PERIODIC INHIBITION OF LIVING PACEMAKER NEURONS (I): LOCKED, INTERMITTENT, MESSY, AND HOPPING BEHAVIORS

This communication is concerned with an embodiment of periodic nonlinear oscillator driving, the synaptic inhibition of one spike-producing pacemaker neuron by another. Data came from a prototypical living synapse. Analyses centered on a prolonged condition between the transients following the onset and cessation of inhibition. Evaluations were guided by point process mathematics and nonlinear dynamics. A rich and exhaustive list of discharge forms, described precisely and canonically, was observed across different inhibitory rates. Previously unrecognized at synapses, most forms were identified with several well known types from nonlinear dynamics. Ordered by decreasing regularities, they were locked, intermittent (including walk-throughs), messy (including erratic and stammerings) and hopping. Each is discussed within physiological and formal contexts. It is conjectured that (i) locked, intermittent and messy forms reflect limit cycles on 2-tori, quasiperiodic orbits and strange attractors, (ii) noise in neurons hovering around threshold contributes to certain intermittent and stammering behaviors, and (iii) hopping either reflects an attractor with several portions or is nonstationary and noise-induced.

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A New Megastable Oscillator with Rational and Irrational Parameters

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127419501761 ◽

2019 ◽

Vol 29 (13) ◽

pp. 1950176 ◽

Cited By ~ 5

Author(s):

Zhen Wang ◽

Ibrahim Ismael Hamarash ◽

Payam Sadeghi Shabestari ◽

Sajad Jafari

Keyword(s):

Dynamical System ◽

Infinite Number ◽

Limit Cycles ◽

Nonlinear Oscillator ◽

Analog Circuit ◽

Strange Attractors ◽

Two Dimensional ◽

System A ◽

New System

In this paper, a new two-dimensional nonlinear oscillator with an unusual sequence of rational and irrational parameters is introduced. This oscillator has endless coexisting limit cycles, which make it a megastable dynamical system. By periodically forcing this system, a new system is designed which is capable of exhibiting an infinite number of coexisting asymmetric torus and strange attractors. This system is implemented by an analog circuit, and its Hamiltonian energy is calculated.

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Nonlinear Dynamics of the BZ Reaction: A Simple Experiment that Illustrates Limit Cycles, Chaos, Bifurcations, and Noise

Journal of Chemical Education ◽

10.1021/ed073p868 ◽

1996 ◽

Vol 73 (9) ◽

pp. 868 ◽

Cited By ~ 17

Author(s):

Peter Strizhak ◽

Michael Menzinger

Keyword(s):

Nonlinear Dynamics ◽

Limit Cycles ◽

Simple Experiment ◽

Bz Reaction

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Various Types of Coexisting Attractors in a New 4D Autonomous Chaotic System

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127417501425 ◽

2017 ◽

Vol 27 (09) ◽

pp. 1750142 ◽

Cited By ~ 48

Author(s):

Qiang Lai ◽

Akif Akgul ◽

Xiao-Wen Zhao ◽

Huiqin Pei

Keyword(s):

Numerical Simulation ◽

Dynamic Behavior ◽

Chaotic System ◽

Limit Cycles ◽

Electronic Circuit ◽

Strange Attractors ◽

Bifurcation Diagrams ◽

Coexisting Attractors ◽

Signum Function ◽

Multiple Coexisting Attractors

An unique 4D autonomous chaotic system with signum function term is proposed in this paper. The system has four unstable equilibria and various types of coexisting attractors appear. Four-wing and four-scroll strange attractors are observed in the system and they will be broken into two coexisting butterfly attractors and two coexisting double-scroll attractors with the variation of the parameters. Numerical simulation shows that the system has various types of multiple coexisting attractors including two butterfly attractors with four limit cycles, two double-scroll attractors with a limit cycle, four single-scroll strange attractors, four limit cycles with regard to different parameters and initial values. The coexistence of the attractors is determined by the bifurcation diagrams. The chaotic and hyperchaotic properties of the attractors are verified by the Lyapunov exponents. Moreover, we present an electronic circuit to experimentally realize the dynamic behavior of the system.

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DYNAMICS OF DELAY-COUPLED EXCITABLE NEURAL SYSTEMS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127409023111 ◽

2009 ◽

Vol 19 (02) ◽

pp. 745-753 ◽

Cited By ~ 39

Author(s):

M. A. DAHLEM ◽

G. HILLER ◽

A. PANCHUK ◽

E. SCHÖLL

Keyword(s):

Nonlinear Dynamics ◽

Fixed Point ◽

Limit Cycles ◽

Coupling Strength ◽

Neural Systems ◽

Stable Fixed Point ◽

Bifurcation Of Limit Cycles ◽

Large Delay ◽

Saddle Node Bifurcation ◽

Delay Times

We study the nonlinear dynamics of two delay-coupled neural systems each modeled by excitable dynamics of FitzHugh–Nagumo type and demonstrate that bistability between the stable fixed point and limit cycle oscillations occurs for sufficiently large delay times τ and coupling strength C. As the mechanism for these delay-induced oscillations, we identify a saddle-node bifurcation of limit cycles.

