Chimera states in networks of biological neurons and coupled damped pendulums
|
|
- Arabella Wilkerson
- 5 years ago
- Views:
Transcription
1 in neural models in networks of pendulum-like elements in networks of biological neurons and coupled damped pendulums J. Hizanidis 1, V. Kanas 2, A. Bezerianos 3, and T. Bountis 4 1 National Center for Scientific Research, Demokritos 2 Department of Electrical and Computer Engineering, University of Patras 3 Cognitive Engineering Lab, Singapore Institute for Neuroengineering, National University of Singapore 4 Department of Mathematics, University of Patras July / 35
2 in neural models in networks of pendulum-like elements Outline 1 Synchronization in complex networks The Kuramoto model 2 History Example of systems exhibiting chimera states 3 in neural models 4 in networks of pendulum-like elements 5 2 / 35
3 in neural models in networks of pendulum-like elements Synchronization in complex networks Synchronization in complex networks The Kuramoto model Emerging phenomenon in which an enormous system of oscillators spontaneously locks to a common frequency, despite the inevitable differences in the natural frequencies of the individual oscillators. Synchronization processes are ubiquitous in nature and play an important role in many contexts (biology, ecology, technology) networks of pacemaker cells in the heart synchronously flashing fireflies and crickets that chirp in unison arrays of lasers and microwave oscillators 3 / 35
4 in neural models in networks of pendulum-like elements The Kuramoto model Synchronization in complex networks The Kuramoto model 1967: Arthur T. Winfree considered biological oscillators as phase oscillators, neglecting the amplitude 1975: Yoshiki Kuramoto developed an analytical theory to explain synchronization of globally coupled phase oscillators θ i = ω i + K N N sin(θ j θ i ), j=1 i = 1... N K: coupling constant N: total numer of oscillators g(ω): natural frequency distribution, symmetric about Ω 4 / 35
5 in neural models in networks of pendulum-like elements Synchronization in complex networks The Kuramoto model The Kuramoto model describes a large population of coupled phase oscillators with distributed natural frequencies. If the coupling strength exceeds a certain threshold, the system exhibits a phase transition: some of the oscillators spontaneously synchronize, while others remain incoherent. Globally coupled oscillators settle in one of a few basic patters: Synchrony Travelling waves (1D), spiral waves (2D) Incoherence (no spatial structure) Spatiotemporal chaos 5 / 35
6 in neural models in networks of pendulum-like elements Kuramoto and Battogtokh (2002) Synchronization in complex networks The Kuramoto model Complex Ginzburg-Landau equation with weak coupling: φ(x, t) t = ω 1 0 G(x x ) sin[φ(x, t) φ(x, t) + α]dx φ(x, t): phase of the oscillator at position x and time t G(x x ) e κ x x nonlocal coupling 6 / 35
7 in neural models in networks of pendulum-like elements Synchronization in complex networks The Kuramoto model Coexistence of Coherence and Incoherence 7 / 35
8 in neural models in networks of pendulum-like elements Abrams and Strogatz (2004) Synchronization in complex networks The Kuramoto model θ(x, t) t π = ω G(x x ) sin[θ(x, t) θ(x, t) + α]dx π G(x) = 1 (1 + A cos x), 0 A 1 2π cosine kernel allows the model to be solved analytically: stability of chimera states dynamics and bifurcations 8 / 35
9 in neural models in networks of pendulum-like elements The chimera state History Example of systems exhibiting chimera states strange mode of syncrhonization: chimera state never been seen in in systems with local or global coupling nothing to do with partially locked/partially incoherent states that occur in populations of nonidentical oscillators 9 / 35
10 in neural models in networks of pendulum-like elements History Example of systems exhibiting chimera states 10 / 35
11 in neural models in networks of pendulum-like elements History Example of systems exhibiting chimera states Dichotomy between synchrony and disorder With local or global coupling, identical oscillators either synchronize or oscillate incoherently, but never do both simultaneously With nonlocal coupling, identical oscillators can split into two coexisting domains, synchronous and asynchronous Chimera state: counterintuitive phenomenon: coexistence of coherent and incoherent oscillations in populations of symmetrically coupled identical oscillators 11 / 35
12 in neural models in networks of pendulum-like elements Coupling schemes in networks History Example of systems exhibiting chimera states r = R/N coupling radius N: total number of elements in the network R: number of coupled nearest neighbors 12 / 35
13 in neural models in networks of pendulum-like elements in two subnetworks Abrams et al PRL, (2008) History Example of systems exhibiting chimera states dθ 1 i dt dθ 2 i dt j=1 j=1 = ω + µ sin(θj 1 θj 1 α) + ν sin(θj 2 θj 1 α) N N N N = ω + µ j=1 sin(θj 2 θj 2 α) + ν j=1 sin(θj 1 θj 2 α) N N N N 13 / 35
