Phase synchronization in the two-dimensional Kuramoto model: Vortices and duality

Sarkar, Mrinal; Gupte, Neelima

doi:10.1103/physreve.103.032204

Cited by 10 publications

(10 citation statements)

References 41 publications

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“…In this sense, one may interpret equation ( 8) as a dynamic Mermin-Wagner theorem of a non-equilibrium system [48]. The analogy between synchronization and the XY model can be made more explicit for the classical Kuramoto model with local sinusoidal coupling [40].…”

Section: Diverging Relaxation Timementioning

confidence: 99%

“…We are interested in dynamic steady-state solutions of the equation of motion, equation (5). As a reference, we first re-visit the classical Kuramoto model with local sinusoidal coupling [40,41]. Specifically, we consider a Kuramoto model of coupled phase oscillators with phases ϕ i at respective lattice positions x i and equation of motion φj (t) = ω 0 − i =j c ij (ϕ i , ϕ j ) with coupling function c ij = ε sin(ϕ j − ϕ i ) for all pairs (i, j) of neighbors and c ij = 0 else.…”

Section: Metachronal Wave Solutionsmentioning

confidence: 99%

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Synchronization in cilia carpets: multiple metachronal waves are stable, but one wave dominates

Соловьев

Friedrich

2022

New J. Phys.

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Carpets of actively bending cilia represent arrays of biological oscillators that can exhibit self-organized metachronal synchronization in the form of traveling waves of cilia phase. This metachronal coordination supposedly enhances fluid transport by cilia carpets. Using a multi-scale model calibrated by an experimental cilia beat pattern, we predict multi-stability of wave modes. Yet, a single mode, corresponding to a dexioplectic wave, has predominant basin-of-attraction. Similar to a ‘dynamic’ Mermin–Wagner theorem, relaxation times diverge with system size, which rules out global order in infinite systems. In finite systems, we characterize a synchronization transition as function of quenched frequency disorder, using generalized Kuramoto order parameters. Our framework termed Lagrangian mechanics of active systems allows to predict the direction and stability of metachronal synchronization for given beat patterns.

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Section: Diverging Relaxation Timementioning

confidence: 99%

Section: Metachronal Wave Solutionsmentioning

confidence: 99%

Synchronization in cilia carpets: multiple metachronal waves are stable, but one wave dominates

Соловьев

Friedrich

2022

New J. Phys.

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“…This equation will approximate the activity around the stationary sets (stationary point or limit cycles) of the cortical columns. It is equivalent to the celebrated Kuramoto model and can also be derived from the Hamiltonian of the XY-model [ 25 ]. This model has been used to describe phase transitions in solid-state physics and has been shown to have interesting topological phase transitions as was shown by Berezinskii, Kosterlitz and Thouless [ 26 ].…”

Section: Interaction Between Columnsmentioning

confidence: 99%

Modelling cortical network dynamics

Cooray,

Rosch,

Friston

2024

Discov Appl Sci

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We have investigated the theoretical constraints of the interactions between coupled cortical columns. Each cortical column consists of a set of neural populations where each population is modelled as a neural mass. The existence of semi-stable states within a cortical column is dependent on the type of interaction between the neuronal populations, i.e., the form of the synaptic kernels. Current-to-current coupling has been shown, in contrast to potential-to-current coupling, to create semi-stable states within a cortical column. The interaction between semi-stable states of the cortical columns is studied where we derive the dynamics for the collected activity. For small excitations the dynamics follow the Kuramoto model; however, in contrast to previous work we derive coupled equations between phase and amplitude dynamics with the possibility of defining connectivity as a stationary and dynamic variable. The turbulent flow of phase dynamics which occurs in networks of Kuramoto oscillators would indicate turbulent changes in dynamic connectivity for coupled cortical columns which is something that has been recorded in epileptic seizures. We used the results we derived to estimate a seizure propagation model which allowed for inversions using the Laplace assumption (Dynamic Causal Modelling). The seizure propagation model was trialed on simulated data, and future work will investigate the estimation of the connectivity matrix from empirical data. This model can be used to predict changes in seizure evolution after virtual changes in the connectivity network, something that could be of clinical use when applied to epilepsy surgical cases.

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“…From a modeling point of view, lattices are very natural to study as they account for 2D and 3D spatial organization among oscillators. Since the pioneering work of Sakaguchi, Shinomoto, and Kuramoto [31], there have been many studies of coupled oscillator systems on lattices investigating synchronization [39,9,13,16,2,26,22,4,3,8], spiral patterns [27,18,28,33,6,40], and chimeras [36,15,34,12,35,23,25].…”

Section: Introductionmentioning

confidence: 99%

Stability of twisted states on lattices of Kuramoto oscillators

Goebel¹,

Mizuhara²,

Stepanoff³

2021

Preprint

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Real world systems comprised of coupled oscillators have the ability to exhibit spontaneous synchronization and other complex behaviors. The interplay between the underlying network topology and the emergent dynamics remains a rich area of investigation for both theory and experiment. In this work we study lattices of coupled Kuramoto oscillators with non-local interactions. Our focus is on the stability of twisted states. These are equilibrium solutions with constant phase shifts between oscillators resulting in spatially linear profiles. Linear stability analysis follows from studying the quadratic form associated with the Jacobian matrix. Novel estimates on both stable and unstable regimes of twisted states are obtained in several cases. Moreover, exploiting the "almost circulant" nature of the Jacobian obtains a surprisingly accurate numerical test for stability. While our focus is on 2D square lattices, we show how our results can be extended to higher dimensions.

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Phase synchronization in the two-dimensional Kuramoto model: Vortices and duality

Cited by 10 publications

References 41 publications

Synchronization in cilia carpets: multiple metachronal waves are stable, but one wave dominates

Synchronization in cilia carpets: multiple metachronal waves are stable, but one wave dominates

Modelling cortical network dynamics

Stability of twisted states on lattices of Kuramoto oscillators

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