Self-organized synchronization of digital phase-locked loops with delayed coupling in theory and experiment

Abstract

Self-organized synchronization occurs in a variety of natural and technical systems but has so far only attracted limited attention as an engineering principle. In distributed electronic systems, such as antenna arrays and multi-core processors, a common time reference is key to coordinate signal transmission and processing. Here we show how the self-organized synchronization of mutually coupled digital phase-locked loops (DPLLs) can provide robust clocking in large-scale systems. We develop a nonlinear phase description of individual and coupled DPLLs that takes into account filter impulse responses and delayed signal transmission. Our phase model permits analytical expressions for the collective frequencies of synchronized states, the analysis of stability properties and the time scale of synchronization. In particular, we find that signal filtering introduces stability transitions that are not found in systems without filtering. To test our theoretical predictions, we designed and carried out experiments using networks of off-the-shelf DPLL integrated circuitry. We show that the phase model can quantitatively predict the existence, frequency, and stability of synchronized states. Our results demonstrate that mutually delay-coupled DPLLs can provide robust and self-organized synchronous clocking in electronic systems.

Travelling waves in somitogenesis: Collective cellular properties emerge from time-delayed juxtacrine oscillation coupling

“…3.3B). A similar pattern has been reported partially in the context of the zebrafish PSM oscillator, HES/HER oscillations in neural differentiation, and a variety of synchronization phenomena across the natural sciences (Wang et al, 2014;Morelli et al, 2009;Herrgen et al, 2010;Momiji & Monk, 2009;Sadeghi & Valizadeh, 2014; Pavlides et al, 2015;Vanag et al, 2016;Wetzel et al, 2017). In contrast to other mathematical descriptions, however, we do not observe oscillation death, a stable non-oscillating state, which has been hypothesized to occur between the synchronization and anti-synchronization phase space regions (Fig.…”

Travelling waves in somitogenesis: collective cellular properties emerge from time-delayed juxtacrine oscillation coupling

“…In each region, the collective period increases monotonically (Figure 5A) and the collective amplitude describes a parabola with a local maximum (Figure 5B). A similar pattern has been reported partially in the context of the zebrafish PSM oscillator, Hes/her oscillations in neural differentiation, and a variety of synchronization phenomena across the natural sciences (Wang et al 2014; Morelli et al, 2009; Herrgen et al, 2010; Momiji & Monk, 2009; Sadeghi & Valizadeh, 2014; Pavlides et al, 2015; Vanag et al, 2016; Wetzel et al, 2017). Nevertheless, in contrast to other mathematical descriptions, we do not observe oscillation death, a stable non-oscillating state, which has been hypothesized to occur between the synchronization and anti-synchronization phase space regions (Figure 5B, Shimojo et al, 2016).…”

“…A wide variety of time-delayed coupling phenomena related to the one investigated here, are currently studied across the natural sciences (Sadeghi & Valizadeh, 2014; Pavlides et al, 2015; Vanag et al, 2016; Wetzel et al, 2017). Recently, a unifying hypothesis for the time-delayed coupling mediated by Notch in neurogenesis and somitogenesis has been proposed (Shimojo et al, 2016; Shimojo & Kageyama, 2016): a difference in coupling delays between biological tissues could explain why dynamical salt-and-pepper patterning is observed in the developing brain and synchronization is observed in the PSM (Shimojo et al, 2016; Shimojo & Kageyama, 2016).…”

Stochastic Kuramoto oscillators with discrete phase states

“…Moreover, we have chosen a generic but contingent discretization scheme for the coupling process (see Appendix B). It will be interesting to unfold the dynamics of different model realizations and to apply the proposed discretization schemes to, e.g., Kuramoto oscillators with inertia [47][48][49] and excitable dynamics [42] as well as time-delayed coupling [41,[50][51][52] and signal filtering [53,54], which goes beyond the Markovian approach. In this Appendix, we derive the linear noise approximation Eqs.…”