Focus on human brain mapping

Dynamic models of large-scale brain activity

Journal name:
Nature Neuroscience
Year published:
Published online


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


  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.


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  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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