The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation, as well as predicting novel network states such as chimeras. However, there are many instances where this theory is expected to break down, say in the presence of strong coupling, or must be carefully interpreted, as in the presence of stochastic forcing. There are also surprises in the dynamical complexity of the attractors that can robustly appear—for example, heteroclinic network attractors. In this review we present a set of mathematical tools that are suitable for addressing the dynamics of oscillatory neural networks, broadening from a standard phase oscillator perspective to provide a practical framework for further successful applications of mathematics to understanding network dynamics in neuroscience.

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Mathematical Frameworks for Oscillatory Network Dynamics in Neuroscience

We study synchronization properties of general uncoupled limit-cycle oscillators driven by common and independent Gaussian white noises. Using phase reduction and averaging methods, we analytically derive the stationary distribution of the phase difference between oscillators for weak noise intensity. We demonstrate that in addition to synchronization, clustering, or more generally coherence, always results from arbitrary initial conditions, irrespective of the details of the oscillators.

/pdf/noise-induced-synchronization-and-clustering-in-ensembles-of-1x5y430w49.pdf

Noise-induced synchronization and clustering in ensembles of uncoupled limit-cycle oscillators.

This paper aims to investigate Gaussian colored-noise-induced stochastic bifurcations and the dynamical influence of correlation time and noise intensity in a bistable Duffing-Van der Pol oscillator. By using the stochastic averaging method, theoretically, one can obtain the stationary probability density function of amplitude for the Duffing-Van der Pol oscillator and can reveal interesting dynamics under the influence of Gaussian colored noise. Stochastic bifurcations are discussed through a qualitative change of the stationary probability distribution, which indicates that system parameters, noise intensity, and noise correlation time, respectively, can be treated as bifurcation parameters. They also imply that the effects of multiplicative noise are rather different from that of additive noise. The results of direct numerical simulation verify the effectiveness of the theoretical analysis. Moreover, the largest Lyapunov exponent calculations indicate that P and D bifurcations have no direct connection.

https://link.aps.org/accepted/10.1103/PhysRevE.83.056215

Stochastic bifurcations in a bistable Duffing-Van der Pol oscillator with colored noise.

We formulate a phase-reduction method for a general class of noisy limit cycle oscillators and find that the phase equation is parametrized by the ratio between time scales of the noise correlation and amplitude relaxation of the limit cycle. The equation naturally includes previously proposed and mutually exclusive phase equations as special cases. The validity of the theory is numerically confirmed. Using the method, we reveal how noise and its correlation time affect limit cycle oscillations.

http://www.k.mei.titech.ac.jp/members/nakao/papers/PhysRevLett_102_194102.pdf

Stochastic phase reduction for a general class of noisy limit cycle oscillators.

We point out that for an oscillator subjected to noise the conventional phase equation is not a proper approximation even for weak noise We present a phase reduction method valid for an oscillator subjected to weak white Gaussian noise Numerical evidence demonstrates that the phase equation properly approximates dynamics of the original oscillator Moreover, we show that, in general, noise causes a shift of the oscillator frequency and discuss its effects on entrainment

Phase reduction of stochastic limit cycle oscillators.

We consider the effect of external noise on the dynamics of limit cycle oscillators. The Lyapunov exponent becomes negative under influence of small white noise, what means synchronization of two or more identical systems subject to common noise. We analytically study the effect of small nonidentities in the oscillators and in the noise, and derive statistical characteristics of deviations from the perfect synchrony. Large white noise can lead to desynchronization of oscillators, provided they are nonisochronous. This is demonstrated for the Van der Pol--Duffing system.

/pdf/synchronization-and-desynchronization-of-self-sustained-3ciqwyvg9w.pdf

Arkadi S. Pikovsky

Papers

Synchronization and desynchronization of self-sustained oscillators by common noise.