III. FUNDAMENTOS 9
1 C OMPUTACIÓN E VOLUTIVA 10
2.3 Metodología de Identificación de Sistemas 50
The Ott-Antonsen ansatz strongly boosted the analysis of Kuramoto models. Net- works are assumed to consist of a continuum of oscillators, whose long-term dynam- ical behavior can be derived in the thermodynamic limit. Of particular interest for this paper is the extension to multiple coupled networks. A simple change of variables may transform two symmetrically coupled networks (with oscillators whose natural frequencies follow a unimodal distribution each) into one global network where the natural frequencies are drawn from a symmetric bimodal dis- tribution. When assuming that internal and external coupling strengths in the two-population case differ, this transformation breaks down, and one is left with an additional degree of freedom. As we have proven in this paper, the additional parameter does not lead to new bifurcations but leaves both systems topologically equivalent. Stability, dynamics, and bifurcations of a symmetric two population system of phase oscillators are equivalent to a single population with a bimodal frequency distribution. This topological equivalence can also be shown when in- troducing small symmetric time-delays that allows for a phase-lag parameter re- duction.
In the second part we aimed for generalizing the equivalence between multi- modal and multiple coupled networks. However, already for the case of three subpopulations, where we adapted the same symmetry assumptions as in the two- population/bimodal case, this equivalence does no longer hold. Our symmetry assumptions are admittedly restrictive. Above all they only represent a slice of possible network configurations. That is, we cannot claim that the dynamics dis- cussed here should be considered generic or not. However, our example clearly shows that the symmetric bidirectional coupling topolgy (cf.K1,2 in Fig. 4.7) does not admit its dynamics to be described by a single network of oscillators whose natural frequencies follow a symmetric trimodal distribution. A detailed analysis in the presence of asymmetries in both the two-population/bimodal approach and the multiple populations/multimodal networks is beyond the scope of the present paper but will be published elsewhere285.
Throughout the paper we based our work on the original Kuramoto model, a net- work of phase oscillators that are all-to-all coupled through the sine of the pairwise phase differences. Coupling two of such networks leads to new long-term behavior such as partially-synchronized states, so-called chimeras in the case of identical oscillators see, e.g., 286. Also, multistable regimes and oscillatory solutions are pos- sible. For sure, non-local coupling, the introduction of phase-lag parameters as in265,266,284, or of more general time-delayssee, e.g., 287, would have further enriched the dynamics. Recently, Martens, Bick and Panaggio investigated how the intro-
duction of heterogeneous phase-lags in our two-population-scenario ofSection 4.3 shapes the dynamics. The additional control parameters were internal versus ex- ternal phase-lag parameters next to (internal and external) coupling strengths and the intrinsic frequency ω. Assuming only homogeneous oscillators in both popu- lations renders the OA ansatz not applicable in a rigorous way. However, it has been argued that in the limit of zero width of the frequency distribution, ∆→0, the assumption of “nearly identical” oscillators enabled the authors to analyze the system analytically288. Interestingly, they found chaotic attractors and reso- nance effects, which shows again the variety of dynamics of a mere two-population system, and highlights the importance to really understand their behavior.
In our two-population/bimodal scenarios the governing dynamics could be re- duced to be effectively two-dimensional. Hence, they cannot exhibit chaos. On the other hand, in the three-population/trimodal network chaotic trajectories should be possible. Though our focus mainly lay on (disproving) the equivalence between the different approaches, a full picture should also take chaos in both systems into account by assessing maximal Lyapunov exponents289 see also 284,288.
Away from the symmetry assumptions considered throughout this work, but also when dealing with non-local coupling, phase-lag parameters, general time-delay or even finite-sized networks, i.e. in particular when the OA ansatz can no longer be applied, topological equivalences, or even (weaker) correspondences between multimodal and multiple coupled networks have to be demonstrated in order to show that coupled networks and networks with multimodal frequency distributions are equivalent, indeed. The analytic tractability of the Ott-Antonsen ansatz helped us to rigorously prove first results about similarities and differences between these two approaches. We believe that, despite the limited range of application of such models, our findings can be assumed seminal for a broader variety of models, and therefore will further enlighten the view on an accurate interchangeability of the notions of multimodal networks and coupled unimodal networks, which in the end will increase the flexibility to derive and specify models in diverse fields of applications.
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5
Parameter-dependent oscillatory
systems
The Ott-Antonsen (OA) ansatz [Chaos 18, 037113 (2008), Chaos 19, 023117 (2009)] has been widely used to describe large systems of coupled phase oscilla- tors. 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 re- main 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 neu- rons, 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.
Adapted from: Pietras B., Daffertshofer A. (2016). Ott-Antonsen attractiveness for parameter-dependent oscillatory networks. Chaos 26, 103101.
5.1 Collective dynamics and parameter dependence
Coupled phase oscillators are being widely used to describe synchronization phe- nomena. The study of their collective dynamics has experienced a major break- through by the results by Ott and Antonsen81–83. The asymptotic behavior of the mean field of infinitely many coupled oscillators can be cast into a reduced, low-dimensional system of ordinary differential equations. The evolution is hence captured by the so-called Ott-Antonsen (OA) manifold.
Very recently, the OA ansatz has been applied to networks of theta neurons see, e.g., 290–296. 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-workers297, Wagemaker and co- workers298, and So and Barreto299. 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 Montbri´o and co-workers293 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[1]. 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 net- works. First, we address non-autonomous systems and show that our proof can be applied to time-varying parameters. An important example here is a biologically [1]Parameter-dependent systems comprise a wide class of systems, from which we here only choose a single family. This family represents a rather weak parameter-dependent system. However, we refrain from this notion since weak parameter dependence would imply that parameter changes have little to no considerable effect. Here, the original proof by Ott and Antonsen has to be changed, such that the parameter effect can be strong. We use the attribute “weak” to highlight that a specific oscillator does not depend on the additional parameter but its mean field dynamics only.
realistic approach to oscillatory systems proposed by Winfree85. Second, we in- clude multiple distributed parameters illustrated by coupled limit-cycle oscillators with shear. Third, we apply our proof to networks with different coupling topolo- gies including non-local coupling by using an heterogeneous mean field approach.