Abstract
Healthy brain function depends on balancing stable integration between brain areas for effective coordinated functioning, with coexisting segregation that allows subsystems to express their functional specialization. Metastability, a concept from the dynamical systems literature, has been proposed as a key signature that characterizes this balance. Building on this principle, the neuroscience literature has leveraged the phenomenon of metastability to investigate various aspects of brain function in health and disease. However, this body of work often uses the notion of metastability heuristically, and sometimes inaccurately, making it difficult to navigate the vast literature, interpret findings and foster further development of theoretical and experimental methodologies. Here, we provide a comprehensive review of metastability and its applications in neuroscience, covering its scientific and historical foundations and the practical measures used to assess it in empirical data. We also provide a critical analysis of recent theoretical developments, clarifying common misconceptions and paving the road for future developments.
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Fig. 1: From physical substrate to attractor landscapes and spatiotemporal patterns in data.
Fig. 2: Different types of stability and attractors.
Fig. 3: Practical signatures of metastability.
Fig. 4: A graphical overview of routes to transient state switching.
Fig. 5: Duration statistics in models of multi-stable and metastable dynamics.
Code availability
All codes used to estimate the signatures of metastability from fMRI data and to generate Videos 1, 2 and 3 are publicly available. See Supplementary information for details.
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Acknowledgements
The authors thank S. Heitmann for providing instruction and guidance on the Brain Dynamics Toolbox.
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Author notes
These authors contributed equally: Fran Hancock, Fernando E. Rosas.
Authors and Affiliations
Department of Neuroimaging, Institute of Psychiatry, Psychology and Neuroscience, King’s College London, London, UK
Fran Hancock & Federico E. Turkheimer
Department of Informatics, University of Sussex, Brighton, UK
Fernando E. Rosas
Sussex Centre for Consciousness Science, University of Sussex, Brighton, UK
Fernando E. Rosas
Centre for Psychedelic Research, Department of Brain Science, Imperial College London, London, UK
Fernando E. Rosas
Centre for Eudaimonia and Human Flourishing, University of Oxford, Oxford, UK
Fernando E. Rosas, Andrea I. Luppi, Joana Cabral & Morten L. Kringelbach
Sussex AI, University of Sussex, Brighton, UK
Fernando E. Rosas
Centre for Complexity Science, Department of Brain Science, Imperial College London, London, UK
Fernando E. Rosas
St John’s College, University of Cambridge, Cambridge, UK
Andrea I. Luppi
Department of Psychiatry, University of Oxford, Oxford, UK
Andrea I. Luppi
Department of Computational Mathematics, Science and Engineering, Michigan State University, East Lansing, MI, USA
Mengsen Zhang
Department of Computing, Imperial College London, London, UK
Pedro A. M. Mediano
Division of Psychology and Language Sciences, University College London, London, UK
Pedro A. M. Mediano
Life and Health Sciences Research Institute School of Medicine, University of Minho, Braga, Portugal
Joana Cabral
Computational Neuroscience Group, Center for Brain and Cognition, Department of Information and Communication Technologies, Universitat Pompeu Fabra, Barcelona, Spain
Gustavo Deco
Institución Catalana de la Recerca i Estudis Avancats (ICREA), Barcelona, Spain
Gustavo Deco
Department of Neuropsychology, Max Planck Institute for Human Cognitive and Brain Sciences, Leipzig, Germany
Gustavo Deco
School of Psychological Sciences, Monash University Clayton, Melbourne, Victoria, Australia
Gustavo Deco
Center for Music in the Brain, Department of Clinical Medicine, Aarhus University, Aarhus, Denmark
Morten L. Kringelbach
School of Psychological Sciences, College of Engineering, Science and the Environment, University of Newcastle, Newcastle, New South Wales, Australia
Michael Breakspear
Center for Complex Systems and Brain Sciences, Florida Atlantic University, Boca Raton, FL, USA
J. A. Scott Kelso
Intelligent Systems Research Centre, Ulster University, Derry~Londonderry, Northern Ireland
J. A. Scott Kelso
The Bath Institute for the Augmented Human, University of Bath, Bath, UK
J. A. Scott Kelso
The Institute for Human and Synthetic Minds, King’s College London, London, UK
Federico E. Turkheimer
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Fran Hancock
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Contributions
F.H. and F.E.R. researched data for the article. F.H., F.E.R., A.I.L., M.Z., M.B. and J.A.S.K. provided substantial contributions to the discussion of the article’s content. F.H., F.E.R., A.I.L., M.Z., G.D., M.L.K., M.B. and J.A.S.K. wrote the article. F.H., F.E.R., A.I.L., M.Z., P.A.M.M., J.C., G.D., M.L.K., M.B., J.A.S.K. and F.E.T. reviewed and/or edited the manuscript before submission.
