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\subsection{Estimating coupling functions from phases}  The reconstruction of the coupling functions $F_1$ and $F_2$ was done in two step. First we inferred the phases $\theta(t)$ and $\phi(t)$from the 37\degree dataset  using our HMMs. HMM's.  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}. phases.  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 then 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.