Chaos in neurons and its application: Perspective of chaos engineering

Hirata, Yoshito; Oku, Makito; Aihara, Kazuyuki

doi:10.1063/1.4738191

Cited by 33 publications

(21 citation statements)

References 53 publications

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“…Important examples of such fast-slow dynamics [6][7][8] are found in chemical reactions, laser dynamics and in mathematical neuron models. Understanding how to generate and control a burst of spikes in neuron cells, and how chaotic behavior can appear in such systems [9][10][11] are some of the most fundamental questions in neuroscience. The key questions that we want to address are: how is this chaotic behavior organized?…”

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confidence: 99%

Symbolic dynamical unfolding of spike-adding bifurcations in chaotic neuron models

Barrio¹,

Lefranc²,

Martínez³

et al. 2015

EPL

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PACS 05.45.Ac -Low-dimensional chaos PACS 05.45.Pq -Numerical simulations of chaotic systems PACS 87.19.ll -Models of single neurons and networks Abstract -We characterize the systematic changes in the topological structure of chaotic attractors that occur as spike-adding and homoclinic bifurcations are encountered in the slow-fast dynamics of neuron models. This phenomenon is detailed in the simple Hindmarsh-Rose neuron model, where we show that the unstable periodic orbits appearing after each spike-adding bifurcation are associated with specific symbolic sequences in the canonical symbolic encoding of the dynamics of the system. This allows us to understand how these bifurcations modify the internal structure of the chaotic attractors.The wide-range assessment of brain and behaviors is one of the pivotal challenges of this century. To understand how an incredibly sophisticated system such as the brain per se functions dynamically, it is imperative to study the dynamics of its constitutive elements -neurons. Therefore, the design of mathematical models for neurons has arisen as a trending topic in science for a few decades, since Hodgkin and Huxley developed the first model of action potentials in the neuron membrane [1]. Starting from that seminal mathematical model, a lot of variant models describing different kinds of neuron cells in numerous animals have been proposed in the literature. For instance, a reduced model [2-4] of the bursting of leech heart neuron is specified by the following three equations derived through the Hodgkin-Huxley gated variable formalism: As control parameters of the system are varied, its dynamics undergoes bifurcations such as classified by dynamical systems theory that match qualitative metamorphoses in terms specific to neuroscience. A broad range of non-stationary activity types can be observed, which includes regular and irregular tonic spiking, bursting and mixed-mode oscillations and combinations of them, as well as oscillatory transients toward quiescent states. In terms of dynamical system theory, these behaviors correspond to stable periodic and deterministically chaotic orbits in the phase space of the model. Figure 1 represents a twoparameter sweep of the leech model in the (τ K2 , I app )-plane, where the number of spikes per period, measured with the spike-counting method (SPC) [5], is color-coded. Banded structures correlated with zones of chaotic behavior appear clearly in this diagram. They are associated with spike-adding bifurcations, where the number of spikes is incremented by one, as indicated in the figure.Spike-adding bifurcations are special bifurcations that are common in fast-slow systems. They lead to the appearance of extra spikes (turns) in the fast manifold region and are quite important in that they progressively p-1

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confidence: 99%

Symbolic dynamical unfolding of spike-adding bifurcations in chaotic neuron models

Barrio¹,

Lefranc²,

Martínez³

et al. 2015

EPL

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“…15 in the RWC Music Database (Musical Instrument Sound). Previous studies on squid giant axons have demonstrated the chaotic nature of the underlying dynamics20212223. These time-series are both scalar and real-valued.…”

Section: Resultsmentioning

confidence: 99%

Predicting disease progression from short biomarker series using expert advice algorithm

Morino

Hirata

Tomioka

et al. 2015

Sci Rep

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Well-trained clinicians may be able to provide diagnosis and prognosis from very short biomarker series using information and experience gained from previous patients. Although mathematical methods can potentially help clinicians to predict the progression of diseases, there is no method so far that estimates the patient state from very short time-series of a biomarker for making diagnosis and/or prognosis by employing the information of previous patients. Here, we propose a mathematical framework for integrating other patients' datasets to infer and predict the state of the disease in the current patient based on their short history. We extend a machine-learning framework of “prediction with expert advice” to deal with unstable dynamics. We construct this mathematical framework by combining expert advice with a mathematical model of prostate cancer. Our model predicted well the individual biomarker series of patients with prostate cancer that are used as clinical samples.

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“…Hirata et al 38 revisit a data set collected from the giant axon of the squid (the preparation used in Hodgkin and Huxley's Nobel Prize winning work in which the dynamics of action potentials were first modeled). The authors show that a relaxed, numerically adapted version of Devaney's criteria for chaos 39 identifies the neuron's voltage time series as chaotic, and also briefly describe a simple mapping that produces deterministic chaos in a neural network model of memory recall.…”

Section: The Papers In This Issuementioning

confidence: 99%

“…While most of these papers were prepared independently (one pair 30,31 does share three authors and another 38,40 shares one), they nicely illustrate common interests in nonlinear dy-namics. In particular, computational methods (rigorously based, as in Refs.…”

Section: The Papers In This Issuementioning

confidence: 99%

“…Symmetries and bifurcations, cross sections and Poincar e maps are prevalent, and increasingly high dimensional phase spaces are being considered. Several papers discuss multiple applications, or present an approach or methods that apply beyond their specific examples, those of Bush et al, 35 Cvitanovič et al, 33 Hirata et al, 38 Lingala et al, 36 and Budi sić et al 37 provide examples.…”

Section: The Papers In This Issuementioning

confidence: 99%

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Introduction to the focus issue: Fifty years of chaos: Applied and theoretical

Hikihara

Holmes

Kambe

et al. 2012

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The discovery of deterministic chaos in the late nineteenth century, its subsequent study, and the development of mathematical and computational methods for its analysis have substantially influenced the sciences. Chaos is, however, only one phenomenon in the larger area of dynamical systems theory. This Focus Issue collects 13 papers, from authors and research groups representing the mathematical, physical, and biological sciences, that were presented at a symposium held at Kyoto University from November 28 to December 2, 2011. The symposium, sponsored by the International Union of Theoretical and Applied Mechanics, was called 50 Years of Chaos: Applied and Theoretical. Following some historical remarks to provide a background for the last 50 years, and for chaos, this Introduction surveys the papers and identifies some common themes that appear in them and in the theory of dynamical systems.

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Chaos in neurons and its application: Perspective of chaos engineering

Cited by 33 publications

References 53 publications

Symbolic dynamical unfolding of spike-adding bifurcations in chaotic neuron models

Symbolic dynamical unfolding of spike-adding bifurcations in chaotic neuron models

Predicting disease progression from short biomarker series using expert advice algorithm

Introduction to the focus issue: Fifty years of chaos: Applied and theoretical

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