14 in neural models in networks of pendulum-like elements in chaotic systems Coupled chaotic logistic maps: History Example of systems exhibiting chimera states z i n+1 = f (z i n ) + σ 2R Coupled Rössler systems: i+r j=i R [f (z j n ) f (z i n )], f (z) = az(1 z) ẋ i = y i z i + σ 2R i+r j=i R (x j x i ) ẏ i = x i + ay i ż i = b + z i (x i c), i = 1... N 14 / 35
15 in neural models in networks of pendulum-like elements Experimental evidence of chimera states History Example of systems exhibiting chimera states 15 / 35
16 in neural models in networks of pendulum-like elements History Example of systems exhibiting chimera states in the context of neuroscience Unihemispheric sleep Many birds as well as dolphins sleep with one eye open, in the sense that one hemisphere of the brain is synchronous while the other is asynchronous. 16 / 35
17 in neural models in networks of pendulum-like elements History Example of systems exhibiting chimera states in the FitzHugh-Nagumo model Omelchencko et al. PRL 2013 ɛ u k = u k u3 k 3 v k v k = u k + a 17 / 35
18 in neural models in networks of pendulum-like elements Single neuron dynamics History Example of systems exhibiting chimera states The FitzHugh-Nagumo system reproduces neuron spiking behaviour and is a simplification of the Hodgkin-Huxley model (1952): Set of nonlinear ordinary differential equations that approximate ionic mechanisms underlying the initiation and propagation of action potentials We will use a more realistic system: the Hindmarhs-Rose model 18 / 35
19 in neural models in networks of pendulum-like elements History Example of systems exhibiting chimera states The Hindmarsh-Rose (HR) model (1984) ẋ = ax 3 + bx 2 + y z + J ẏ = c dx 2 y ż = r(s(x x 0 ) z) x: membrane potential y: activity of fast gated ion channels (Na + and K + ) z: activity of slow gated ion channels (Ca and Cl ) system parameters: s = 4, x 0 = 1.6, a = 1, b = 3, c = 1, d = 5, r = / 35
20 in neural models in networks of pendulum-like elements Typical HR bursting patterns History Example of systems exhibiting chimera states external stimulus J = 2 20 / 35
21 in neural models in networks of pendulum-like elements Two-dimensional HR model (1982) ẋ = ax 3 + bx 2 + y + J ẏ = c dx 2 y z variable eliminated no ability to produce bursting modes no firing frequency adaptation 21 / 35
22 in neural models in networks of pendulum-like elements Network of 2D-HR oscillators ẋ k = y k xk 3 + 3x 2 + J + σ j=k+r x [b xx (x j x k ) + b xy (y j y k )] 2R j=k R ẏ k = 1 5xk 2 y k + σ j=k+r y [b yx (x j x k ) + b yy (y j y k )] 2R j=k R Coupling matrix (Omelchencko et al. 2013): ( ) ( buu b B = uv cosφ sinφ = sinφ cosφ b vu b vv ) 22 / 35
23 in neural models in networks of pendulum-like elements Snapshots at t = 3000, N = 1000, σ x = σ y = 0.1, J = 0 Mixed oscillatory states (MOS), wave-like patterns Chimera state for diagonal coupling Multi-chimera states for smaller R 23 / 35
24 in neural models in networks of pendulum-like elements Bistability in 2D-HR model (J = 0) MOS due to bistability Uncoupled neuron: LC coexists with 3 fixed points For J > 0 the x-nullcline is lowered and LC is the only stable attractor 24 / 35
25 in neural models in networks of pendulum-like elements Coupling in x variable only (σ y = 0, N = 1000, R = 350) Coupling in voltage variable is more realistic Chimera state with one incoherent domain appears as intermediate pattern between desych and synch as σ increases 25 / 35
26 in neural models in networks of pendulum-like elements Network of 3D-HR oscillators ẋ k = y k x 3 k + bx 2 k + J z + σ j=k+r x (x j x k ) 2R j=k R j=k R ẏ k = 1 5x 2 k y k + σ j=k+r y (y j y k ) 2R ż k = r(s(x k + 1.6) z k ) s = 4: firing frequency adaptation and burst production r = 0.01: spiking frequency, number of spikes per bursting b: transition between spiking and bursting 26 / 35
27 in neural models in networks of pendulum-like elements Rich bifurcation scenarios: Storace et al. (2008) prepare system in spiking regime (b = 3, J = 5) 27 / 35
28 in neural models in networks of pendulum-like elements Snapshots at t = 5000 (N = 1000, R = 350, σ x = σ y ) MOS: regularly spiking neurons synchronize, irregularly spiking neurons do not Complete syncrhonization for large σ No chimera states observed 28 / 35
29 in neural models in networks of pendulum-like elements Coupling in x variable only (σ y = 0, N = 1000, R = 350) Desynch alternates with full synch Chimera state associated with change in dynamics Mulistability and sensitive dependence on initial conditions 29 / 35
30 in neural models in networks of pendulum-like elements Chimera States in Mechanical Oscillator Networks Large spring coupling k: Synchronization Low k values: Antiphase synchronization Intermediate k values: Chimeras emerge naturally Martens et al. (2013) 30 / 35
31 in neural models in networks of pendulum-like elements Networks of pendulum-like elements m d 2 θi 1 dt 2 +ɛdθ1 i dt = ω 1 i + µ N N sin(θj 1 θi 1 α)+ ν N j=1 N sin(θj 2 θi 1 α) j=1 m d 2 θi 2 dt 2 +ɛdθ1 i dt = ω 2 i + µ N N sin(θj 2 θi 2 α)+ ν N j=1 N sin(θj 1 θi 2 α) j=1 ω i taken from Lorentzian distribution ɛ: damping parameter m: inertia term 31 / 35
32 in neural models in networks of pendulum-like elements µ = 0.6 strong coupling within each population ν = 0.4 weak coupling between populations 32 / 35