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Correspondence to Fran Hancock or Fernando E. Rosas.
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Glossary
Asymmetric
A systematic imbalance in some property of the system.
Attractor
A set of states on which many trajectories converge.
Basin of attraction
All the points in phase space that flow onto the attractor.
Bifurcation
A qualitative change in dynamics produced when a control parameter reaches a critical point.
Bistability
A form of dynamic stability wherein two attractors exist in the dynamics, that is, when there are two stable solutions of the differential equation describing the dynamics.
Chaos
A form of dynamical behaviour that can arise in a time-invariant nonlinear system. Chaos is characterized by sustained aperiodic (nonrepeating) oscillations, leading to extreme sensitivity of future states to small changes in present values of the system.
Chaotic attractor
An attractor that holds dynamics that are highly sensitive to their initial conditions.
Chaotic itinerancy
The behaviour of complicated systems with weakly attracting sets, in which destabilized attractors allow the system to leave its basin of attraction for another through a trajectory of connected saddles.
Control parameters
Parameters that modify a system of differential or difference equations, hence deforming the corresponding flows through phase space.
Crisis
The collision of an unstable periodic orbit and a coexisting chaotic attractor.
Critical fluctuations
Stochastic fluctuations that are orders of magnitude larger than normal, which occur when a system is close to a critical point. They may be sufficient to kick the system out of its basin and into the region of another attractor. A switch will occur, even though the original fixed point may still be classified as stable.
Critical point
The value of a control parameter at which a bifurcation occurs.
Dynamical systems theory
A branch of mathematics that studies how the state of systems evolves over time based on either an analytical (pencil and paper), a geometric (shapes) or a numeric (approximations using a computer) study of deterministic evolution equations.
Dynamic instabilities
Behavioural changes of the system in the vicinity of a bifurcation.
Fixed point
A point in the state space wherein the rate of change of the system with respect to time is equal to zero, corresponding to states at which the system remains unchanged unless perturbed.
Ghost attractors
Regions of phase space wherein the memory of a fixed point is attractive for the system. The memory is created by the annihilation of a fixed point and a repeller when a control parameter is changed.
Hidden Markov models
Statistical models that are used to describe the probabilistic relationship between sequences of observations and sequences of hidden states. They are used to classify sequences or predict future observations based on the underlying hidden processes that generate the data.
Metastability
A specific type of dynamics that may take place in a system with coexisting tendencies of attraction and repulsion, and is characterized by patterns that recur either in repeatable sequences (pattern) or flexible alternation (no pattern).
Milnor attractor
An attractor that no longer attracts all trajectories in its basin of attraction following an arbitrary small perturbation.
Monostability
A form of dynamic stability wherein a single attractor exists in the dynamics, that is, when there is one stable solution to the differential equation describing the dynamics.
Multi-stability
A form of dynamic stability wherein multiple stable attractors exist in the dynamics, that is, when there are two or more stable solutions to the differential equation describing the dynamics.
Order parameter
A single variable that captures the collective or macro-behaviour of a system composed of microscopic elements.
Phase space
The set of all possible states and, hence, contains all the allowed combinations of values of the variables of a system (also known as state variables).
Repeller
A set of states from which many trajectories migrate.
Saddles
Fixed points that are stable in one direction but unstable in another. Trajectories approach a saddle and are repelled away from it at the fixed point.
States
A state is a configuration of the variables of a system that is a solution to the equations.
Trajectory
A sequence of states within the phase space that satisfies the dynamics of the system as defined by its differential equation.
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Hancock, F., Rosas, F.E., Luppi, A.I. et al. Metastability demystified — the foundational past, the pragmatic present and the promising future. Nat. Rev. Neurosci. (2024). https://doi.org/10.1038/s41583-024-00883-1
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Accepted:01 November 2024
Published:11 December 2024
DOI:https://doi.org/10.1038/s41583-024-00883-1
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