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In the last decades, mathematical modeling and signal processing techniques have played an important role in the study of cardiovascular control physiology and heartbeat nonlinear dynamics. In particular, nonlinear models have been devised for the assessment of the cardiovascular system by accounting for short-memory second-order nonlinearities. In this paper, we introduce a novel inverse Gaussian point process model with Laguerre expansion of the nonlinear Volterra kernels. Within the model, the second-order nonlinearities also account for the long-term information given by the past events of the nonstationary non-Gaussian time series. In addition, the mathematical link to an equivalent cubic input-output Wiener-Volterra model allows for a novel instantaneous estimation of the dynamic spectrum, bispectrum and trispectrum of the considered inter-event intervals. The proposed framework is tested with synthetic simulations and two experimental heartbeat interval datasets. Applications on further heterogeneous datasets such as milling inserts, neural spikes, gait from short walks, and geyser geologic events are also reported. Results show that our model improves on previously developed models and, at the same time, it is able to provide a novel instantaneous characterization and tracking of the inherent nonlinearity of heartbeat dynamics.