Inferring the connectivity of coupled oscillators from event timing analysis

Understanding the coupling structure of interacting systems is an important open problem, and many methods have been proposed to reconstruct a network from observed data. Most require continuous observation of the nodes’ dynamics; however, in many situations, we can only monitor the times when some events occur (e.g., in neural systems, spike times). Here, we propose a method for network reconstruction based on the analysis of event times at the network's nodes. First, from the event times, we generate phase time series. Then, we assimilate the phase time series to the Kuramoto model by using the unscented Kalman filter (UKF) that returns the inferred coupling coefficients. Finally, we use a clustering algorithm to discriminate the coupling coefficients into two groups that we associate with existing and non-existing links. We demonstrate the method with synthetic data from small networks of Izhikevich neurons, where we analyze the spike times, and with experimental data from a larger network of chaotic electronic circuits, where the events are voltage threshold-crossings. We also compare the UKF with the performance of the cross-correlation (CC), and the mutual information (MI). We show that, for neural network reconstruction, UKF often outperforms CC and MI, while for electronic network reconstruction, UKF shows similar performance to MI, and both methods outperform CC. Altogether, our results suggest that when event times are the only information available, the UKF can give a good reconstruction of small networks. However, as the network size increases, the method becomes computationally demanding.