Dynamics of Delayed Neuroendocrine Systems and Their Reconstructions Using Sparse Identification and Reservoir Computing

Abstract

Neuroendocrine system mainly consists of hypothalamus, anterior pituitary, and target organ. In this paper, a three-state-variable delayed Goodwin model with two Hill functions is considered, where the Hill functions with delays denote the hormonal feedback suppressions from target organ to hypothalamus and to anterior in the reproductive hormonal axis. The existence of Hopf bifurcation shows the circadian rhythms of neuroendocrine system. The direction and stability of Hopf bifurcation are also analyzed using the normal form theory and the center manifold theorem for functional differential equations. Furthermore, based on the sparse identification algorithm, it is verified that the transient time series generated from the delayed Goodwin model cannot be equivalently presented by ordinary differential equations from the viewpoint of data when considering that a library of candidates are at most cubic terms. The reason is because the solution space of delayed differential equations is of infinite dimensions. Finally, we report that reservoir computing can predict the periodic behaviors of the delayed Goodwin model accurately if the size of reservoir and the length of data used for training are large enough. The predicting performances are evaluated by the mean squared errors between the trajectories generated from the numerical simulations and the reservoir computing.