman, B. & Worth, R.M. A coupled ordinary differential equation lattice model for the simulation of epileptic seizures. Chaos 9, 795804 (1999).
  49. Breakspear, M., Terry, J.R. & Friston, K.J. Modulation of excitatory synaptic coupling facilitates synchronization and complex dynamics in a nonlinear model of neuronal dynamics. Network 14, 703732 (2003).
  50. Stefanescu, R.A. & Jirsa, V.K. Reduced representations of heterogeneous mixed neural networks with synaptic coupling. Phys. Rev. E 83, 026204 (2011).
  51. Jirsa, V.K. & Stefanescu, R.A. Neural population modes capture biologically realistic large scale network dynamics. Bull. Math. Biol. 73, 325343 (2011).
  52. Miller, P., Brody, C.D., Romo, R. & Wang, X.J. A recurrent network model of somatosensory parametric working memory in the prefrontal cortex. Cereb. Cortex 13, 12081218 (2003).
  53. Wong, K.-F. & Wang, X.-J. A recurrent network mechanism of time integration in perceptual decisions. J. Neurosci. 26, 13141328 (2006).
  54. Breakspear, M., Williams, L.M. & Stam, C.J. A novel method for the topographic analysis of neural activity reveals formation and dissolution of 'dynamic cell assemblies'. J. Comput. Neurosci. 16, 4968 (2004).
  55. Breakspear, M. & Stam, C.J. Dynamics of a neural system with a multiscale architecture. Phil. Trans. R. Soc. Lond. B 360, 10511074 (2005).
  56. Stephan, K.E. et al. Advanced database methodology for the Collation of Connectivity data on the Macaque brain (CoCoMac). Phil. Trans. R. Soc. Lond. B 356, 11591186 (2001).
  57. Horvát, S. et al. Spatial embedding and wiring cost constrain the functional layout of the cortical network of rodents and primates. PLoS Biol. 14, e1002512 (2016).
  58. Woolrich, M.W. & Stephan, K.E. Biophysical network models and the human connectome. Neuroimage 80, 330338 (2013).
  59. Mejias, J.F., Murray, J.D., Kennedy, H. & Wang, X.-J. Feedforward and feedback frequency-dependent interactions in a large-scale laminar network of the primate cortex. Sci. Adv. 2, e1601335 (2016).
  60. Beurle, R.L. Properties of a mass of cells capable of regenerating pulses. Phil. Trans. R. Soc. Lond. B 240, 5594 (1956).
  61. Amari, S. Dynamics of pattern formation in lateral-inhibition type neural fields. Biol. Cybern. 27, 7787 (1977).
  62. Nunez, P.L. The brain wave equation: a model for the EEG. Math. Biosci. 21, 279297 (1974).
  63. Jirsa, V.K. & Haken, H. A derivation of a macroscopic field theory of the brain from the quasi-microscopic neural dynamics. Physica D 99, 503526 (1997).
  64. Robinson, P.A. Patchy propagators, brain dynamics, and the generation of spatially structured gamma oscillations. Phys. Rev. E 73, 041904 (2006).
  65. Laing, C.R. Waves in spatially-disordered neural fields: a case study in uncertainty quantification. in Uncertainty in Biology 367391 (Springer International, 2016).
  66. Rennie, C.J., Robinson, P.A. & Wright, J.J. Unified neurophysical model of EEG spectra and evoked potentials. Biol. Cybern. 86, 457471 (2002).
  67. Robinson, P.A. et al. Eigenmodes of brain activity: neural field theory predictions and comparison with experiment. Neuroimage 142, 7998 (2016).
  68. Muller, L., Reynaud, A., Chavane, F. & Destexhe, A. The stimulus-evoked population response in visual cortex of awake monkey is a propagating wave. Nat. Commun. 5, 3675 (2014).
