scholarly journals Critical synchronization dynamics of the Kuramoto model on connectome and small world graphs

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Géza Ódor ◽  
Jeffrey Kelling
Keyword(s):  
Control Parameter ◽  
Mean Field ◽  
Small World ◽  
Kuramoto Model ◽  
Crossover Point ◽  
Small World Graphs ◽  
Synchronization Phase ◽  
Experimental Values ◽  
The Kuramoto Model ◽  
Parameter Dependent

AbstractThe hypothesis, that cortical dynamics operates near criticality also suggests, that it exhibits universal critical exponents which marks the Kuramoto equation, a fundamental model for synchronization, as a prime candidate for an underlying universal model. Here, we determined the synchronization behavior of this model by solving it numerically on a large, weighted human connectome network, containing 836733 nodes, in an assumed homeostatic state. Since this graph has a topological dimension d < 4, a real synchronization phase transition is not possible in the thermodynamic limit, still we could locate a transition between partially synchronized and desynchronized states. At this crossover point we observe power-law–tailed synchronization durations, with τt ≃ 1.2(1), away from experimental values for the brain. For comparison, on a large two-dimensional lattice, having additional random, long-range links, we obtain a mean-field value: τt ≃ 1.6(1). However, below the transition of the connectome we found global coupling control-parameter dependent exponents 1 < τt ≤ 2, overlapping with the range of human brain experiments. We also studied the effects of random flipping of a small portion of link weights, mimicking a network with inhibitory interactions, and found similar results. The control-parameter dependent exponent suggests extended dynamical criticality below the transition point.

scholarly journals Investigation of Preferred Orientations in Planar Polycrystals

2008 ◽  
Vol 75 (5) ◽  
Author(s):  
M. R. Tonks ◽  
A. J. Beaudoin ◽  
F. Schilder ◽  
D. A. Tortorelli
Keyword(s):  
Texture Evolution ◽  
Mean Field ◽  
Homogenization Method ◽  
Process Models ◽  
Kuramoto Model ◽  
Taylor Model ◽  
Crystal Plasticity Finite Element ◽  
Polycrystal Plasticity ◽  
Taylor Hypothesis ◽  
The Kuramoto Model

More accurate manufacturing process models come from better understanding of texture evolution and preferred orientations. We investigate the texture evolution in the simplified physical framework of a planar polycrystal with two slip systems used by Prantil et al. (1993, “An Analysis of Texture and Plastic Spin for Planar Polycrystal,” J. Mech. Phys. Solids, 41(8), pp. 1357–1382). In the planar polycrystal, the crystal orientations behave in a manner similar to that of a system of coupled oscillators represented by the Kuramoto model. The crystal plasticity finite element method and the stochastic Taylor model (STM), a stochastic method for mean-field polycrystal plasticity, predict the development of a steady-state texture not shown when employing the Taylor hypothesis. From this analysis, the STM appears to be a useful homogenization method when using representative standard deviations.

scholarly journals Dynamical Aspects of Mean Field Plane Rotators and the Kuramoto Model

2009 ◽  
Vol 138 (1-3) ◽  
pp. 270-290 ◽  
Author(s):  
Lorenzo Bertini ◽  
Giambattista Giacomin ◽  
Khashayar Pakdaman
Keyword(s):  
Mean Field ◽  
Kuramoto Model ◽  
The Kuramoto Model ◽  
Field Plane ◽  
Plane Rotators

scholarly journals HYDRODYNAMICS OF THE KURAMOTO–VICSEK MODEL OF ROTATING SELF-PROPELLED PARTICLES

2013 ◽  
Vol 24 (02) ◽  
pp. 277-325 ◽  
Author(s):  
PIERRE DEGOND ◽  
GIACOMO DIMARCO ◽  
THI BICH NGOC MAC
Keyword(s):  
Angular Velocity ◽  
Hydrodynamic Model ◽  
Mean Field ◽  
Slight Modification ◽  
Kuramoto Model ◽  
Local Alignment ◽  
Vicsek Model ◽  
Linearized Stability ◽  
The Kuramoto Model ◽  
Rotation Motion

We consider an Individual-Based Model for self-rotating particles interacting through local alignment and investigate its macroscopic limit. This model describes self-propelled particles moving in the plane and trying to synchronize their rotation motion with their neighbors. It combines the Kuramoto model of synchronization and the Vicsek model of swarm formation. We study the mean-field kinetic and hydrodynamic limits of this system within two different scalings. In the small angular velocity regime, the resulting model is a slight modification of the "Self-Organized Hydrodynamic" model which has been previously introduced by the first author. In the large angular velocity case, a new type of hydrodynamic model is obtained. A preliminary study of the linearized stability is proposed.

Network Dynamics

2019 ◽  
pp. 207-242
Author(s):  
David D. Nolte
Keyword(s):  
Random Graphs ◽  
Small World ◽  
Distance Matrix ◽  
Kuramoto Model ◽  
Global Synchronization ◽  
Scale Free ◽  
Wide Range ◽  
Network Topologies ◽  
The Kuramoto Model ◽  
Static Connectivity

A language of nodes and links, degree and moments, and adjacency matrix and distance matrix, among others, is defined and used to capture the wide range of different types and properties of network topologies. Regular graphs and random graphs have fundamentally different connectivities that play a role in dynamic processes such as diffusion and synchronization on a network. Three common random graphs are the Erdös–Rényi (ER) graph, the small-world (SW) graph, and the scale-free (SF) graph. Random graphs give rise to critical phenomena based on static connectivity properties, such as the percolation threshold, but also exhibit dynamical thresholds for the diffusion of states across networks and the synchronization of oscillators. The vaccination threshold for diseases propagating on networks and the global synchronization transition in the Kuramoto model are examples of dynamical processes that can be used to probe network topologies.

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