Phase transition to frequency entrainment in a long chain of pulse-coupled oscillators

Östborn, Per

doi:10.1103/physreve.66.016105

Cited by 13 publications

(12 citation statements)

References 26 publications

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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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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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“…These correspond to the segments of 25 oscillators within which the integration was exact in Ref. [9]. The two phase transitions separate three phases.…”

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confidence: 98%

“…For a long time, model studies only revealed negative or inconclusive results regarding the existence of such a phase transition [8]. Recently, we proved that it was present in a one-dimensional chain of pulse-coupled oscillators [9]. A finite critical coupling strength g c was found, at which global frequency entrainment settles.…”

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confidence: 98%

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Phase transitions towards frequency entrainment in large oscillator lattices

Östborn

Åberg²,

Öhlén³

2003

Phys. Rev. E

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We simulate two-dimensional lattices of pulse-coupled oscillators with random natural frequencies, resembling pacemaker cells in the heart. As coupling increases, this system seems to undergo two phase transitions in the thermodynamic limit. First, the largest cluster of frequency entrained oscillators becomes macroscopic. Second, all oscillators frequency entrain, except possibly some isolated ones. Between the two transitions, the system has features indicating self-organized criticality. To our knowledge, the first transition and the intermediate phase have never been observed before. It remains to be seen if they are generic for large lattices of limit cycle oscillators.

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Analysis of heterogeneous cardiac pacemaker tissue models and traveling wave dynamics

Weinberg

2018

Journal of Theoretical Biology

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The sinoatrial-node (SAN) is a complex heterogeneous tissue that generates a stable rhythm in healthy hearts, yet a general mechanistic explanation for when and how this tissue remains stable is lacking. Although computational and theoretical analyses could elucidate these phenomena, such methods have rarely been used in realistic (large-dimensional) gap-junction coupled heterogeneous pacemaker tissue models. In this study, we adapt a recent model of pacemaker cells (Severi et al., 2012), incorporating biophysical representations of ion channel and intracellular calcium dynamics, to capture physiological features of a heterogeneous population of pacemaker cells, in particular "center" and "peripheral" cells with distinct intrinsic frequencies and action potential morphology. Large-scale simulations of the SAN tissue, represented by a heterogeneous tissue structure of pacemaker cells, exhibit a rich repertoire of behaviors, including complete synchrony, traveling waves of activity originating from periphery to center, and transient traveling waves originating from the center. We use phase reduction methods that do not require fully simulating the large-scale model to capture these observations. Moreover, the phase reduced models accurately predict key properties of the tissue electrical dynamics, including wave frequencies when synchronization occurs, and wave propagation direction in a variety of tissue models. With the reduced phase models, we analyze the relationship between cell distributions and coupling strengths and the resulting transient dynamics. Further, the reduced phase model predicts parameter regimes of irregular electrical dynamics. Thus, we demonstrate that phase reduced oscillator models applied to realistic pacemaker tissue is a useful tool for investigating the spatial-temporal dynamics of cardiac pacemaker activity.

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Phase transition to frequency entrainment in a long chain of pulse-coupled oscillators

Cited by 13 publications

References 26 publications

Frequency spirals

Frequency spirals

Phase transitions towards frequency entrainment in large oscillator lattices

Analysis of heterogeneous cardiac pacemaker tissue models and traveling wave dynamics

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