  69. Rubino, D., Robbins, K.A. & Hatsopoulos, N.G. Propagating waves mediate information transfer in the motor cortex. Nat. Neurosci. 9, 15491557 (2006).
  70. Heitmann, S., Boonstra, T. & Breakspear, M. A dendritic mechanism for decoding traveling waves: principles and applications to motor cortex. PLoS Comput. Biol. 9, e1003260 (2013).
  71. Henderson, J.A. & Robinson, P.A. Geometric effects on complex network structure in the cortex. Phys. Rev. Lett. 107, 018102 (2011).
  72. Roberts, J.A. et al. The contribution of geometry to the human connectome. Neuroimage 124, 379393 (2016).
  73. Coombes, S. & Byrne, Á. Next generation neural mass models. Preprint at https://arxiv.org/abs/1607.06251 (2016).
  74. Moran, R.J. et al. A neural mass model of spectral responses in electrophysiology. Neuroimage 37, 706720 (2007).
  75. Wolf, A., Swift, J.B., Swinney, H.L. & Vastano, J.A. Determining Lyapunov exponents from a time series. Physica D 16, 285317 (1985).
  76. Grassberger, P. & Procaccia, I. Characterization of strange attractors. Phys. Rev. Lett. 50, 346 (1983).
  77. Soong, A.C. & Stuart, C.I. Evidence of chaotic dynamics underlying the human alpha-rhythm electroencephalogram. Biol. Cybern. 62, 5562 (1989).
  78. Pritchard, W.S. & Duke, D.W. Dimensional analysis of no-task human EEG using the Grassberger-Procaccia method. Psychophysiology 29, 182192 (1992).
  79. Babloyantz, A., Salazar, J. & Nicolis, C. Evidence of chaotic dynamics of brain activity during the sleep cycle. Phys. Lett. A 111, 152156 (1985).
  80. Gregson, R.A., Britton, L.A., Campbell, E.A. & Gates, G.R. Comparisons of the nonlinear dynamics of electroencephalograms under various task loading conditions: a preliminary report. Biol. Psychol. 31, 173191 (1990).
  81. Babloyantz, A. & Destexhe, A. Low-dimensional chaos in an instance of epilepsy. Proc. Natl. Acad. Sci. USA 83, 35133517 (1986).
  82. Theiler, J. Spurious dimension from correlation algorithms applied to limited time-series data. Phys. Rev. A Gen. Phys. 34, 24272432 (1986).
  83. Osborne, A.R. & Provenzale, A. Finite correlation dimension for stochastic systems with power-law spectra. Physica D 35, 357381 (1989).
  84. Rapp, P.E., Albano, A.M., Schmah, T.I. & Farwell, L.A. Filtered noise can mimic low-dimensional chaotic attractors. Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 47, 22892297 (1993).
  85. Pritchard, W.S., Duke, D.W. & Krieble, K.K. Dimensional analysis of resting human EEG. II: Surrogate-data testing indicates nonlinearity but not low-dimensional chaos. Psychophysiology 32, 486491 (1995).
  86. Paluš, M. Nonlinearity in normal human EEG: cycles, temporal asymmetry, nonstationarity and randomness, not chaos. Biol. Cybern. 75, 389396 (1996).
  87. Theiler, J., Eubank, S., Longtin, A., Galdrikian, B. & Doyne Farmer, J. Testing for nonlinearity in time series: the method of surrogate data. Physica D 58, 7794 (1992).
  88. Prichard, D. & Theiler, J. Generating surrogate data for time series with several simultaneously measured variables. Phys. Rev. Lett. 73, 951954 (1994).
  89. Stam, C.J., Pijn, J.P., Suffczynski, P. & Lopes da Silva, F.H. Dynamics of the human alpha rhythm: evidence for non-linearity? Clin. Neurophysiol. 110, 18011813 (1999).
  90. Breakspear, M. Nonlinear phase desynchronization in human electroencephalographic data. Hum. Brain Mapp. 15, 175198 (2002).