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Poincaré Bifurcation of Some Nonlinear Oscillator of Generalized Liénard Type Using Symbolic Computation Method

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127418500967 ◽

2018 ◽

Vol 28 (08) ◽

pp. 1850096 ◽

Cited By ~ 3

Author(s):

Hongying Zhu ◽

Bin Qin ◽

Sumin Yang ◽

Minzhi Wei

Keyword(s):

Limit Cycle ◽

Limit Cycles ◽

Nonlinear Oscillator ◽

Melnikov Function ◽

Computation Method ◽

Rigorous Proof ◽

Heteroclinic Loop ◽

Period Annulus ◽

Domain Iii ◽

Weak Damping

In this paper, we study the Poincaré bifurcation of a nonlinear oscillator of generalized Liénard type by using the Melnikov function. The oscillator has weak damping terms. When the damping terms vanish, the oscillator has a heteroclinic loop connecting a nilpotent cusp to a hyperbolic saddle. Our results reveal that: (i) the oscillator can have at most four limit cycles bifurcating from the corresponding period annulus. (ii) There are some parameters such that three limit cycles emerge in the original periodic orbit domain. (iii) Especially, we give a rigorous proof that [Formula: see text] limit cycle(s) can emerge near the original singular loop and [Formula: see text] limit cycle(s) can emerge near the original elementary center with [Formula: see text].

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Revealing Nonlinear Dynamics Phenomena in Mooring Due to Slowly-Varying Drift

24th International Conference on Offshore Mechanics and Arctic Engineering: Volume 1, Parts A and B ◽

10.1115/omae2005-67289 ◽

2005 ◽

Author(s):

Michael M. Bernitsas ◽

Joa˜o Paulo J. Matsuura

Keyword(s):

Nonlinear Dynamics ◽

Limit Cycles ◽

Harmonic Balance ◽

Harmonic Balance Method ◽

Resonance Phenomenon ◽

Nonlinear Phenomena ◽

Mooring System ◽

Stable Limit ◽

Dynamic Interactions ◽

Wave Drift

The effects of slowly-varying wave drift forces on the nonlinear dynamics of mooring systems have been studied extensively in the past 30 years. It has been concluded that slowly-varying wave drift may resonate with mooring system natural frequencies. In recent work, we have shown that this resonance phenomenon is only one of several possible nonlinear dynamic interactions between slowly-varying wave drift and mooring systems. We were able to reveal new phenomena based on the design methodology developed at the University of Michigan for autonomous mooring systems and treating slowly-varying wave drift as an external time-varying force in systematic simulations. This methodology involves exhaustive search regarding the nonautonomous excitation, however, and approximations in defining response bifurcations. In this paper, a new approach is developed based on the harmonic balance method, where the response to the slowly-varying wave drift spectrum is modeled by limit cycles of frequency estimated from a limited number of simulations. Thus, it becomes possible to rewrite the nonautonomous system as autonomous and reveal stability properties of the nonautonomous response. Catastrophe sets of the symmetric principal equilibrium, serving as design charts, define regions in the design space where the trajectories of the mooring system are asymptotically stable, limit cycles, or non-periodic. This methodology reveals and proves that mooring systems subjected to slowly-varying wave drift exhibit many nonlinear phenomena, which lead to motions with amplitudes 2–3 orders of magnitude larger than those resulting from linear resonance. A turret mooring system (TMS) is used to demonstrate the harmonic balance methodology developed. The produced catastrophe sets are then compared with numerical results obtained from systematic simulations of the TMS dynamics.

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NEURONAL OSCILLATIONS AND STOCHASTIC LIMIT CYCLES

International Journal of Neural Systems ◽

10.1142/s0129065796000373 ◽

1996 ◽

Vol 07 (04) ◽

pp. 399-402 ◽

Cited By ~ 2

Author(s):

CHRISTIAN KURRER ◽

KLAUS SCHULTEN

Keyword(s):

Nonlinear Dynamics ◽

Limit Cycles ◽

Neural Activity ◽

Oscillatory Behavior ◽

Neuronal Oscillations ◽

Coupled Oscillator ◽

Stochastic Limit ◽

Excitable Systems ◽

The Individual ◽

Coupled Oscillator Systems

We investigate a model for synchronous neural activity in networks of coupled neurons. The individual systems are governed by nonlinear dynamics and can continuously vary between excitable and oscillatory behavior. Analytical calculations and computer simulations show that coupled excitable systems can undergo two different phase transitions from synchronous to asynchronous firing behavior. One of the transitions is akin to the synchronization transitions in coupled oscillator systems, while the second transition can only be found in coupled excitable systems. Using the concept of Stochastic Limit Cycles, we present an analytical derivation of the two transitions and discuss implications for synchronization transitions in biological neural networks.

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