Ott-Antonsen attractiveness for parameter-dependent oscillatory systems

Abstract

The Ott-Antonsen (OA) ansatz [Chaos 18, 037113 (2008), Chaos 19, 023117 (2009)] has been widely used to describe large systems of coupled phase oscillators. If the coupling is sinusoidal and if the phase dynamics does not depend on the specific oscillator, then the macroscopic behavior of the systems can be fully described by a low-dimensional dynamics. Does the corresponding manifold remain attractive when introducing an intrinsic dependence between an oscillator's phase and its dynamics by additional, oscillator specific parameters? To answer this we extended the OA ansatz and proved that parameter-dependent oscillatory systems converge to the OA manifold given certain conditions. Our proof confirms recent numerical findings that already hinted at this convergence. Furthermore we offer a thorough mathematical underpinning for networks of so-called theta neurons, where the OA ansatz has just been applied. In a final step we extend our proof by allowing for time-dependent and multi-dimensional parameters as well as for network topologies other than global coupling. This renders the OA ansatz an excellent starting point for the analysis of a broad class of realistic settings. Very recently, the OA ansatz has been applied to networks of theta neurons, see, e.g., Refs. 4-10. A particular property of coupled, inhomogeneous theta neurons is that both the phase of a single neuron as well as its dynamics depend on a parameter, which establishes an intrinsic relation between them. While numerical results suggest the attractiveness of the OA manifold in the presence of such a parameter dependence, it has as to yet not been proven whether the dynamics really converges to it. For a certain class of parameter dependencies we here extend the existing theory of the OA ansatz and show that the OA manifold continues to asymptotically attract the mean field dynamics.Parameter-dependent systems and their description through the OA ansatz have been considered by, e.g., Strogatz and co-workers 11 , Wagemaker and coworkers 12 , and So and Barreto 13 . There, parameters seemingly did not yield a correlation between an oscillator's phase and its dynamics but a rigorous proof for this is still missing. We explicitly address this last point. In particular,we prove a conjecture later formulated by Montbrió and co-workers 7 on the attractiveness of the OA manifold for parameter-dependent systems. The case of parameters serving as mere auxiliary variables readily follows from our result -we will refer to this as "weak" parameter-dependence 14 . By showing that a network of theta neurons can be treated as a parameter-dependent oscillatory system, our result establishes an immediate link to networks of quadratic integrate-and-fire (QIF) neurons: That is, the so-called Lorentzian ansatz as an equivalent approach to the OA ansatz is analytically substantiated. By this we may exert an important impact in mathematical neuroscience.Finally, we extend the parameter-dependence for more general classes of networks. First, we address non-autonomou...

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