Frequency entrainment in long chains of oscillators with random natural frequencies in the weak coupling limit

Östborn, Per

doi:10.1103/physreve.70.016120

Cited by 11 publications

(11 citation statements)

References 12 publications

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“…In general, synchronization is achieved through a mutual interaction due to a coupling between two or more oscillatory systems. One of the most extensively studied coupling mechanisms is global coupling (or all-to-all coupling) through which each individual oscillator interacts with equal strength with all the other oscillators in the system (Ostborn 2004). Another type of coupling mechanism in many physical systems is local coupling or diffusive coupling (or nearest-neighbour coupling).…”

Section: Diffusive Coupling Can Induce Periodic Activity In Neural Nementioning

confidence: 99%

Potassium diffusive coupling in neural networks

Durand

Park

Jensen

2010

Phil. Trans. R. Soc. B

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Conventional neural networks are characterized by many neurons coupled together through synapses. The activity, synchronization, plasticity and excitability of the network are then controlled by its synaptic connectivity. Neurons are surrounded by an extracellular space whereby fluctuations in specific ionic concentration can modulate neuronal excitability. Extracellular concentrations of potassium ([K þ ] o ) can generate neuronal hyperexcitability. Yet, after many years of research, it is still unknown whether an elevation of potassium is the cause or the result of the generation, propagation and synchronization of epileptiform activity. An elevation of potassium in neural tissue can be characterized by dispersion (global elevation of potassium) and lateral diffusion (local spatial gradients). Both experimental and computational studies have shown that lateral diffusion is involved in the generation and the propagation of neural activity in diffusively coupled networks. Therefore, diffusion-based coupling by potassium can play an important role in neural networks and it is reviewed in four sections. Section 2 shows that potassium diffusion is responsible for the synchronization of activity across a mechanical cut in the tissue. A computer model of diffusive coupling shows that potassium diffusion can mediate communication between cells and generate abnormal and/or periodic activity in small ( §3) and in large networks of cells ( §4). Finally, in §5, a study of the role of extracellular potassium in the propagation of axonal signals shows that elevated potassium concentration can block the propagation of neural activity in axonal pathways. Taken together, these results indicate that potassium accumulation and diffusion can interfere with normal activity and generate abnormal activity in neural networks.

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Section: Diffusive Coupling Can Induce Periodic Activity In Neural Nementioning

confidence: 99%

Potassium diffusive coupling in neural networks

Durand

Park

Jensen

2010

Phil. Trans. R. Soc. B

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“…Importantly, the existence of a synchronization transition in the continuum regime implies the phe-nomenon is universal, and not dependent on the details of the lattice or disorder. We note that a synchronization transition for one dimensional chains of oscillators with non-odd nearest-neighbor coupling was previously identified by Östborn [28]. That analysis, however, predicts a critical J c which differs from Eq.…”

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confidence: 45%

“…The presence of the cosine term in the coupling function of our model has a significant effect however. Unlike the Kuramoto model, our coupling is non-odd in its arguments, and it has been suggested that this may bring about synchronization more readily [27,28]. Numerical simulations of Eq.…”

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confidence: 99%

Synchronization in disordered oscillator lattices: Nonequilibrium phase transition for driven-dissipative bosons

Moroney

Eastham

2021

Phys. Rev. Research

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We show that a lattice of phase oscillators with random natural frequencies, described by a generalization of the nearest-neighbor Kuramoto model with an additional cosine coupling term, undergoes a phase transition from a desynchronized to a synchronized state. This model may be derived from the complex Ginzburg-Landau equations describing a disordered lattice of driven-dissipative Bose-Einstein condensates of exciton polaritons. We derive phase diagrams that classify the desynchronized and synchronized states that exist in both one and two dimensions. This is achieved by outlining the connection of the oscillator model to the quantum description of localization of a particle in a random potential through a mapping to a modified Kardar-Parisi-Zhang equation.Our results indicate that long-range order in polariton condensates, and other systems of coupled oscillators, is not destroyed by randomness in their natural frequencies.

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“…The next wave of work relaxed the all-to-all assumption by allowing oscillators to be connected to their nearest neighbors on a one-dimensional chain or ring, a two-dimensional square grid, or a higherdimensional cubic lattice [8][9][10][11][12][13][14][15][16][17][18][19][20][21] . More recently, many researchers have explored the Kuramoto model on more complex topologies, allowing for nonlocal coupling 22 and small-world, scale-free, or other network architectures 6,7 .…”

Section: Introduction a The Kuramoto Modelmentioning

confidence: 99%

Frequency spirals

Ottino-Loffler

Strogatz

2016

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We study the dynamics of coupled phase oscillators on a two-dimensional Kuramoto lattice with periodic boundary conditions. For coupling strengths just below the transition to global phase-locking, we find localized spatiotemporal patterns that we call "frequency spirals." These patterns cannot be seen under time averaging; they become visible only when we examine the spatial variation of the oscillators' instantaneous frequencies, where they manifest themselves as two-armed rotating spirals. In the more familiar phase representation, they appear as wobbly periodic patterns surrounding a phase vortex. Unlike the stationary phase vortices seen in magnetic spin systems, or the rotating spiral waves seen in reaction-diffusion systems, frequency spirals librate: the phases of the oscillators surrounding the central vortex move forward and then backward, executing a periodic motion with zero winding number. We construct the simplest frequency spiral and characterize its properties using analytical and numerical methods. Simulations show that frequency spirals in large lattices behave much like this simple prototype.Spirals have long been objects of fascination in many parts of human culture, from art and architecture to science and mathematics. Within nonlinear dynamics, spirals have been studied in connection with such diverse phenomena as spiral density waves in galaxies, spiral waves of electrical activity in the heart and nervous system, growth spirals caused by screw dislocations in crystals, and spiral patterns of florets on the heads of sunflowers, to name just a few. Here we consider a spiral pattern of a type that, as far as we know, has not been discussed previously. Instead of a pattern of density or phase, it is a pattern of instantaneous frequencies: hence, a "frequency spiral." We found it in a two-dimensional array of coupled oscillators whose natural frequencies were not quite identical, a regime in which the oscillators could no longer maintain global frequency entrainment and instead self-organized into one or more frequency spirals. We explore the properties of frequency spirals by analytical and geometrical methods and illustrate them by videotaped simulations of their dynamics in time and space.

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Frequency entrainment in long chains of oscillators with random natural frequencies in the weak coupling limit

Cited by 11 publications

References 12 publications

Potassium diffusive coupling in neural networks

Potassium diffusive coupling in neural networks

Synchronization in disordered oscillator lattices: Nonequilibrium phase transition for driven-dissipative bosons

Frequency spirals

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