Stationary and Traveling Wave States of the Kuramoto Model with an Arbitrary Distribution of Frequencies and Coupling Strengths

We consider the Kuramoto model of an ensemble of interacting oscillators allowing for an arbitrary distribution of frequencies and coupling strengths. We define a family of traveling wave states as stationary in a rotating frame, and derive general equations for their parameters. We suggest empirical stability conditions which, for the case of incoherence, become exact. In addition to making new theoretical predictions, we show that many earlier results follow naturally from our general framework. The results are applicable in scientific contexts ranging from physics to biology.Comment: 5 pages, 1 figur

Understanding the enhanced synchronization of delay-coupled networks with fluctuating topology

We study the dynamics of networks with coupling delay, from which the connectivity changes over time. The synchronization properties are shown to depend on the interplay of three time scales: the internal time scale of the dynamics, the coupling delay along the network links and time scale at which the topology changes. Concentrating on a linearized model, we develop an analytical theory for the stability of a synchronized solution. In two limit cases the system can be reduced to an “effective” topology: In the fast switching approximation, when the network fluctuations are much faster than the internal time scale and the coupling delay, the effective network topology is the arithmetic mean over the different topologies. In the slow network limit, when the network fluctuation time scale is equal to the coupling delay, the effective adjacency matrix is the geometric mean over the adjacency matrices of the different topologies. In the intermediate regime the system shows a sensitive dependence on the ratio of time scales, and specific topologies, reproduced as well by numerical simulations. Our results are shown to describe the synchronization properties of fluctuating networks of delay-coupled chaotic maps

A biophysical model of dynamic balancing of excitation and inhibition in fast oscillatory large-scale networks

Over long timescales, neuronal dynamics can be robust to quite large perturbations, such as changes in white matter connectivity and grey matter structure through processes including learning, aging, development and certain disease processes. One possible explanation is that robust dynamics are facilitated by homeostatic mechanisms that can dynamically rebalance brain networks. In this study, we simulate a cortical brain network using the Wilson-Cowan neural mass model with conduction delays and noise, and use inhibitory synaptic plasticity (ISP) to dynamically achieve a spatially local balance between excitation and inhibition. Using MEG data from 55 subjects we find that ISP enables us to simultaneously achieve high correlation with multiple measures of functional connectivity, including amplitude envelope correlation and phase locking. Further, we find that ISP successfully achieves local E/I balance, and can consistently predict the functional connectivity computed from real MEG data, for a much wider range of model parameters than is possible with a model without ISP

25th Annual Computational Neuroscience Meeting: CNS-2016

Abstracts of the 25th Annual Computational Neuroscience Meeting: CNS-2016 Seogwipo City, Jeju-do, South Korea. 2–7 July 201

Modelling human choices: MADeM and decision‑making

Research supported by FAPESP 2015/50122-0 and DFG-GRTK 1740/2. RP and AR are also part of the Research, Innovation and Dissemination Center for Neuromathematics FAPESP grant (2013/07699-0). RP is supported by a FAPESP scholarship (2013/25667-8). ACR is partially supported by a CNPq fellowship (grant 306251/2014-0)

25th annual computational neuroscience meeting: CNS-2016

The same neuron may play different functional roles in the neural circuits to which it belongs. For example, neurons in the Tritonia pedal ganglia may participate in variable phases of the swim motor rhythms [1]. While such neuronal functional variability is likely to play a major role the delivery of the functionality of neural systems, it is difficult to study it in most nervous systems. We work on the pyloric rhythm network of the crustacean stomatogastric ganglion (STG) [2]. Typically network models of the STG treat neurons of the same functional type as a single model neuron (e.g. PD neurons), assuming the same conductance parameters for these neurons and implying their synchronous firing [3, 4]. However, simultaneous recording of PD neurons shows differences between the timings of spikes of these neurons. This may indicate functional variability of these neurons. Here we modelled separately the two PD neurons of the STG in a multi-neuron model of the pyloric network. Our neuron models comply with known correlations between conductance parameters of ionic currents. Our results reproduce the experimental finding of increasing spike time distance between spikes originating from the two model PD neurons during their synchronised burst phase. The PD neuron with the larger calcium conductance generates its spikes before the other PD neuron. Larger potassium conductance values in the follower neuron imply longer delays between spikes, see Fig. 17.Neuromodulators change the conductance parameters of neurons and maintain the ratios of these parameters [5]. Our results show that such changes may shift the individual contribution of two PD neurons to the PD-phase of the pyloric rhythm altering their functionality within this rhythm. Our work paves the way towards an accessible experimental and computational framework for the analysis of the mechanisms and impact of functional variability of neurons within the neural circuits to which they belong

Characterizing an ensemble of interacting oscillators:the mean-field variability index

We introduce a way of characterizing an ensemble of interacting oscillators in terms of their mean-field variability index kappa, a dimensionless parameter defined as the variance of the oscillators' mean field r divided by the mean square of r. Based on the assumption that the overall mean field is the sum of a very large number of oscillators, each giving a small contribution to the total signal, we show that kappa depends on the mutual interactions between the oscillators, independently of their number or spectral properties. For purely random phasors, or a noninteracting ensemble of oscillators, kappa converges on 0.215. Interactions push kappa in different directions: lower where there is interoscillator phase coherence, tending to zero for complete phase synchronization, or higher for amplitude synchronization or intermittent synchronization. We calculate kappa for several different cases to illustrate its utility, using both numerically simulated data and electroencephalograph signals from the brains of human subjects while awake, while anesthetized, and while undergoing an epileptic fit. DOI: 10.1103/PhysRevE.87.01290

Kuramoto model with time-varying parameters

We analyze the Kuramoto model generalized by explicit consideration of deterministically time-varying parameters. The oscillators' natural frequencies and/or couplings are influenced by external forces with constant or distributed strengths. A dynamics of the collective rhythms is observed, consisting of the external system superimposed on the autonomous one, a characteristic feature of many thermodynamically open systems. This deterministic, stable, continuously time-dependent, collective behavior is fully described, and the external impact to the original system is defined in both the adiabatic and the nonadiabatic limits

Mean-field and mean-ensemble frequencies of a system of coupled oscillators

We investigate interacting phase oscillators whose mean field is at a different frequency from the mean or mode of their natural frequencies. The associated asymmetries lead to a macroscopic traveling wave. We show that the mean-ensemble frequency of such systems differs from their entrainment frequency. In some scenarios these frequencies take values that, counterintuitively, lie beyond the limits of the natural frequencies. The results indicate that a clear distinction should be drawn between the two variables describing the macroscopic dynamics of cooperative systems. This has important implications for real systems where a nontrivial distribution of parameters is common