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Relaxation oscillator

About: Relaxation oscillator is a(n) research topic. Over the lifetime, 1952 publication(s) have been published within this topic receiving 22326 citation(s).


Papers
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Journal ArticleDOI
TL;DR: The results suggest that the emergent synchronization behavior of oscillating neural networks can be dramatically influenced by the intrinsic properties of the network components.
Abstract: Synchronization properties of locally coupled neural oscillators were investigated analytically and by computer simulation. When coupled in a manner that mimics excitatory chemical synapses, oscillators having more than one time scale (relaxation oscillators) are shown to approach synchrony using mechanisms very different from that of oscillators with a more sinusoidal waveform. The relaxation oscillators make critical use of fast modulations of their thresholds, leading to a rate of synchronization relatively independent of coupling strength within some basin of attraction; this rate is faster for oscillators that have conductance-based features than for neural caricatures such as the FitzHugh-Nagumo equations that lack such features. Computer simulations of one-dimensional arrays show that oscillators in the relaxation regime synchronize much more rapidly than oscillators with the same equations whose parameters have been modulated to yield a more sinusoidal waveform. We present a heuristic explanation of this effect based on properties of the coupling mechanisms that can affect the way the synchronization scales with array length. These results suggest that the emergent synchronization behavior of oscillating neural networks can be dramatically influenced by the intrinsic properties of the network components. Possible implications for perceptual feature binding and attention are discussed.

375 citations

Journal ArticleDOI
Abstract: An architecture of locally excitatory, globally inhibitory oscillator networks is proposed and investigated both analytically and by computer simulation The model for each oscillator corresponds to a standard relaxation oscillator with two time scales Oscillators are locally coupled by a scheme that resembles excitatory synaptic coupling, and each oscillator also inhibits other oscillators through a common inhibitor Oscillators are driven to be oscillatory by external stimulation The network exhibits a mechanism of selective gating, whereby an oscillator jumping up to its active phase rapidly recruits the oscillators stimulated by the same pattern, while preventing the other oscillators from jumping up We show analytically that with the selective gating mechanism, the network rapidly achieves both synchronization within blocks of oscillators that are stimulated by connected regions and desynchronization between different blocks Computer simulations demonstrate the model's promising ability for segmenting multiple input patterns in real time This model lays a physical foundation for the oscillatory correlation theory of feature binding and may provide an effective computational framework for scene segmentation and figure/ ground segregation

375 citations

Journal ArticleDOI
TL;DR: A learning rule for oscillators which adapts their frequency to the frequency of any periodic or pseudo-periodic input signal, which is easily generalizable to a large class of oscillators, from phase oscillators to relaxation oscillators and strange attractors with a generic learning rule.
Abstract: Nonlinear oscillators are widely used in biology, physics and engineering for modeling and control. They are interesting because of their synchronization properties when coupled to other dynamical systems. In this paper, we propose a learning rule for oscillators which adapts their frequency to the frequency of any periodic or pseudo-periodic input signal. Learning is done in a dynamic way: it is part of the dynamical system and not an offline process. An interesting property of our model is that it is easily generalizable to a large class of oscillators, from phase oscillators to relaxation oscillators and strange attractors with a generic learning rule. One major feature of our learning rule is that the oscillators constructed can adapt their frequency without any signal processing or the need to specify a time window or similar free parameters. All the processing is embedded in the dynamics of the adaptive oscillator. The convergence of the learning is proved for the Hopf oscillator, then numerical experiments are carried out to explore the learning capabilities of the system. Finally, we generalize the learning rule to non-harmonic oscillators like relaxation oscillators and strange attractors.

298 citations

Journal ArticleDOI
TL;DR: A novel class of locally excitatory, globally inhibitory oscillator networks (LEGION) is proposed and investigated, which lays a physical foundation for the oscillatory correlation theory of feature binding and may provide an effective computational framework for scene segmentation and figure/ground segregation in real time.
Abstract: A novel class of locally excitatory, globally inhibitory oscillator networks (LEGION) is proposed and investigated. The model of each oscillator corresponds to a standard relaxation oscillator with two time scales. In the network, an oscillator jumping up to its active phase rapidly recruits the oscillators stimulated by the same pattern, while preventing other oscillators from jumping up. Computer simulations demonstrate that the network rapidly achieves both synchronization within blocks of oscillators that are stimulated by connected regions and desynchronization between different blocks. This model lays a physical foundation for the oscillatory correlation theory of feature binding and may provide an effective computational framework for scene segmentation and figure/ground segregation in real time. >