  91. Freyer, F., Aquino, K., Robinson, P.A., Ritter, P. & Breakspear, M. Bistability and non-Gaussian fluctuations in spontaneous cortical activity. J. Neurosci. 29, 85128524 (2009).
  92. Valdes, P.A., Jiménez, J.C., Riera, J., Biscay, R. & Ozaki, T. Nonlinear EEG analysis based on a neural mass model. Biol. Cybern. 81, 415424 (1999).
  93. Altenburg, J., Vermeulen, R.J., Strijers, R.L., Fetter, W.P. & Stam, C.J. Seizure detection in the neonatal EEG with synchronization likelihood. Clin. Neurophysiol. 114, 5055 (2003).
  94. Lopes da Silva, F. et al. Epilepsies as dynamical diseases of brain systems: basic models of the transition between normal and epileptic activity. Epilepsia 44 (Suppl. 12): 7283 (2003).
  95. Suffczynski, P., Kalitzin, S. & Lopes Da Silva, F.H. Dynamics of non-convulsive epileptic phenomena modeled by a bistable neuronal network. Neuroscience 126, 467484 (2004).
  96. Jirsa, V.K., Stacey, W.C., Quilichini, P.P., Ivanov, A.I. & Bernard, C. On the nature of seizure dynamics. Brain 137, 22102230 (2014).
  97. Aquino, K.M., Schira, M.M., Robinson, P.A., Drysdale, P.M. & Breakspear, M. Hemodynamic traveling waves in human visual cortex. PLoS Comput. Biol. 8, e1002435 (2012).
  98. Nguyen, V.T., Breakspear, M. & Cunnington, R. Reciprocal interactions of the SMA and cingulate cortex sustain premovement activity for voluntary actions. J. Neurosci. 34, 1639716407 (2014).
  99. Bojak, I. & Breakspear, M. Encyclopedia of Computational Neuroscience 19191944 (Springer, 2015).
  100. Friston, K.J., Harrison, L. & Penny, W. Dynamic causal modelling. Neuroimage 19, 12731302 (2003).
  101. Stephan, K.E. et al. Nonlinear dynamic causal models for fMRI. Neuroimage 42, 649662 (2008).
  102. Baker, A.P. et al. Fast transient networks in spontaneous human brain activity. Elife 3, e01867 (2014).
  103. Chang, C. & Glover, G.H. Time-frequency dynamics of resting-state brain connectivity measured with fMRI. Neuroimage 50, 8198 (2010).
  104. Zalesky, A., Fornito, A., Cocchi, L., Gollo, L.L. & Breakspear, M. Time-resolved resting-state brain networks. Proc. Natl. Acad. Sci. USA 111, 1034110346 (2014).
  105. Hansen, E.C., Battaglia, D., Spiegler, A., Deco, G. & Jirsa, V.K. Functional connectivity dynamics: modeling the switching behavior of the resting state. Neuroimage 105, 525535 (2015).
  106. Friston, K., Breakspear, M. & Deco, G. Perception and self-organized instability. Front. Comput. Neurosci. 6, 44 (2012).
  107. Cabral, J., Kringelbach, M.L. & Deco, G. Exploring the network dynamics underlying brain activity during rest. Prog. Neurobiol. 114, 102131 (2014).
  108. Gollo, L.L. & Breakspear, M. The frustrated brain: from dynamics on motifs to communities and networks. Philos. Trans. R. Soc. Lond. B. Biol. Sci. 369, 20130532 (2014).
  109. Deco, G. & Jirsa, V.K. Ongoing cortical activity at rest: criticality, multistability, and ghost attractors. J. Neurosci. 32, 33663375 (2012).
  110. Tsuda, I. Toward an interpretation of dynamic neural activity in terms of chaotic dynamical systems. Behav. Brain Sci. 24, 793810 discussion 810–848 (2001).
