The Kuramoto model with distributed shear

We uncover a solvable generalization of the Kuramoto model in which shears (or nonisochronicities) and natural frequencies are distributed and statistically dependent. We show that the strength and sign of this dependence greatly alter synchronization and yield qualitatively different phase diagrams. The Ott-Antonsen ansatz allows us to obtain analytical results for a specific family of joint distributions. We also derive, using linear stability analysis, general formulae for the stability border of incoherence.Comment: 6 page

Shear diversity prevents collective synchronization

Large ensembles of heterogeneous oscillators often exhibit collective synchronization as a result of mutual interactions. If the oscillators have distributed natural frequencies and common shear (or nonisochronicity), the transition from incoherence to collective synchronization is known to occur at large enough values of the coupling strength. However, here we demonstrate that shear diversity cannot be counterbalanced by diffusive coupling leading to synchronization. We present the first analytical results for the Kuramoto model with distributed shear, and show that the onset of collective synchronization is impossible if the width of the shear distribution exceeds a precise threshold

Synchrony-induced modes of oscillation of a neural field model

We investigate the modes of oscillation of heterogeneous ring-networks of quadratic integrate-and-fire (QIF) neurons with non-local, space-dependent coupling. Perturbations of the equilibrium state with a particular wave number produce transient standing waves with a specific temporal frequency, analogous to those in a tense string. In the neuronal network, the equilibrium corresponds to a spatially homogeneous, asynchronous state. Perturbations of this state excite the network’s oscillatory modes, which reflect the interplay of episodes of synchronous spiking with the excitatory-inhibitory spatial interactions. In the thermodynamic limit, an exact low-dimensional neural field model (QIF-NFM) describing the macroscopic dynamics of the network is derived. This allows us to obtain formulas for the Turing eigenvalues of the spatially-homogeneous state, and hence to obtain its stability boundary. We find that the frequency of each Turing mode depends on the corresponding Fourier coefficient of the synaptic pattern of connectivity. The decay rate instead, is identical for all oscillation modes as a consequence of the heterogeneity-induced desynchronization of the neurons. Finally, we numerically compute the spectrum of spatially-inhomogeneous solutions branching from the Turing bifurcation, showing that similar oscillatory modes operate in neural bump states, and are maintained away from onset

A mean field model for movement induced changes in the beta rhythm

In electrophysiological recordings of the brain, the transition from high amplitude to low amplitude signals are most likely caused by a change in the synchrony of underlying neuronal population firing patterns. Classic examples of such modulations are the strong stimulus-related oscillatory phenomena known as the movement related beta decrease (MRBD) and post-movement beta rebound (PMBR). A sharp decrease in neural oscillatory power is observed during movement (MRBD) followed by an increase above baseline on movement cessation (PMBR). MRBD and PMBR represent important neuroscientific phenomena which have been shown to have clinical relevance. Here, we present a parsimonious model for the dynamics of synchrony within a synaptically coupled spiking network that is able to replicate a human MEG power spectrogram showing the evolution from MRBD to PMBR. Importantly, the high-dimensional spiking model has an exact mean field description in terms of four ordinary differential equations that allows considerable insight to be obtained into the cause of the experimentally observed time-lag from movement termination to the onset of PMBR (~ 0.5 s), as well as the subsequent long duration of PMBR (~ 1-10 s). Our model represents the first to predict these commonly observed and robust phenomena and represents a key step in their understanding, in health and disease

Community Structure and Multi-Modal Oscillations in Complex Networks

In many types of network, the relationship between structure and function is of great significance. We are particularly interested in community structures, which arise in a wide variety of domains. We apply a simple oscillator model to networks with community structures and show that waves of regular oscillation are caused by synchronised clusters of nodes. Moreover, we show that such global oscillations may arise as a direct result of network topology. We also observe that additional modes of oscillation (as detected through frequency analysis) occur in networks with additional levels of topological hierarchy and that such modes may be directly related to network structure. We apply the method in two specific domains (metabolic networks and metropolitan transport) demonstrating the robustness of our results when applied to real world systems. We conclude that (where the distribution of oscillator frequencies and the interactions between them are known to be unimodal) our observations may be applicable to the detection of underlying community structure in networks, shedding further light on the general relationship between structure and function in complex systems

26th Annual Computational Neuroscience Meeting (CNS*2017): Part 3 - Meeting Abstracts - Antwerp, Belgium. 15–20 July 2017

This work was produced as part of the activities of FAPESP Research,\ud Disseminations and Innovation Center for Neuromathematics (grant\ud 2013/07699-0, S. Paulo Research Foundation). NLK is supported by a\ud FAPESP postdoctoral fellowship (grant 2016/03855-5). ACR is partially\ud supported by a CNPq fellowship (grant 306251/2014-0)