Transients Versus Attractors in Complex Networks

Muezzinoglu, Mehmet K.; Tristan, Irma; Huerta, Ramón; Afraimovich, Valentin; Rabinovich, M. I.

doi:10.1142/s0218127410026745

Cited by 22 publications

(19 citation statements)

References 39 publications

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“…The non-symmetric inhibitory interaction between the modes helps to solve an apparent paradox related to the notion that sensitivity and reliability in a network can coexist: the joint action of the external input and a stimulus-dependent connectivity matrix defines the stimulus-specific sequence. Dynamical chaos can also be observed in this case [46]. Furthermore, a specific kind of the dynamical chaos, where the order of the switching is deterministic, but the lifetime of the metastable states is random, is possible [47].…”

Section: Methodsmentioning

confidence: 95%

Dynamical Principles of Emotion-Cognition Interaction: Mathematical Images of Mental Disorders

et al. 2010

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The key contribution of this work is to introduce a mathematical framework to understand self-organized dynamics in the brain that can explain certain aspects of itinerant behavior. Specifically, we introduce a model based upon the coupling of generalized Lotka-Volterra systems. This coupling is based upon competition for common resources. The system can be regarded as a normal or canonical form for any distributed system that shows self-organized dynamics that entail winnerless competition. Crucially, we will show that some of the fundamental instabilities that arise in these coupled systems are remarkably similar to endogenous activity seen in the brain (using EEG and fMRI). Furthermore, by changing a small subset of the system's parameters we can produce bifurcations and metastable sequential dynamics changing, which bear a remarkable similarity to pathological brain states seen in psychiatry. In what follows, we will consider the coupling of two macroscopic modes of brain activity, which, in a purely descriptive fashion, we will label as cognitive and emotional modes. Our aim is to examine the dynamical structures that emerge when coupling these two modes and relate them tentatively to brain activity in normal and non-normal states.

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Section: Methodsmentioning

confidence: 95%

Dynamical Principles of Emotion-Cognition Interaction: Mathematical Images of Mental Disorders

et al. 2010

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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

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Perl

et al. 2019

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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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“…A widely-accepted rate model of competition among N agents is the Generalized Lotka–Volterra (GLV) system (Muezzinoglu et al 2010; Murray 2002)

\frac{d x_{i}}{d t} = x_{i} true(σ_{i} (I) - \sum_{j = 1}^{N} ρ_{i j} (I) x_{j} + η (t) true), i = 1, \dots, N,

where x i ≥ 0 denotes the i th competitor, I summarizes all observable environmental factors that influence competition, σ i ≥ 0 is the resources available for the competitor i to prosper, ρ ij is the competition matrix with nonnegative entries, and η is a noise process, capturing all unpredictable effects from the environment.…”

Section: Competition In the Brainmentioning

confidence: 99%

Nonlinear Dynamics of Emotion-Cognition Interaction: When Emotion Does not Destroy Cognition?

et al. 2010

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Emotion (i.e., spontaneous motivation and subsequent implementation of a behavior) and cognition (i.e., problem solving by information processing) are essential to how we, as humans, respond to changes in our environment. Recent studies in cognitive science suggest that emotion and cognition are subserved by different, although heavily integrated, neural systems. Understanding the time-varying relationship of emotion and cognition is a challenging goal with important implications for neuroscience. We formulate here the dynamical model of emotion-cognition interaction that is based on the following principles: (1) the temporal evolution of cognitive and emotion modes are captured by the incoming stimuli and competition within and among themselves (competition principle); (2) metastable states exist in the unified emotion-cognition phase space; and (3) the brain processes information with robust and reproducible transients through the sequence of metastable states. Such a model can take advantage of the often ignored temporal structure of the emotion-cognition interaction to provide a robust and generalizable method for understanding the relationship between brain activation and complex human behavior. The mathematical image of the robust and reproducible transient dynamics is a Stable Heteroclinic Sequence (SHS), and the Stable Heteroclinic Channels (SHCs). These have been hypothesized to be possible mechanisms that lead to the sequential transient behavior observed in networks. We investigate the modularity of SHCs, i.e., given a SHS and a SHC that is supported in one part of a network, we study conditions under which the SHC pertaining to the cognition will continue to function in the presence of interfering activity with other parts of the network, i.e., emotion.

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Transients Versus Attractors in Complex Networks

Cited by 22 publications

References 39 publications

Dynamical Principles of Emotion-Cognition Interaction: Mathematical Images of Mental Disorders

Dynamical Principles of Emotion-Cognition Interaction: Mathematical Images of Mental Disorders

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

Nonlinear Dynamics of Emotion-Cognition Interaction: When Emotion Does not Destroy Cognition?

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