Reconstruction of Network Phase Dynamics from Data - Gebrezabher, Zeray Hagos (2023)
Many dynamical systems, both natural and man-made, are composed of interacting parts. Isolated dynamical systems such as the spiking of neurons, cardiac cells, and electrical circuits are periodic in nature. Mathematically, such periodic systems can be described by a limit cycle oscillator, which can be parameterized in terms of phases. Nowadays it is possible to collect and process enormous amounts of data from the units of many such interacting limit cycle oscillators. However, we do not have enough models of such systems to identify and parameterize the crucial features that must be incorporated into the model. The main objective of this thesis is to reconstruct models of dynamical systems from available time-series data. In this context, we considered the case where the data comes from a network of oscillatory units that interact weakly. To this end, we aim to reconstruct phase dynamics from time series in terms of phases. The phases can be estimated from each time series of such oscillatory systems. Theoretically, the phase reduction framework is discussed for the case of a weakly perturbed dynamical system with an exponentially stable limit cycle when unperturbed, where this was also extended to weakly interacting oscillatory systems, using the concept of isochrons. The influence that one dynamical system exerts on another is described by a coupling function, and the coupling functions extracted from the time series of interacting dynamical systems are often found to be time-varying. Motivated by the time-variability of biological interactions, including neural delta-alpha interac- tion functions which were reconstructed based on Bayesian inference, we studied the existence of synchronization transitions caused by time-varying coupling functions, even though the net coupling strength is invariant. We also studied the emergence of hypernetworks when recon- structing models of nonlinearly coupled oscillators from data. In particular, when the data comes from a network of weakly coupled Stuart-Landau oscillators, we showed that sparse recovery methods reveal hypernetworks. This result is verified theoretically using second-order phase reduction theory via the perturbation method.