33 in neural models in networks of pendulum-like elements ɛ = 1, increase m 33 / 35
34 in neural models in networks of pendulum-like elements ɛ = 1, increase m 34 / 35
35 in neural models in networks of pendulum-like elements Discussion and future work Hindmarh-Rose model Consider two populations and look for chimera states Bifurcation diagram in (σ, R) plane Effect of inhomogeneities in single dynamics and network topology on chimera states Pendulum-like model Check stability of chimera state for decreasing damping Scan (ɛ, m) plane and identify dynamical behaviours Check role of phase lag α Analytic results? 35 / 35
arxiv: v1 [nlin.cd] 20 Jul 2013
International Journal of Bifurcation and Chaos c World Scientific Publishing Company Chimera states in networs of nonlocally coupled Hindmarsh-Rose neuron models arxiv:37.545v [nlin.cd] Jul 3 Johanne Hizanidis
More informationarxiv: v4 [nlin.cd] 20 Feb 2014
EPJ manuscript No. (will be inserted by the editor) Chimera States in a Two Population Network of Coupled Pendulum Like Elements Tassos Bountis 1,a, Vasileios G. Kanas, Johanne Hizanidis 3, and Anastasios
More informationCollective behavior in networks of biological neurons: mathematical modeling and software development
RESEARCH PROJECTS 2014 Collective behavior in networks of biological neurons: mathematical modeling and software development Ioanna Chitzanidi, Postdoctoral Researcher National Center for Scientific Research
More informationSynchronization in delaycoupled bipartite networks
Synchronization in delaycoupled bipartite networks Ram Ramaswamy School of Physical Sciences Jawaharlal Nehru University, New Delhi February 20, 2015 Outline Ø Bipartite networks and delay-coupled phase
More informationChimera State Realization in Chaotic Systems. The Role of Hyperbolicity
Chimera State Realization in Chaotic Systems. The Role of Hyperbolicity Vadim S. Anishchenko Saratov State University, Saratov, Russia Nizhny Novgorod, July 20, 2015 My co-authors Nadezhda Semenova, PhD
More informationFrom neuronal oscillations to complexity
1/39 The Fourth International Workshop on Advanced Computation for Engineering Applications (ACEA 2008) MACIS 2 Al-Balqa Applied University, Salt, Jordan Corson Nathalie, Aziz Alaoui M.A. University of
More informationSpiral Wave Chimeras
Spiral Wave Chimeras Carlo R. Laing IIMS, Massey University, Auckland, New Zealand Erik Martens MPI of Dynamics and Self-Organization, Göttingen, Germany Steve Strogatz Dept of Maths, Cornell University,
More informationarxiv: v1 [nlin.cd] 4 Dec 2017
Chimera at the phase-flip transition of an ensemble of identical nonlinear oscillators R. Gopal a, V. K. Chandrasekar a,, D. V. Senthilkumar c,, A. Venkatesan d, M. Lakshmanan b a Centre for Nonlinear
More informationarxiv:nlin/ v1 [nlin.cd] 4 Oct 2005
Synchronization of Coupled Chaotic Dynamics on Networks R. E. Amritkar and Sarika Jalan Physical Research Laboratory, Navrangpura, Ahmedabad 380 009, India. arxiv:nlin/0510008v1 [nlin.cd] 4 Oct 2005 Abstract
More informationSaturation of Information Exchange in Locally Connected Pulse-Coupled Oscillators
Saturation of Information Exchange in Locally Connected Pulse-Coupled Oscillators Will Wagstaff School of Computer Science, Georgia Institute of Technology, Atlanta, Georgia 30332, USA (Dated: 13 December
More informationLearning Cycle Linear Hybrid Automata for Excitable Cells
Learning Cycle Linear Hybrid Automata for Excitable Cells Sayan Mitra Joint work with Radu Grosu, Pei Ye, Emilia Entcheva, I V Ramakrishnan, and Scott Smolka HSCC 2007 Pisa, Italy Excitable Cells Outline
More informationAnalysis of Chimera Behavior in Coupled Logistic Maps on Different Connection Topologies
Analysis of Chimera Behavior in Coupled Logistic Maps on Different Connection Topologies Thesis submitted in partial fulfillment of the requirements for the degree of Master of Science by Research in Computational
More informationUniversity of Colorado. The Kuramoto Model. A presentation in partial satisfaction of the requirements for the degree of MSc in Applied Mathematics
University of Colorado The Kuramoto Model A presentation in partial satisfaction of the requirements for the degree of MSc in Applied Mathematics Jeff Marsh 2008 April 24 1 The Kuramoto Model Motivation:
More informationModelling biological oscillations
Modelling biological oscillations Shan He School for Computational Science University of Birmingham Module 06-23836: Computational Modelling with MATLAB Outline Outline of Topics Van der Pol equation Van
More information1 Introduction and neurophysiology
Dynamics of Continuous, Discrete and Impulsive Systems Series B: Algorithms and Applications 16 (2009) 535-549 Copyright c 2009 Watam Press http://www.watam.org ASYMPTOTIC DYNAMICS OF THE SLOW-FAST HINDMARSH-ROSE
More informationDynamical systems in neuroscience. Pacific Northwest Computational Neuroscience Connection October 1-2, 2010
Dynamical systems in neuroscience Pacific Northwest Computational Neuroscience Connection October 1-2, 2010 What do I mean by a dynamical system? Set of state variables Law that governs evolution of state
More informationDynamics and complexity of Hindmarsh-Rose neuronal systems
