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Amplitude dynamics favors synchronization in complex networks

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, Articlenumber:24915 (2016) | Download Citation

In this paper we study phase synchronization in random complex networks of coupled periodic oscillators. In particular, we show that, when amplitude dynamics is not negligible, phase synchronization may be enhanced. To illustrate this, we compare the behavior of heterogeneous units with both amplitude and phase dynamics and pure (Kuramoto) phase oscillators. We find that in small network motifs the behavior crucially depends on the topology and on the node frequency distribution. Surprisingly, the microscopic structures for which the amplitude dynamics improves synchronization are those that are statistically more abundant in random complex networks. Thus, amplitude dynamics leads to a general lowering of the synchronization threshold in arbitrary random topologies. Finally, we show that this synchronization enhancement is generic of oscillators close to Hopf bifurcations. To this aim we consider coupled FitzHugh-Nagumo units modeling neuron dynamics.

Synchronization of interacting units is a universal collective behavior appearing in a variety of natural and artificial systems. The most used mathematical formalisms to study how synchronization shows up, such as the paradigmatic Kuramoto model, consider each dynamical unit as a pure phase oscillator. This coarse-grained dynamical description lies in the following assumption: since each isolated unit has a stable limit cycle oscillation at a given frequency, it is enough to study the variable accounting for the motion along this limit cycle (the phase), while the other dynamical variables may be ignored. This approach is justified provided the attraction to the limit cycle is strong compared to the coupling between dynamical units. On the contrary, if the coupling is strong or the attraction to the limit cycle is weak, different phenomena, whose analysis demands both amplitude and phase dynamics, such as oscillation death and remote synchronization, can occur.

The Stuart-Landau (SL) model allows to bridge the gap between the simplicity of the Kuramoto model and the completeness of the amplitude-phase frameworks. In particular the SL model may describe, by means of a single parameter, both the behavior close to the Hopf bifurcation, which represents an instance where the attraction to the limit cycle is weak, and the case where, on the contrary, the motion is purely confined to the limit cycle thus behaving as a phase oscillator.

Synchronization of pure phase oscillators interacting according to a complex network topology has been widely studied during the last decade. These studies have covered different topics such as the conditions for the onset of synchronization, the path towards it, the interplay between topology and dynamics, the emergence of first-order transition, the effect of noise on the robustness of the global state, among others. However, interesting issues about the interplay of a network topology of interactions and the coupled dynamics of the amplitude and phase of the oscillators are often ignored from scratch, due to the underlying dynamical framework chosen.

In this paper, we adopt the framework of SL oscillators coupled according to different network configurations and compare with its reduction to pure phase oscillators. Our goal is to investigate the effect of the amplitude dynamics near the Hopf bifurcation on the synchronization in random complex networks. The most striking result is that, when the amplitude dynamics is not negligible and the natural oscillation frequencies of the nodes are not homogeneous, synchronization is enhanced regardless of the topology of the underlying network. The result holds for oscillators close to the Hopf bifurcation and in particular we also discuss its implications for neuron-like dynamics, by illustrating the behavior for the FitzHugh-Nagumo model.

Our analysis is based on a system of coupled SL oscillators formally described by:

where is a complex variable so that in polar coordinates: , being the amplitude of oscillator and its phase. is the Hopf bifurcation parameter which controls how fast the trajectory decays onto the attractor, is the frequency of oscillator when uncoupled (natural frequency), the coupling strength between interacting units and is the network adjacency matrix (defined as  =  = 1 if and are connected,  = 0 otherwise).

In this work, system (1) is studied with respect to the parameters and and different frequency distributions. Importantly, when is large compared to coupling , the amplitude dynamics vanishes, , and . In this limiting case Eq. (1) reduce to:

, the Kuramoto model in a network. As introduced above, our aim is to compare phase synchronization in networks of amplitude and phase oscillators with that in networks of pure phase oscillators. In doing so, we unveil the key role of the parameter ruling the Hopf bifurcation, which transforms a generic oscillator into a pure phase one driving the system far from the critical point. To this aim, we consider SL oscillators [Eq. (1)] with small (in particular,  = 1), while for Kuramoto (K) oscillators we study Eq. (2) or, equivalently, Eq. (1) with large . We will then discuss the universality of our findings by studying dynamics other than the SL in the proximity of the Hopf bifurcation.

Phase synchronization in the model is monitored through the order parameter . This parameter, rigorously defined in the Methods section, represents the degree of global synchronization in the whole network. Values of close to 1 indicates that all the pairs of oscillators are phase synchronized, whereas  = 0 in the absence of synchronization.

We start our discussion by considering the onset of synchronization in small undirected network motifs. In particular we analyze the 2-node motif (representing a pair of coupled oscillators) and two 3-node motifs (the open and the closed triangle). In the case of the open triangle we have three different configurations depending on how natural frequencies are assigned to each of the three oscillators. In particular, if we indicate as ω 1 the natural frequency of the degree-2 (central) node we have: i ) ω 1  >  ω 2 , ω 3 ; ii ) ω 1  <  ω 2 , ω 3 ; and iii ) ω 2  <  ω 1  <  ω 3 (or equivalently ω 2  >  ω 1  >  ω 3 ).

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George A. Schier; Wayne D. Shepperd; John R. Jones


There are basically two approaches to regenerating aspen stands-sexual reproduction using seed, or vegetative regeneration by root suckering. In the West, root suckering is the most practical method. The advantage of having an existing, well established root system capable of producing numerous root suckers easily outweighs natural or artificial reforestation in the...

Macroscopic in vivo imaging of facial nerve regeneration in Thy1-GFP rats .

Placheta, Eva; Wood, Matthew D; Lafontaine, Christine; Frey, Manfred; Gordon, Tessa; Borschel, Gregory H

Facial nerve injury leads to severe functional and aesthetic deficits. The transgenic Thy1-GFP rat is a new model for facial nerve injury and reconstruction research that will help improve clinical outcomes through translational facial nerve injury research. To determine whether serial in vivo imaging of nerve regeneration in the transgenic rat model is possible, facial nerve regeneration was imaged under the main paradigms of facial nerve injury and reconstruction. Fifteen male Thy1-GFP rats , which express green fluorescent protein (GFP) in their neural structures, were divided into 3 groups in the laboratory: crush-injury, direct repair, and cross-face nerve grafting (30-mm graft length). The distal nerve stump or nerve graft was predegenerated for 2 weeks. The facial nerve of the transgenic rats was serially imaged at the time of operation and after 2, 4, and 8 weeks of regeneration . The imaging was performed under a GFP-MDS-96/BN excitation stand (BLS Ltd). Facial nerve injury. Optical fluorescence of regenerating facial nerve axons. Serial in vivo imaging of the regeneration of GFP-positive axons in the Thy1-GFP rat model is possible. All animals survived the short imaging procedures well, and nerve regeneration was followed over clinically relevant distances. The predegeneration of the distal nerve stump or the cross-face nerve graft was, however, necessary to image the regeneration front at early time points. Crush injury was not suitable to sufficiently predegenerate the nerve (and to allow for degradation of the GFP through Wallerian degeneration). After direct repair, axons regenerated over the coaptation site in between 2 and 4 weeks. The GFP-positive nerve fibers reached the distal end of the 30-mm-long cross-face nervegrafts after 4 to 8 weeks of regeneration . The time course of facial nerve regeneration was studied by serial in vivo imaging in the transgenic rat model. Nerve regeneration was followed over clinically relevant distances in a small

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