We study the Kuramoto model of globally coupled oscillators with a biharmonic coupling function. We develop an analytic self-consistency approach to find stationary synchronous states in the thermodynamic limit and demonstrate that there is a huge multiplicity of such states, which differ microscopically in the distributions of locked phases. These synchronous regimes already exist prior to the linear instability transition of the fully asynchronous state. In the presence of white Gaussian noise, the multiplicity is lifted, but the dependence of the order parameters on coupling constants remains nontrivial.

Sleep plays an important role in consolidation of recent memories. However, the mechanisms of consolidation remain poorly understood. In this study, using a realistic computational model of the thalamocortical network, we demonstrated that sleep spindles (the hallmark of N2 stage sleep) and slow oscillations (the hallmark of N3 stage sleep) both facilitate spike sequence replay as necessary for consolidation. When multiple memories were trained, the local nature of spike sequence replay during spindles allowed replay of the memories independently, while during slow oscillations replay of the weak memory was competing to the strong memory replay. This led to the weak memory extinction unless when sleep spindles (N2 sleep) preceded slow oscillations (N3 sleep), as observed during natural sleep. Our study presents a mechanistic explanation for the role of sleep rhythms in memory consolidation and proposes a testable hypothesis how the natural structure of sleep stages provides an optimal environment to consolidate memories.peer-reviewed) is the author/funder. All rights reserved. No reuse allowed without permission.The copyright holder for this preprint (which was not . http://dx.doi.org/10.1101/153007 doi: bioRxiv preprint first posted online 3 Significant StatementNumerous studies suggest importance of NREM sleep rhythms -spindles and slow oscillations -in sleep related memory consolidation. However, synaptic mechanisms behind the role of these rhythms in memory and learning are still unknown. Our new study predicts that sleep replay -the neuronal substrate of memory consolidation -is organized within the sleep spindles and coordinated by the Down to Up state transitions of the slow oscillation. For multiple competing memories, slow oscillations facilitated only strongest memory replay, while sleep spindles allowed a consolidation of the multiple competing memories independently. Our study predicts how the basic structure of the natural sleep stages provides an optimal environment for consolidation of multiple memories.peer-reviewed)

We study a generic model of globally coupled rotors that includes the effects of noise, phase shift in the coupling, and distributions of moments of inertia and natural frequencies of oscillation. As particular cases, the setup includes previously studied Sakaguchi-Kuramoto, Hamiltonian and Brownian mean-field, and Tanaka-Lichtenberg-Oishi and Acebrón-Bonilla-Spigler models. We derive an exact solution of the self-consistent equations for the order parameter in the stationary state, valid for arbitrary parameters in the dynamics, and demonstrate nontrivial phase transitions to synchrony that include reentrant synchronous regimes.

We study synchronization in a Kuramoto model of globally coupled phase oscillators with a bi-harmonic coupling function, in the thermodynamic limit of large populations. We develop a method for an analytic solution of self-consistent equations describing uniformly rotating complex order parameters, both for single-branch (one possible state of locked oscillators) and multi-branch (two possible values of locked phases) entrainment. We show that synchronous states coexist with the neutrally linearly stable asynchronous regime. The latter has a finite life time for finite ensembles, this time grows with the ensemble size as a power law.