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\section{Estimating coupling functions from phases}  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 $(\theta,\phi)$ plane from the time derivative of the phases \cite{Rosenblum2001}.   The velocities are computed by taking the finite difference of the phases $v_{\theta}(\phi,\theta) = (\theta(t+\Delta t)-\theta(t))/\Delta t$, with $\Delta t$ begin equal to $0.5h$. 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.  In total we selected $2753$ time traces with a minimum duration of $24h$ and at least two divisions. These gave $208'762$ velocities for each function. The data density (number of data points per discrete position in the phase plane) is shown in Figure \ref{fig:dataDensity}. An example of reconstructed functions from synthetic data is also shown in Figure \ref{fig:example_of_reconstructed_coupling_functions}./