Dynamics and complexity of Hindmarsh-Rose neuronal systems Nathalie Corson and M.A. Aziz-Alaoui Laboratoire de Mathématiques Appliquées du Havre, 5 rue Philippe Lebon, 766 Le Havre, FRANCE nathalie.corson@univ-lehavre.fr
More informationBursting Oscillations of Neurons and Synchronization
Bursting Oscillations of Neurons and Synchronization Milan Stork Applied Electronics and Telecommunications, Faculty of Electrical Engineering/RICE University of West Bohemia, CZ Univerzitni 8, 3064 Plzen
More informationHysteretic Transitions in the Kuramoto Model with Inertia
Rostock 4 p. Hysteretic Transitions in the uramoto Model with Inertia A. Torcini, S. Olmi, A. Navas, S. Boccaletti http://neuro.fi.isc.cnr.it/ Istituto dei Sistemi Complessi - CNR - Firenze, Italy Istituto
More informationSingle neuron models. L. Pezard Aix-Marseille University
Single neuron models L. Pezard Aix-Marseille University Biophysics Biological neuron Biophysics Ionic currents Passive properties Active properties Typology of models Compartmental models Differential
More information8 Example 1: The van der Pol oscillator (Strogatz Chapter 7)
8 Example 1: The van der Pol oscillator (Strogatz Chapter 7) So far we have seen some different possibilities of what can happen in two-dimensional systems (local and global attractors and bifurcations)
More informationSynchronization Transitions in Complex Networks
Synchronization Transitions in Complex Networks Y. Moreno 1,2,3 1 Institute for Biocomputation and Physics of Complex Systems (BIFI) University of Zaragoza, Zaragoza 50018, Spain 2 Department of Theoretical
More informationAnalysis of burst dynamics bound by potential with active areas
NOLTA, IEICE Paper Analysis of burst dynamics bound by potential with active areas Koji Kurose 1a), Yoshihiro Hayakawa 2, Shigeo Sato 1, and Koji Nakajima 1 1 Laboratory for Brainware/Laboratory for Nanoelectronics
More informationSynchronization and Phase Oscillators
1 Synchronization and Phase Oscillators Richard Bertram Department of Mathematics and Programs in Neuroscience and Molecular Biophysics Florida State University Tallahassee, Florida 32306 Synchronization
More informationTitle. Author(s)Fujii, Hiroshi; Tsuda, Ichiro. CitationNeurocomputing, 58-60: Issue Date Doc URL. Type.
Title Neocortical gap junction-coupled interneuron systems exhibiting transient synchrony Author(s)Fujii, Hiroshi; Tsuda, Ichiro CitationNeurocomputing, 58-60: 151-157 Issue Date 2004-06 Doc URL http://hdl.handle.net/2115/8488
More informationConsider the following spike trains from two different neurons N1 and N2:
About synchrony and oscillations So far, our discussions have assumed that we are either observing a single neuron at a, or that neurons fire independent of each other. This assumption may be correct in
More information6.3.4 Action potential
I ion C m C m dφ dt Figure 6.8: Electrical circuit model of the cell membrane. Normally, cells are net negative inside the cell which results in a non-zero resting membrane potential. The membrane potential
More informationarxiv: v1 [nlin.cd] 6 Mar 2018
Constructive approach to limiting periodic orbits with exponential and power law dynamics A. Provata Institute of Nanoscience and Nanotechnology, National Center for Scientific Research Demokritos, GR-15310
More informationControlling the cortex state transitions by altering the oscillation energy
Controlling the cortex state transitions by altering the oscillation energy Valery Tereshko a and Alexander N. Pisarchik b a School of Computing, University of Paisley, Paisley PA 2BE, Scotland b Centro
More informationAn Introductory Course in Computational Neuroscience
An Introductory Course in Computational Neuroscience Contents Series Foreword Acknowledgments Preface 1 Preliminary Material 1.1. Introduction 1.1.1 The Cell, the Circuit, and the Brain 1.1.2 Physics of
More informationNonlinear and Collective Effects in Mesoscopic Mechanical Oscillators
Dynamics Days Asia-Pacific: Singapore, 2004 1 Nonlinear and Collective Effects in Mesoscopic Mechanical Oscillators Alexander Zumdieck (Max Planck, Dresden), Ron Lifshitz (Tel Aviv), Jeff Rogers (HRL,
More informationPhase Synchronization of Coupled Rossler Oscillators: Amplitude Effect
Commun. Theor. Phys. (Beijing, China) 47 (2007) pp. 265 269 c International Academic Publishers Vol. 47, No. 2, February 15, 2007 Phase Synchronization of Coupled Rossler Oscillators: Amplitude Effect
More informationLecture 4: Importance of Noise and Fluctuations
Lecture 4: Importance of Noise and Fluctuations Jordi Soriano Fradera Dept. Física de la Matèria Condensada, Universitat de Barcelona UB Institute of Complex Systems September 2016 1. Noise in biological
More informationTraveling chimera states for coupled pendula
Nonlinear Dyn https://doi.org/10.1007/s11071-018-4664-5 ORIGINAL PAPER raveling chimera states for coupled pendula Dawid Dudkowski Krzysztof Czołczyński omasz Kapitaniak Received: 10 July 2018 / Accepted:
More informationSlow Manifold of a Neuronal Bursting Model
Slow Manifold of a Neuronal Bursting Model Jean-Marc Ginoux 1 and Bruno Rossetto 2 1 PROTEE Laboratory, Université du Sud, B.P. 2132, 83957, La Garde Cedex, France, ginoux@univ-tln.fr 2 PROTEE Laboratory,
