Transient Dynamics in Complex Systems: Heteroclinic Sequences with Multidimensional Unstable Manifolds

Afraimovich, Valentin; Tristan, Irma; Varona, Pablo; Rabinovich, M. I.

doi:10.5890/dnc.2012.11.001

Cited by 20 publications

(15 citation statements)

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“…It has been suggested that the brain phase space displays funnel-like locations where trajectories converge towards the shortest path as time progresses Sengupta et al, 2016). Other neuroscientists suggested that brain function does not exhibit erratic brain dynamics nor attractors, rather a stable sequence of transient heteroclinic channels (Afraimovich et al, 2013). Further, concepts such as communication-through-coherence (Deco and Jirsa, 2012) and collective movements (Touboul 2012; Tozzi, 2015) must be taken into account.…”

Section: Discussionmentioning

confidence: 99%

Non-Equilibrium Belousov–zhabotinsky Reaction Describes the Spontaneous Activity of the Brain

Zhang

Guo

Sun

et al. 2019

Preprint

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Belousov-Zhabotinsky (BZ) reactions describe chemical patterns in terms of non-equilibrium thermodynamics, chaotic evolution of nonlinear oscillators, excitability under the influence of peculiar chemical stimulations. In touch with this chemical model, we hypothesized that the nonlinear brain electric activity at the edge of the chaos could be triggered by neural oscillation equipped with BZ dynamics, and that changes in neural activity might be correlated with the transient occurrence of (either short and long-range) BZ-like reactions in cortical subareas. To prove our hypothesis of neuronal waves driven by BZ-like processes, we evaluated fMRI movies that assess in vivo BOLD resting state activity of the human brain. We found that the spontaneous activity of the brain display features fully overlapping the recentlyintroduced local circuits based on BZ chemical reaction. Therefore, neuronal paths during spontaneous activity of the brain match BZ dynamics' previsions. Our results point towards the brain as crossed by diffusive nonlinear neural oscillations patterns that are predictable. Furthermore, our results suggest that chaotic dynamics arise from nothing else than the network arrangements subtending physical and biological systems.

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Section: Discussionmentioning

confidence: 99%

Non-Equilibrium Belousov–zhabotinsky Reaction Describes the Spontaneous Activity of the Brain

Zhang

Guo

Sun

et al. 2019

Preprint

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“…To analyze the robustness of the heteroclinic sequence against small perturbations we consider here the case where metastable states are just saddle equilibrium points N S S , , 1  of a system ), (x F x   where x is n-dimensional vector (Afraimovich et al, 2012 (Afraimovich et al, 2013)) leave a vicinity of  and go to other attracting sets in the phase space. The temporal characteristics of the transients are related to the exit problem for small random perturbations of dynamical systems with saddle sets.…”

Section: Robustness Of Sequential Cognitive Dynamics Based On Heterocmentioning

confidence: 99%

Hierarchical nonlinear dynamics of human attention

Rabinovich

Tristan

Varona

2015

Neuroscience & Biobehavioral Reviews

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Attention is the process of focusing mental resources on a specific cognitive/behavioral task. Such brain dynamics involves different partially overlapping brain functional networks whose interconnections change in time according to the performance stage, and can be stimulus-driven or induced by an intrinsically generated goal. The corresponding activity can be described by different families of spatiotemporal discrete patterns or sequential dynamic modes. Since mental resources are finite, attention modalities compete with each other at all levels of the hierarchy, from perception to decision making and behavior. Cognitive activity is a dynamical process and attention possesses some universal dynamical characteristics. Thus, it is time to apply nonlinear dynamical theory for the description and prediction of hierarchical attentional tasks. Such theory has to include the analyses of attentional control stability, the time cost of attention switching, the finite capacity of informational resources in the brain, and the normal and pathological bifurcations of attention sequential dynamics. In this paper we have integrated today's knowledge, models and results in these directions.

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“…The autonomous LV equations have been used to generate reproducible transient sequences in neural circuits [60][61][62][63][64] , but global brain dynamics do not converge or stabilize around a fixed set of invariants, and might be described as a continuous flow of quick and irregular oscillations 27,65 . Thus, instead of describing dynamics in terms of asymptotic behavior, an alternative 3/19 consist of introducing the dynamical informational structures (DISs), defined as the ISs indexed continuously by time (see Fig.2).…”

Section: Measuring Energy Levels From Fmri Datamentioning

confidence: 99%

Dynamical informational structures characterize the different human brain states of wakefulness and deep sleep

Galadí

Pereira

Perl

et al. 2019

Preprint

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The dynamical activity of the human brain describes an extremely complex energy landscape changing over time and its characterisation is central unsolved problem in neuroscience. We propose a novel mathematical formalism for characterizing how the landscape of attractors sustained by a dynamical system evolves in time. This mathematical formalism is used to distinguish quantitatively and rigorously between the different human brain states of wakefulness and deep sleep. In particular, by using a whole-brain dynamical ansatz integrating the underlying anatomical structure with the local node dynamics based on a Lotka-Volterra description, we compute analytically the global attractors of this cooperative system and their associated directed graphs, here called the informational structures. The informational structure of the global attractor of a dynamical system describes precisely the past and future behaviour in terms of a directed graph composed of invariant sets (nodes) and their corresponding connections (links). We characterize a brain state by the time variability of these informational structures. This theoretical framework is potentially highly relevant for developing reliable biomarkers of patients with e.g. neuropsychiatric disorders or different levels of coma. of the phase space described by a set of selected invariant global solutions of the associated dynamical system, such as stationary points (equilibria), connecting orbits among them, periodic solutions and limit cycles. The informational structure of the GA is defined as a directed graph composed of nodes associated with those invariants and links establishing their connections (see Methods and Supplementary Information for a formal rigorous definition). Informational Structure has been used to show the dependence between the topology, the value of the parameters and the state with respect to its level of integration 14 , as such pointing for the small world configuration of the brain 16 (although see recent controversies 17 ).Previous research has applied Informational Structure to population dynamics in complex networks 18,19 . Equally used for modelling mutualistic systems in Theoretical Ecology and Economy 20, 21 , Informational Structure can relate the topology of complex networks and their dynamics. In mutualistic systems when achieving robustness and life abundance -the so-called architecture of biodiversity 22, 23 -the dynamics is determined not only by the topology 24 but also by modularity 25 and the strength of the parameters 26 .These kinds of relationships are also at the heart of many important open questions in neuroscience 27,28 . Informational Structures and their continuous time-dependence on the strength of connections can allow to understand the dynamics of the system as a coherent process, whose information is structured, and potentially providing new insights into sudden bifurcations 29 .Here, we were interested in characterising human sleep, which is traditionally subdivided into different stages that alternate in the course...

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Transient Dynamics in Complex Systems: Heteroclinic Sequences with Multidimensional Unstable Manifolds

Cited by 20 publications

References 0 publications

Non-Equilibrium Belousov–zhabotinsky Reaction Describes the Spontaneous Activity of the Brain

Non-Equilibrium Belousov–zhabotinsky Reaction Describes the Spontaneous Activity of the Brain

Hierarchical nonlinear dynamics of human attention

Dynamical informational structures characterize the different human brain states of wakefulness and deep sleep

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