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\subsection{Reconstruction coupling functions}  The reconstruction of the coupling functions $F_1$ and $F_2$ was done in two step. First we infered the phases $\theta(t)$ and $\phi(t)$ from the 37 \degree dataset using our HMMs. Secondly we estimated the coupling functions from the phases by computing the instantaneous phase velocity in the (\th,\phi) $(\th,\phi)$  plane from the time derivative of the phases. The velocities are computed by taking the finite difference of the phases $v_{\theta}(\phi,\theta) = (\theta(t)-\theta(t+\Delta t))/\Delta t$. We estimated the functions $F1$ and $F_2$ by doing a least square fit of a 2D Fourier serie with 10 harmonics on the collection of velocities $\{v_{\theta}\}$ and $\{v_{\phi} \}$. Before doing so we verified that the distribution of velocities at every position in the phase plane was mainly unimodal. The functions were estimated on a $40$ by $40$ grid, given by the discrete hidden states of our HMMs.