More informationNeuroscience applications: isochrons and isostables. Alexandre Mauroy (joint work with I. Mezic)
Neuroscience applications: isochrons and isostables Alexandre Mauroy (joint work with I. Mezic) Outline Isochrons and phase reduction of neurons Koopman operator and isochrons Isostables of excitable systems
More informationEntrainment Alex Bowie April 7, 2004
Entrainment Alex Bowie April 7, 2004 Abstract The driven Van der Pol oscillator displays entrainment, quasiperiodicity, and chaos. The characteristics of these different modes are discussed as well as
More informationEvaluating Bistability in a Mathematical Model of Circadian Pacemaker Neurons
International Journal of Theoretical and Mathematical Physics 2016, 6(3): 99-103 DOI: 10.5923/j.ijtmp.20160603.02 Evaluating Bistability in a Mathematical Model of Circadian Pacemaker Neurons Takaaki Shirahata
More informationDynamical Systems and Chaos Part II: Biology Applications. Lecture 10: Coupled Systems. Ilya Potapov Mathematics Department, TUT Room TD325
Dynamical Systems and Chaos Part II: Biology Applications Lecture 10: Coupled Systems. Ilya Potapov Mathematics Department, TUT Room TD325 Foreword In order to model populations of physical/biological
More informationAn analysis of how coupling parameters influence nonlinear oscillator synchronization
An analysis of how coupling parameters influence nonlinear oscillator synchronization Morris Huang, 1 Ben McInroe, 2 Mark Kingsbury, 2 and Will Wagstaff 3 1) School of Mechanical Engineering, Georgia Institute
More informationProceedings of Neural, Parallel, and Scientific Computations 4 (2010) xx-xx PHASE OSCILLATOR NETWORK WITH PIECEWISE-LINEAR DYNAMICS
Proceedings of Neural, Parallel, and Scientific Computations 4 (2010) xx-xx PHASE OSCILLATOR NETWORK WITH PIECEWISE-LINEAR DYNAMICS WALTER GALL, YING ZHOU, AND JOSEPH SALISBURY Department of Mathematics
More information2013 NSF-CMACS Workshop on Atrial Fibrillation
2013 NSF-CMACS Workshop on A Atrial Fibrillation Flavio H. Fenton School of Physics Georgia Institute of Technology, Atlanta, GA and Max Planck Institute for Dynamics and Self-organization, Goettingen,
More informationFirefly Synchronization. Morris Huang, Mark Kingsbury, Ben McInroe, Will Wagstaff
Firefly Synchronization Morris Huang, Mark Kingsbury, Ben McInroe, Will Wagstaff Biological Inspiration Why do fireflies flash? Mating purposes Males flash to attract the attention of nearby females Why
More informationDESYNCHRONIZATION TRANSITIONS IN RINGS OF COUPLED CHAOTIC OSCILLATORS
Letters International Journal of Bifurcation and Chaos, Vol. 8, No. 8 (1998) 1733 1738 c World Scientific Publishing Company DESYNCHRONIZATION TRANSITIONS IN RINGS OF COUPLED CHAOTIC OSCILLATORS I. P.
More informationDynamical modelling of systems of coupled oscillators
Dynamical modelling of systems of coupled oscillators Mathematical Neuroscience Network Training Workshop Edinburgh Peter Ashwin University of Exeter 22nd March 2009 Peter Ashwin (University of Exeter)
More informationTracking the State of the Hindmarsh-Rose Neuron by Using the Coullet Chaotic System Based on a Single Input
ISSN 1746-7659, England, UK Journal of Information and Computing Science Vol. 11, No., 016, pp.083-09 Tracking the State of the Hindmarsh-Rose Neuron by Using the Coullet Chaotic System Based on a Single
More informationHSND-2015, IPR. Department of Physics, University of Burdwan, Burdwan, West Bengal.
New kind of deaths: Oscillation Death and Chimera Death HSND-2015, IPR Dr. Tanmoy Banerjee Department of Physics, University of Burdwan, Burdwan, West Bengal. Points to be discussed Oscillation suppression
More informationCanonical Neural Models 1
Canonical Neural Models 1 Frank Hoppensteadt 1 and Eugene zhikevich 2 ntroduction Mathematical modeling is a powerful tool in studying fundamental principles of information processing in the brain. Unfortunately,
More informationAutomatic control of phase synchronization in coupled complex oscillators
Physica D xxx (2004) xxx xxx Automatic control of phase synchronization in coupled complex oscillators Vladimir N. Belykh a, Grigory V. Osipov b, Nina Kuckländer c,, Bernd Blasius c,jürgen Kurths c a Mathematics
More informationSPATIOTEMPORAL CHAOS IN COUPLED MAP LATTICE. Itishree Priyadarshini. Prof. Biplab Ganguli
SPATIOTEMPORAL CHAOS IN COUPLED MAP LATTICE By Itishree Priyadarshini Under the Guidance of Prof. Biplab Ganguli Department of Physics National Institute of Technology, Rourkela CERTIFICATE This is to
More informationLIMIT CYCLE OSCILLATORS
MCB 137 EXCITABLE & OSCILLATORY SYSTEMS WINTER 2008 LIMIT CYCLE OSCILLATORS The Fitzhugh-Nagumo Equations The best example of an excitable phenomenon is the firing of a nerve: according to the Hodgkin
More informationDynamical Systems in Neuroscience: The Geometry of Excitability and Bursting
Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting Eugene M. Izhikevich The MIT Press Cambridge, Massachusetts London, England Contents Preface xv 1 Introduction 1 1.1 Neurons
More informationPhase Oscillators. and at r, Hence, the limit cycle at r = r is stable if and only if Λ (r ) < 0.
