Collective canard explosions of globally-coupled rotators with adaptive coupling

TL;DR

Problem: understanding how canard phenomena can emerge at the collective level in large populations of globally-coupled rotators with adaptive coupling. Approach: derive a 3D slow-fast mean-field description via Ott-Antonsen for a bimodal frequency distribution, analyze the 1D critical manifold and canard trajectories, and validate with network simulations across system sizes. Findings: the system exhibits collective canard explosions and bursting organized by the slow manifold, with irregular canards near the fold and finite-size effects depending on frequency sampling. Significance: reveals emergent macroscopic slow-fast dynamics not present in single rotators, with potential applicability to neuroscience and to Kuramoto systems with inertia or other hysteretic transitions.

Abstract

Canards, special trajectories that follow invariant repelling slow manifolds for long time intervals, have been frequently observed in slow-fast systems of either biological, chemical and physical nature. Here, collective canard explosions are demonstrated in a population of globally-coupled phase-rotators subject to adaptive coupling. In particular, we consider a bimodal Kuramoto model displaying coexistence of asynchronous and partially synchronized dynamics subject to a linear global feedback. A detailed geometric singular perturbation analysis of the associated mean-field model allows us to explain the emergence of collective canards in terms of the stability properties of the one-dimensional critical manifold, near which the slow macroscopic dynamics takes place. We finally show how collective canards and related manifolds gradually emerge in the globally-coupled system for increasing system sizes, in spite of the trivial dynamics of the uncoupled rotators.

Figures (6)

• Figure 1: Time-series of $R$ for the network model (\ref{['network']}) (black) and the mean-field model (\ref{['mf']}) (red). In each panel from (a) to (e) the control parameter $K$ used in both models is the same: (a) $K=$9.044, (b) $K=$9.017137, (c) $K=$9.017, (d) $K=$8.9, (e) $K=$8.5. In panels (f)-(j) we plot the same network time-traces as in (a)-(e), while the control parameter in the mean-field model has been adjusted to reproduce the dynamics of the network: (f) $K=$9.06, (g) $K=$9.04441, (h) $K=$9.044288, (i) $K=$8.925, (j) $K=$8.505. Other parameters: $\epsilon=$0.02, $\Delta_0$=1.4 $\omega_0$=1.8, $\alpha$=7. For the network we use a population of $N=5 \times 10^5$ rotators and a randomly-generated bimodal Lorentzian distribution.
• Figure 2: Hopf and canard cycles of the network model (\ref{['network']}) for (a) $\epsilon=$0.02 and (b) $\epsilon=$0.07. The limit cycles are illustrated in the ($S$,$R$) plane together with the critical manifold (\ref{['smr']}, whose attracting (repelling) branches are plotted as black solid (dashed) lines. (a) Hopf cycle $K=$9.017137 (green) and canard cycles $K=$9.0165 (red), $K=$9.017 (blue), $K=$9.0171 (magenta). (b) Hopf cycle $K=8.98$ (green) and canard cycles $K=$8.97 (red), $K=$8.9725 (blue), $K=$8.973. Other parameters as in Fig. \ref{['figure1']}.
• Figure 3: Irregular canard cycles (red traces) for (a) the network model (\ref{['network']}) ($K=9.017134$) and (b,c) the mean-field model (\ref{['mf']}): (b) $K=$9.0444014469 (c) $K=$9.044401447113997. The cycles are illustrated in the ($S$,$R$) plane together with the critical manifold (\ref{['smr']}, whose attracting (repelling) branches are plotted as black solid (dashed) lines. Other parameters as in Fig. \ref{['figure1']}.
• Figure 4: (a) Time-dependent behavior of $R$ and $\lambda^t$. (b) Running average of the maximal Lyapunov exponent $\lambda_M$ vs time. In the inset is shown an enlargement of the time-dependent behavior for long simulation times. Parameters: time step $\Delta t=0.001$, transient time $t_{tr}=2000$, simulation time $t_s=5 \times 10^7$, $K=9.044401447113997$, $\epsilon=0.02$. Other parameters as in Fig. \ref{['figure1']}.
• Figure 5: Time-series of $R$ for the network model (\ref{['network']}) for $N=1000$ rotators and different sets of their natural frequencies. In (a-d) the frequencies are generated in a random way while in (e) are generated deterministically accordingly to the rule (\ref{['gend']}). In panel (e) we also show the corresponding mean-field prediction (blue solid trace). All parameters are fixed: $\epsilon=$0.02, $\Delta_0=$1.4 $\omega_0=$1.8, $\alpha$=7, $K=$8.955. The initial conditions are $\rho=$0.01,$\phi$=0, $S=$K.