Dynamics of hierarchical weighted networks of Van der Pol oscillators

We investigate the dynamics of regular fractal-like networks of hierarchically coupled van der Pol oscillators. The hierarchy is imposed in terms of the coupling strengths or link weights. We study the low frequency modes, as well as frequency and phase synchronization, in the network by a process of repeated coarse-graining of oscillator units. At any given stage of this process, we sum over the signals from the oscillator units of a clique to obtain a new oscillating unit. The frequencies and the phases for the coarse-grained oscillators are found to progressively synchronize with the number of coarse-graining steps. Furthermore, the characteristic frequency is found to decrease and finally stabilize to a value that can be tuned via the parameters of the system. We compare our numerical results with those of an approximate analytic solution and find good qualitative agreement. Our study on this idealized model shows how oscillations with a precise frequency can be obtained in systems with heterogeneous couplings. It also demonstrates the effect of imposing a hierarchy in terms of link weights instead of one that is solely topological, where the connectivity between oscillators would be the determining factor, as is usually the case.
<br>
Many oscillatory signaling processes in nature are multi-scale, where the characteristic macro-scale frequency of the signal is an emergent property of the coupling of oscillators of different frequencies acting on the micro-scale. Here, we show, using a combination of numerical simulations and analytical investigations, how oscillations of a precise frequency can be obtained from a fractal network of oscillators when the coupling is heterogeneous. Our results on various hierarchically coupled networks shed light on how the frequencies synchronize and the phases become locked at various scales. Our model also shows how the concept of hierarchy can be realized when studying synchronization in weighted networks.