1 Phase Oscillators Richard Bertram Department of Mathematics and Programs in Neuroscience and Molecular Biophysics Florida State University Tallahassee, Florida 32306 2 Phase Oscillators Oscillations
More information2.152 Course Notes Contraction Analysis MIT, 2005
2.152 Course Notes Contraction Analysis MIT, 2005 Jean-Jacques Slotine Contraction Theory ẋ = f(x, t) If Θ(x, t) such that, uniformly x, t 0, F = ( Θ + Θ f x )Θ 1 < 0 Θ(x, t) T Θ(x, t) > 0 then all solutions
More informationData assimilation with and without a model
Data assimilation with and without a model Tim Sauer George Mason University Parameter estimation and UQ U. Pittsburgh Mar. 5, 2017 Partially supported by NSF Most of this work is due to: Tyrus Berry,
More informationNonlinear systems, chaos and control in Engineering
Nonlinear systems, chaos and control in Engineering Module 1 block 3 One-dimensional nonlinear systems Cristina Masoller Cristina.masoller@upc.edu http://www.fisica.edu.uy/~cris/ Schedule Flows on the
More informationCollective and Stochastic Effects in Arrays of Submicron Oscillators
DYNAMICS DAYS: Long Beach, 2005 1 Collective and Stochastic Effects in Arrays of Submicron Oscillators Ron Lifshitz (Tel Aviv), Jeff Rogers (HRL, Malibu), Oleg Kogan (Caltech), Yaron Bromberg (Tel Aviv),
More informationarxiv: v1 [math.ds] 13 Jul 2018
Heterogeneous inputs to central pattern generators can shape insect gaits. Zahra Aminzare Philip Holmes arxiv:1807.05142v1 [math.ds] 13 Jul 2018 Abstract In our previous work [1], we studied an interconnected
More informationNonlinear Observer Design and Synchronization Analysis for Classical Models of Neural Oscillators
Nonlinear Observer Design and Synchronization Analysis for Classical Models of Neural Oscillators Ranjeetha Bharath and Jean-Jacques Slotine Massachusetts Institute of Technology ABSTRACT This work explores
More informationarxiv: v1 [nlin.ao] 2 Mar 2018
Stable amplitude chimera states in a network of locally coupled Stuart-Landau oscillators K. Premalatha, V. K. Chandrasekar, 2 M. Senthilvelan, and M. Lakshmanan ) Centre for Nonlinear Dynamics, School
More informationNonlinear Control Systems 1. - Introduction to Nonlinear Systems
Nonlinear Control Systems 1. - Introduction to Nonlinear Systems Dept. of Electrical Engineering Department of Electrical Engineering University of Notre Dame, USA EE658-1 Nonlinear Control Systems 1.
More informationTWO DIMENSIONAL FLOWS. Lecture 5: Limit Cycles and Bifurcations
TWO DIMENSIONAL FLOWS Lecture 5: Limit Cycles and Bifurcations 5. Limit cycles A limit cycle is an isolated closed trajectory [ isolated means that neighbouring trajectories are not closed] Fig. 5.1.1
More informationChimeras in networks with purely local coupling
Chimeras in networks with purely local coupling Carlo R. Laing Institute of Natural and Mathematical Sciences, Massey University, Private Bag -94 NSMC, Auckland, New Zealand. phone: +64-9-44 8 extn. 435
More informationSynchrony in Neural Systems: a very brief, biased, basic view
Synchrony in Neural Systems: a very brief, biased, basic view Tim Lewis UC Davis NIMBIOS Workshop on Synchrony April 11, 2011 components of neuronal networks neurons synapses connectivity cell type - intrinsic
More informationarxiv: v2 [nlin.ao] 25 Feb 2016
A Tweezer for Chimeras in Small Networs Iryna Omelcheno, Oleh E. Omel cheno, Anna Zaharova, Matthias Wolfrum, and Ecehard Schöll, Institut für Theoretische Physi, Technische Universität Berlin, Hardenbergstraße
More informationElectrophysiology of the neuron
School of Mathematical Sciences G4TNS Theoretical Neuroscience Electrophysiology of the neuron Electrophysiology is the study of ionic currents and electrical activity in cells and tissues. The work of
More informationSynchronization Phenomena and Chimera States in Networks of Coupled Oscillators
Synchronization Phenomena and Chimera States in Networks of Coupled Oscillators A. Provata National Center for Scientific Research Demokritos, Athens 24th SummerSchool-Conference on Dynamical Systems and
More informationModels Involving Interactions between Predator and Prey Populations
Models Involving Interactions between Predator and Prey Populations Matthew Mitchell Georgia College and State University December 30, 2015 Abstract Predator-prey models are used to show the intricate
More informationPart II. Dynamical Systems. Year
Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 34 Paper 1, Section II 30A Consider the dynamical system where β > 1 is a constant. ẋ = x + x 3 + βxy 2, ẏ = y + βx 2
More informationSurvey of Synchronization Part I: Kuramoto Oscillators
Survey of Synchronization Part I: Kuramoto Oscillators Tatsuya Ibuki FL11-5-2 20 th, May, 2011 Outline of My Research in This Semester Survey of Synchronization - Kuramoto oscillator : This Seminar - Synchronization
More informationFast neural network simulations with population density methods
Fast neural network simulations with population density methods Duane Q. Nykamp a,1 Daniel Tranchina b,a,c,2 a Courant Institute of Mathematical Science b Department of Biology c Center for Neural Science
More informationSpontaneous Synchronization in Complex Networks
B.P. Zeegers Spontaneous Synchronization in Complex Networks Bachelor s thesis Supervisors: dr. D. Garlaschelli (LION) prof. dr. W.Th.F. den Hollander (MI) August 2, 25 Leiden Institute of Physics (LION)
More informationDynamic Characteristics of Neuron Models and Active Areas in Potential Functions K. Nakajima a *, K. Kurose a, S. Sato a, Y.