  111. Laumann, T.O. et al. On the stability of BOLD fMRI correlations. Cereb. Cortex http://dx.doi.org/10.1093/cercor/bhw265 (2016).
  112. Allen, M. et al. Unexpected arousal modulates the influence of sensory noise on confidence. Elife 5, e18103 (2016).
  113. Chang, C. & Glover, G.H. Relationship between respiration, end-tidal CO2, and BOLD signals in resting-state fMRI. Neuroimage 47, 13811393 (2009).
  114. Nguyen, V.T., Breakspear, M., Hu, X. & Guo, C.C. The integration of the internal and external milieu in the insula during dynamic emotional experiences. Neuroimage 124, 455463 (2016).
  115. Zalesky, A. & Breakspear, M. Towards a statistical test for functional connectivity dynamics. Neuroimage 114, 466470 (2015).
  116. Leonardi, N. & Van De Ville, D. On spurious and real fluctuations of dynamic functional connectivity during rest. Neuroimage 104, 430436 (2015).
  117. Rombouts, S., Keunen, R. & Stam, C. Investigation of nonlinear structure in multichannel EEG. Phys. Lett. A 202, 352358 (1995).
  118. Breakspear, M. “Dynamic” connectivity in neural systems: theoretical and empirical considerations. Neuroinformatics 2, 205226 (2004).
  119. Haimovici, A., Tagliazucchi, E., Balenzuela, P. & Chialvo, D.R. Brain organization into resting state networks emerges at criticality on a model of the human connectome. Phys. Rev. Lett. 110, 178101 (2013).
  120. Roberts, J.A., Iyer, K.K., Vanhatalo, S. & Breakspear, M. Critical role for resource constraints in neural models. Front. Syst. Neurosci. 8, 154159 (2014).
  121. Petkov, G., Goodfellow, M., Richardson, M.P. & Terry, J.R. A critical role for network structure in seizure onset: a computational modeling approach. Front. Neurol. 5, 261 (2014).
  122. Jirsa, V. et al. The virtual epileptic patient: individualized whole-brain models of epilepsy spread. Neuroimage 145, 377388 (2017).
  123. Iyer, K.K. et al. Cortical burst dynamics predict clinical outcome early in extremely preterm infants. Brain 138, 22062218 (2015).
  124. Ching, S., Purdon, P.L., Vijayan, S., Kopell, N.J. & Brown, E.N. A neurophysiological-metabolic model for burst suppression. Proc. Natl. Acad. Sci. USA 109, 30953100 (2012).
  125. Liley, D.T. & Walsh, M. The mesoscopic modeling of burst suppression during anesthesia. Front. Comput. Neurosci. 7, 46 (2013).
  126. Stam, C.J. Nonlinear dynamical analysis of EEG and MEG: review of an emerging field. Clin. Neurophysiol. 116, 22662301 (2005).
  127. Stephan, K.E., Friston, K.J. & Frith, C.D. Dysconnection in schizophrenia: from abnormal synaptic plasticity to failures of self-monitoring. Schizophr. Bull. 35, 509527 (2009).
  128. Breakspear, M. et al. A disturbance of nonlinear interdependence in scalp EEG of subjects with first episode schizophrenia. Neuroimage 20, 466478 (2003).
  129. Wagner, G. et al. Structural and functional dysconnectivity of the fronto-thalamic system in schizophrenia: a DCM-DTI study. Cortex 66, 3545 (2015).
  130. Hyett, M.P., Breakspear, M.J., Friston, K.J., Guo, C.C. & Parker, G.B. Disrupted effective connectivity of cortical systems supporting attention and interoception in melancholia. JAMA Psychiatry 72, 350358 (2015).
  131. Stephan, K.E., Iglesias, S., Heinzle, J. & Diaconescu, A.O. Translational perspectives for computational neuroimaging. Neuron 87, 716732 (2015).
