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\subsection{Inferring coupling function from synthetic data}  \label{sec:Inferring_coupling_function_from_synthetic_data}  In order to test our coupling function inference method we simulated phases from our stochastic model (Section \ref{sec:stochasticModel}) using unidirectional coupling functions. We randomly generated the coupling functions such that the synchronization index $R(\theta-\phi)$ was close to $0.6$, setting the system in a synchronized regime similar to the one observed in the data. We generated 800 traces with a duration of 80 hours by using the measured waveforms (Eq. \ref{eqn:circ_HMM4} and \ref{eqn:cellcylce_HMM3}, Methods).  Next we inferred the phases from the simulated data using our HMM's (Section \ref{sec:HMM_methods}) and reconstructed the coupling functions. The resulting functions are shown in Figure \ref{fig:example_of_reconstructed_coupling_functions}, \ref{fig:example_of_reconstructed_coupling_functions2} and \ref{fig:example_of_reconstructed_coupling_functions3}.