Available online at www.sciencedirect.com Procedia IUTAM 5 (2012 ) 49 53 IUTAM Symposium on 50 Years of Chaos: Applied and Theoretical Dynamic Characteristics of Neuron Models and Active Areas in Potential
More informationChaos. Lendert Gelens. KU Leuven - Vrije Universiteit Brussel Nonlinear dynamics course - VUB
Chaos Lendert Gelens KU Leuven - Vrije Universiteit Brussel www.gelenslab.org Nonlinear dynamics course - VUB Examples of chaotic systems: the double pendulum? θ 1 θ θ 2 Examples of chaotic systems: the
More informationBursting and Chaotic Activities in the Nonlinear Dynamics of FitzHugh-Rinzel Neuron Model
Bursting and Chaotic Activities in the Nonlinear Dynamics of FitzHugh-Rinzel Neuron Model Abhishek Yadav *#, Anurag Kumar Swami *, Ajay Srivastava * * Department of Electrical Engineering, College of Technology,
More informationUNIVERSITY OF CALIFORNIA SANTA BARBARA. Neural Oscillator Identification via Phase-Locking Behavior. Michael J. Schaus
UNIVERSITY OF CALIFORNIA SANTA BARBARA Neural Oscillator Identification via Phase-Locking Behavior by Michael J. Schaus A thesis submitted in partial satisfaction of the requirements for the degree of
More informationNeuronal Dynamics: Computational Neuroscience of Single Neurons
Week 4 part 5: Nonlinear Integrate-and-Fire Model 4.1 From Hodgkin-Huxley to 2D Neuronal Dynamics: Computational Neuroscience of Single Neurons Week 4 Recing detail: Two-dimensional neuron models Wulfram
More informationComputational Neuroscience. Session 4-2
Computational Neuroscience. Session 4-2 Dr. Marco A Roque Sol 06/21/2018 Two-Dimensional Two-Dimensional System In this section we will introduce methods of phase plane analysis of two-dimensional systems.
More informationHow fast elements can affect slow dynamics
Physica D 180 (2003) 1 16 How fast elements can affect slow dynamics Koichi Fujimoto, Kunihiko Kaneko Department of Pure and Applied Sciences, Graduate school of Arts and Sciences, University of Tokyo,
More information46th IEEE CDC, New Orleans, USA, Dec , where θ rj (0) = j 1. Re 1 N. θ =
Proceedings of the 46th IEEE Conference on Decision and Control New Orleans, LA, USA, Dec. 12-14, 2007 INVITED SESSION NUMBER: 157 Event-Based Feedback Control of Nonlinear Oscillators Using Phase Response
More informationCHIMERA STATES IN A RING OF NONLOCALLY COUPLED OSCILLATORS
Papers International Journal of Bifurcation and Chaos, Vol. 16, No. 1 (26) 21 37 c World Scientific Publishing Company CHIMERA STATES IN A RING OF NONLOCALLY COUPLED OSCILLATORS DANIEL M. ABRAMS and STEVEN
More informationOscillatory Motion. Simple pendulum: linear Hooke s Law restoring force for small angular deviations. small angle approximation. Oscillatory solution
Oscillatory Motion Simple pendulum: linear Hooke s Law restoring force for small angular deviations d 2 θ dt 2 = g l θ small angle approximation θ l Oscillatory solution θ(t) =θ 0 sin(ωt + φ) F with characteristic
More informationProblem Set Number 02, j/2.036j MIT (Fall 2018)
Problem Set Number 0, 18.385j/.036j MIT (Fall 018) Rodolfo R. Rosales (MIT, Math. Dept., room -337, Cambridge, MA 0139) September 6, 018 Due October 4, 018. Turn it in (by 3PM) at the Math. Problem Set
More informationPhysics: spring-mass system, planet motion, pendulum. Biology: ecology problem, neural conduction, epidemics
Applications of nonlinear ODE systems: Physics: spring-mass system, planet motion, pendulum Chemistry: mixing problems, chemical reactions Biology: ecology problem, neural conduction, epidemics Economy:
More informationOscillatory Motion. Simple pendulum: linear Hooke s Law restoring force for small angular deviations. Oscillatory solution
Oscillatory Motion Simple pendulum: linear Hooke s Law restoring force for small angular deviations d 2 θ dt 2 = g l θ θ l Oscillatory solution θ(t) =θ 0 sin(ωt + φ) F with characteristic angular frequency
More information1. < 0: the eigenvalues are real and have opposite signs; the fixed point is a saddle point