  132. Stephan, K.E. et al. Charting the landscape of priority problems in psychiatry, part 2: pathogenesis and aetiology. Lancet Psychiatry 3, 8490 (2016).
  133. Breakspear, M. et al. Network dysfunction of emotional and cognitive processes in those at genetic risk of bipolar disorder. Brain 138, 34273439 (2015).
  134. Friston, K., Breakspear, M. & Deco, G. Critical slowing and perception. Criticality in Neural Systems (eds. Plenz, D. & Niebur, E.) 191226 (Wiley, 2014).
  135. Loh, M., Rolls, E.T. & Deco, G. A dynamical systems hypothesis of schizophrenia. PLoS Comput. Biol. 3, e228 (2007).
  136. Murray, J.D. et al. Linking microcircuit dysfunction to cognitive impairment: effects of disinhibition associated with schizophrenia in a cortical working memory model. Cereb. Cortex 24, 859872 (2014).
  137. Ruff, C.C. et al. Concurrent TMS-fMRI and psychophysics reveal frontal influences on human retinotopic visual cortex. Curr. Biol. 16, 14791488 (2006).
  138. Cocchi, L. et al. A hierarchy of timescales explains distinct effects of local inhibition of primary visual cortex and frontal eye fields. Elife 5, e15252 (2016).
  139. Kunze, T., Hunold, A., Haueisen, J., Jirsa, V. & Spiegler, A. Transcranial direct current stimulation changes resting state functional connectivity: A large-scale brain network modeling study. Neuroimage 140, 174187 (2016).
  140. Gollo, L.L., Roberts, J.A. & Cocchi, L. Mapping how local perturbations influence systems-level brain dynamics. Neuroimage http://dx.doi.org/%2010.1016/j.neuroimage.2017.01.057 (2016).
  141. Petermann, T. et al. Spontaneous cortical activity in awake monkeys composed of neuronal avalanches. Proc. Natl. Acad. Sci. USA 106, 1592115926 (2009).
  142. Eguíluz, V.M., Chialvo, D.R., Cecchi, G.A., Baliki, M. & Apkarian, A.V. Scale-free brain functional networks. Phys. Rev. Lett. 94, 018102 (2005).
  143. Rubinov, M., Sporns, O., Thivierge, J.P. & Breakspear, M. Neurobiologically realistic determinants of self-organized criticality in networks of spiking neurons. PLoS Comput. Biol. 7, e1002038 (2011).
  144. Levina, A., Herrmann, J.M. & Geisel, T. Dynamical synapses causing self-organized criticality in neural networks. Nat. Phys. 3, 857860 (2007).
  145. Millman, D., Mihalas, S., Kirkwood, A. & Niebur, E. Self-organized criticality occurs in non-conservative neuronal networks during Up states. Nat. Phys. 6, 801805 (2010).
  146. Moran, R.J. et al. Losing control under ketamine: suppressed cortico-hippocampal drive following acute ketamine in rats. Neuropsychopharmacology 40, 268277 (2015).
  147. Breakspear, M. & Knock, S. Kinetic models of brain activity. Brain Imaging Behav. 2, 270288 (2008).
  148. Gatlin, L.L. Information Theory and the Living System (Columbia Univ. Press, 1972).
  149. Linkenkaer-Hansen, K., Nikouline, V.V., Palva, J.M. & Ilmoniemi, R.J. Long-range temporal correlations and scaling behavior in human brain oscillations. J. Neurosci. 21, 13701377 (2001).
  150. Lundstrom, B.N., Higgs, M.H., Spain, W.J. & Fairhall, A.L. Fractional differentiation by neocortical pyramidal neurons. Nat. Neurosci. 11, 13351342 (2008).

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Affiliations

  1. QIMR Berghofer Medical Research Institute, Herston, Queensland, Australia.

    • Michael Breakspear
  2. Metro North Mental Health Service, Herston, Queensland, Australia.

    • Michael Breakspear

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