Sequences by Metastable Attractors: Interweaving Dynamical Systems and Experimental Data

Hutt, Axel; Graben, Peter beim

doi:10.3389/fams.2017.00011

Cited by 20 publications

(14 citation statements)

References 55 publications

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“…The third and last proposed mechanism shown in panels Figure 9C corresponds to a system possessing a robust heteroclinic cycle between two saddle equilibria (Rabinovich et al, 2008; Rodrigues and Labouriau, 2014; beim Graben and Hutt, 2015; Hutt and beim Graben, 2017). In this case, the metastable states are in fact saddles where the unstable manifold of each saddle connects to the other saddle, hence allowing for a stable robust heteroclinic cycle to exist.…”

Section: Discussionmentioning

confidence: 99%

“…which induces another recurrence grammar from which a Hausdorff clustering of metastable states is obtained; for further details see Hutt and beim Graben (2017). The threshold θ must be chosen in such a way that the entire ensemble E is covered by a minimal set of metastable states.…”

Section: Methodsmentioning

confidence: 99%

“…For each metastable state S i , which is actually given as a spherical distribution of sampling points, we select a representative point. Hutt and beim Graben (2017) suggested to use the ensemble mean, i.e., the barycenter of the distribution. Here, however, we choose the vector…”

Section: Methodsmentioning

confidence: 99%

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Metastable Resting State Brain Dynamics

Graben

Jiménez-Marín

Diez

et al. 2019

Front. Comput. Neurosci.

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Metastability refers to the fact that the state of a dynamical system spends a large amount of time in a restricted region of its available phase space before a transition takes place, bringing the system into another state from where it might recur into the previous one. beim Graben and Hutt (2013) suggested to use the recurrence plot (RP) technique introduced by Eckmann et al. (1987) for the segmentation of system's trajectories into metastable states using recurrence grammars. Here, we apply this recurrence structure analysis (RSA) for the first time to resting-state brain dynamics obtained from functional magnetic resonance imaging (fMRI). Brain regions are defined according to the brain hierarchical atlas (BHA) developed by Diez et al. (2015), and as a consequence, regions present high-connectivity in both structure (obtained from diffusion tensor imaging) and function (from the blood-level dependent-oxygenation—BOLD—signal). Remarkably, regions observed by Diez et al. were completely time-invariant. Here, in order to compare this static picture with the metastable systems dynamics obtained from the RSA segmentation, we determine the number of metastable states as a measure of complexity for all subjects and for region numbers varying from 3 to 100. We find RSA convergence toward an optimal segmentation of 40 metastable states for normalized BOLD signals, averaged over BHA modules. Next, we build a bistable dynamics at population level by pooling 30 subjects after Hausdorff clustering. In link with this finding, we reflect on the different modeling frameworks that can allow for such scenarios: heteroclinic dynamics, dynamics with riddled basins of attraction, multiple-timescale dynamics. Finally, we characterize the metastable states both functionally and structurally, using templates for resting state networks (RSNs) and the automated anatomical labeling (AAL) atlas, respectively.

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Section: Discussionmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

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Metastable Resting State Brain Dynamics

Graben

Jiménez-Marín

Diez

et al. 2019

Front. Comput. Neurosci.

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“…Facchini, Kantz, and Tiezzi [123] show that characteristic curved patterns in an RP can be caused by nonstationarities in the signal, in particular by a linearly increasing or a periodically modulated carrier frequency, and that such nonstationarities are not captured in a traditional spectrogram. More recently, Hutt, and beim Graben [124] propose a new RP-based approach that can identify metastable attractors and heteroclinic orbits, a task that is typically challenging using standard methods.…”

Section: Quantification Based On Recurrence Patternsmentioning

confidence: 99%

A Brief Introduction to Nonlinear Time Series Analysis and Recurrence Plots

Goswami

2019

Vibration

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Nonlinear time series analysis gained prominence from the late 1980s on, primarily because of its ability to characterize, analyze, and predict nontrivial features in data sets that stem from a wide range of fields such as finance, music, human physiology, cognitive science, astrophysics, climate, and engineering. More recently, recurrence plots, initially proposed as a visual tool for the analysis of complex systems, have proven to be a powerful framework to quantify and reveal nontrivial dynamical features in time series data. This tutorial review provides a brief introduction to the fundamentals of nonlinear time series analysis, before discussing in greater detail a few (out of the many existing) approaches of recurrence plot-based analysis of time series. In particular, it focusses on recurrence plot-based measures which characterize dynamical features such as determinism, synchronization, and regime changes. The concept of surrogate-based hypothesis testing, which is crucial to drawing any inference from data analyses, is also discussed. Finally, the presented recurrence plot approaches are applied to two climatic indices related to the equatorial and North Pacific regions, and their dynamical behavior and their interrelations are investigated.

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“…Transitions between attractors. The concept of attractor is actually central in describing brain dynamics [178,179]. Especially, a current trend in neuroscience is to associate to brain states attractors (or ghost attractors, see [180] and references therein).…”

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confidence: 99%

Thermodynamic Formalism in Neuronal Dynamics and Spike Train Statistics

Cofré

Maldonado

Cessac

2020

Preprint

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The Thermodynamic Formalism provides a rigorous mathematical framework to study quantitative and qualitative aspects of dynamical systems. At its core there is a variational principle and corresponding, in its simplest form, to the Maximum Entropy principle, used as a statistical inference procedure to represent, by specific probability measures (Gibbs measures), the collective behaviour of complex systems. This framework has found applications in different domains of scienThe Thermodynamic Formalism provides a rigorous mathematical framework to study quantitative and qualitative aspects of dynamical systems. At its core there is a variational principle and corresponding, in its simplest form, to the Maximum Entropy principle, used as a statistical inference procedure to represent, by specific probability measures (Gibbs measures), the collective behaviour of complex systems. This framework has found applications in different domains of science, in particular, has been fruitful and influential in neurosciences. In this article, we review how the Thermodynamic Formalism can be exploited in the field of theoretical neuroscience, as a conceptual and operational tool, to link the dynamics of interacting neurons and the statistics of action potentials from either experimental data or mathematical models. We comment on perspectives and open problems in theoretical neuroscience that could be addressed within this formalism.ce, in particular, has been fruitful and influential in neurosciences. In this article, we review how the Thermodynamic Formalism can be exploited in the field of theoretical neuroscience, as a conceptual and operational tool, to link the dynamics of interacting neurons and the statistics of action potentials from either experimental data or mathematical models. We comment on perspectives and open problems in theoretical neuroscience that could be addressed within this formalism.

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Sequences by Metastable Attractors: Interweaving Dynamical Systems and Experimental Data

Cited by 20 publications

References 55 publications

Metastable Resting State Brain Dynamics

Metastable Resting State Brain Dynamics

A Brief Introduction to Nonlinear Time Series Analysis and Recurrence Plots

Thermodynamic Formalism in Neuronal Dynamics and Spike Train Statistics

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