Stability and multistability of synchronization in networks of coupled phase oscillators

Coupled phase oscillators usually achieve synchronization as the coupling strength among oscillators is increased beyond a critical value. The stability of synchronous state remains an open issue. In this paper, we study the stability of the synchronous state in coupled phase oscillators. It is found that numerical integration of differential equations of coupled phase oscillators with a finite time step may induce desynchronization at strong couplings. The mechanism behind this instability is that numerical accumulated errors in simulations may trigger the loss of stability of the synchronous state.Desynchronization critical couplings are found to increase and diverge as a power law with decreasing the integral time step. Theoretical analysis supports the local stability of the synchronized state. Globally the emergence of synchronous state depends on the initial conditions. Other metastable ordered states such as twisted states can coexist with the synchronous mode. These twisted states keep locally stable on a sparse network but lose their stability when the network becomes dense.

Synchronization–desynchronization transitions in networks of circle maps with sinusoidal coupling

Coupled phase oscillators are adopted as powerful platforms in studying synchrony behaviors emerged in various systems with rhythmic dynamics. Much attention has been focused on coupled time-continuous oscillators described by differential equations. In this paper, we study the synchronization dynamics of networks of coupled circle maps as the discrete version of the Kuramoto model. Despite of its simplicity in mathematical form, it is found that discreteness may induce many interesting synchronization behaviors. Multiple synchronization and desynchronization transitions of both phases and frequencies are found with varying the coupling among circle-map oscillators. The mechanisms of these transitions are interpreted in terms of the mean-field approach, where collective bifurcation cascades are revealed for coupled circle-map oscillators.

USP21 deubiquitylates Nanog to regulate protein stability and stem cell pluripotency

The homeobox transcription factor Nanog has a vital role in maintaining pluripotency and self-renewal of embryonic stem cells(ESCs).Stabilization of Nanog proteins is essential for ESCs.The ubiquitin–proteasome pathway mediated by E3 ubiquitin ligases and deubiquitylases is one of the key ways to regulate protein levels and functions.Although ubiquitylation of Nanog catalyzed by the ligase FBXW8 has been demonstrated,the deubiquitylase that maintains the protein levels of Nanog in ESCs yet to be defined.In this study,we identify the ubiquitin-specific peptidase 21(USP21)as a deubiquitylase for Nanog,but not for Oct4 or Sox2.USP21 interacts with Nanog protein in ESCs in vivo and in vitro.The C-terminal USP domain of USP21 and the C-domain of Nanog are responsible for this interaction.USP21 deubiquitylates the K48-type linkage of the ubiquitin chain of Nanog,stabilizing Nanog.USP21-mediated Nanog stabilization is enhanced in mouse ESCs and this stabilization is required to maintain the pluripotential state of the ESCs.Depletion of USP21 in mouse ESCs leads to Nanog degradation and ESC differentiation.Overall,our results demonstrate that USP21 maintains the stemness of mouse ESCs through deubiquitylating and stabilizing Nanog.