Solving a Linear System τ = trace(a) = a + d = λ 1 + λ 2 λ 1,2 = τ± = det(a) = ad bc = λ 1 λ 2 Classification of Fixed Points τ 2 4 1. < 0: the eigenvalues are real and have opposite signs; the fixed point
More informationWeierstraß-Institut. für Angewandte Analysis und Stochastik. Leibniz-Institut im Forschungsverbund Berlin e. V. Preprint ISSN
Weierstraß-Institut für Angewandte Analysis und Stochasti Leibniz-Institut im Forschungsverbund Berlin e. V. Preprint ISSN 98-5855 Optimal design of the tweezer control for chimera states Iryna Omelcheno,
More informationData assimilation with and without a model
Data assimilation with and without a model Tyrus Berry George Mason University NJIT Feb. 28, 2017 Postdoc supported by NSF This work is in collaboration with: Tim Sauer, GMU Franz Hamilton, Postdoc, NCSU
More informationNeural Modeling and Computational Neuroscience. Claudio Gallicchio
Neural Modeling and Computational Neuroscience Claudio Gallicchio 1 Neuroscience modeling 2 Introduction to basic aspects of brain computation Introduction to neurophysiology Neural modeling: Elements
More informationarxiv: v2 [nlin.cd] 8 Nov 2017
Non-identical multiplexing promotes chimera states Saptarshi Ghosh a, Anna Zakharova b, Sarika Jalan a a Complex Systems Lab, Discipline of Physics, Indian Institute of Technology Indore, Simrol, Indore
More informationSYNCHRONIZATION IN SMALL-WORLD DYNAMICAL NETWORKS
International Journal of Bifurcation and Chaos, Vol. 12, No. 1 (2002) 187 192 c World Scientific Publishing Company SYNCHRONIZATION IN SMALL-WORLD DYNAMICAL NETWORKS XIAO FAN WANG Department of Automation,
More informationPHASE-LOCKED SOLUTIONS IN A HUB CONNECTED OSCILLATOR RING NETWORK
Copyright c 29 by ABCM PHASE-LOCKED SOLUTIONS IN A HUB CONNECTED OSCILLATOR RING NETWORK Jacqueline Bridge, Jacqueline.Bridge@sta.uwi.edu Department of Mechanical Engineering, The University of the West
More informationCoupling in Networks of Neuronal Oscillators. Carter Johnson
Coupling in Networks of Neuronal Oscillators Carter Johnson June 15, 2015 1 Introduction Oscillators are ubiquitous in nature. From the pacemaker cells that keep our hearts beating to the predator-prey
More informationSynchronization Bound for Networks of Nonlinear Oscillators
Synchronization Bound for Networs of Nonlinear Oscillators Elizabeth N. Davison, Biswadip Dey and Naomi Ehrich Leonard Abstract Investigation of synchronization phenomena in networs of coupled nonlinear
More informationThe Kuramoto Model. Gerald Cooray. U.U.D.M. Project Report 2008:23. Department of Mathematics Uppsala University
U.U.D.M. Project Report 008:3 The Kuramoto Model Gerald Cooray Examensarbete i matematik, 30 hp Handledare och examinator: David Sumpter September 008 Department of Mathematics Uppsala University THE
More informationOn the Response of Neurons to Sinusoidal Current Stimuli: Phase Response Curves and Phase-Locking
On the Response of Neurons to Sinusoidal Current Stimuli: Phase Response Curves and Phase-Locking Michael J. Schaus and Jeff Moehlis Abstract A powerful technique for analyzing mathematical models for
More informationLecture 10 : Neuronal Dynamics. Eileen Nugent
Lecture 10 : Neuronal Dynamics Eileen Nugent Origin of the Cells Resting Membrane Potential: Nernst Equation, Donnan Equilbrium Action Potentials in the Nervous System Equivalent Electrical Circuits and
More informationUSING COUPLED OSCILLATORS TO MODEL THE SINO-ATRIAL NODE IN THE HEART
USING COUPLED OSCILLATORS TO MODEL THE SINO-ATRIAL NODE IN THE HEART A Thesis Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree
More informationMechanical Resonance and Chaos
Mechanical Resonance and Chaos You will use the apparatus in Figure 1 to investigate regimes of increasing complexity. Figure 1. The rotary pendulum (from DeSerio, www.phys.ufl.edu/courses/phy483l/group_iv/chaos/chaos.pdf).
More informationOscillator synchronization in complex networks with non-uniform time delays
Oscillator synchronization in complex networks with non-uniform time delays Jens Wilting 12 and Tim S. Evans 13 1 Networks and Complexity Programme, Imperial College London, London SW7 2AZ, United Kingdom
More information