260 citations

Book
03 Apr 1987
Abstract: 1. Introduction.- 1.1 The Van der Pol oscillator.- 1.2 Mechanical prototypes of relaxation oscillators.- 1.3 Relaxation oscillations in physics and biology.- 1.4 Discontinuous approximations.- 1.5 Matched asymptotic expansions.- 1.6 Forced oscillations.- 1.7 Mutual entrainment.- 2 Free oscillation.- 2.1 Autonomous relaxation oscillation: definition and existence.- 2.1.1 A mathematical characterization of relaxation oscillations.- 2.1.2 Application of the Poincare-Bendixson theorem.- 2.1.3 Application of the extension theorem.- 2.1.4 Application of Tikhonov's theorem.- 2.1.5 The analytical method of Cartwright.- 2.2 Asymptotic solution of the Van der Pol equation.- 2.2.1 The physical plane.- 2.2.2 The phase plane.- 2.2.3 The Lienard plane.- 2.2.4 Approximations of amplitude and period.- 2.3 The Volterra-Lotka equations.- 2.3.1 Modeling prey-predator systems.- 2.3.2 Oscillations with both state variables having a large amplitude.- 2.3.3 Oscillations with one state variable having a large amplitude.- 2.3.4 The period for large amplitude oscillations by inverse Laplace asymptotics.- 2.4 Chemical oscillations.- 2.4.1 The Brusselator.- 2.4.2 The Belousov-Zhabotinskii reaction and the Oregonator.- 2.5 Bifurcation of the Van der Pol equation with a constant forcing term.- 2.5.1 Modeling nerve excitation the Bonhoeffer-Van der Pol equation.- 2.5.2 Canards.- 2.6 Stochastic and chaotic oscillations.- 2.6.1 Chaotic relaxation oscillations.- 2.6.2 Randomly perturbed oscillations.- 2.6.3 The Van der Pol oscillator with a random forcing term.- 2.6.4 Distinction between chaos and noise.- 3. Forced oscillation and mutual entrainment.- 3.1 Modeling coupled oscillations.- 3.1.1 Oscillations in the applied sciences.- 3.1.2 The system of differential equations and the method of analysis.- 3.2 A rigorous theory for weakly coupled oscillators.- 3.2.1 Validity of the discontinuous approximation.- 3.2.2 Construction of the asymptotic solution.- 3.2.3 Existence of a periodic solution.- 3.2.4 Formal extension to oscillators coupled with delay.- 3.3 Coupling of two oscillators.- 3.3.1 Piece-wise linear oscillators.- 3.3.2 Van der Pol oscillators.- 3.3.3 Entrainment with frequency ratio 1:3.- 3.3.4 Oscillators with different limit cycles.- Modeling biological oscillations.- 3.4.1 Entrainment with frequency ratio n:m.- 3.4.2 A chain of oscillators with decreasing autonomous frequency.- 3.4.3 A large population of coupled oscillators with widely different frequencies.- 3.4.4 A large population of coupled oscillators with frequencies having a Gaussian distribution.- 3.4.5 Periodic structures of coupled oscillators.- 3.4.6 Nonlinear phase diffusion equations.- 4. The Van der Pol oscillator with a sinusoidal forcing term.- 4.1 Qualitative methods of analysis.- 4.1.1 Global behavior and the Poincare mapping.- 4.1.2 The use of symbolic dynamics.- 4.1.3 Some remarks on the annulus mapping.- 4.2 Asymptotic solution of the Van der Pol equation with a moderate forcing term.- 4.2 Asymptotic solution of the Van der Pol equation with a large forcing term.- 4.2.1 Subharmonic solutions.- 4.2.2 Dips slices and chaotic solutions.- 4.3 Asymptotic solution of the Van der Pol equation with a large forcing term.- 4.3.1 Subharmonic solutions.- 4.3.2 Dips and slices.- 4.3.3 Irregular solutions.- Appendices.- A: Asymptotics of some special functions.- B: Asymptotic ordering and expansions.- C: Concepts of the theory of dynamical systems.- D: Stochastic differential equations and diffusion approximations.- Literature.- Author Index.

242 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202127
202044
201962
201855
201756
201656