Reconstruction of ensembles of nonlinear neurooscillators with sigmoid coupling function

Inferring information about interactions between oscillatory systems from their time series is a highly debated problem. However, many approaches for solving this problem consider either linear systems or linear couplings. We propose a method for the reconstruction of ensembles of nonlinearly coupled neurooscillators described by first-order nonlinear differential equations. The method is based on the minimization of a special target function for each oscillator in the ensemble separately. To find the solution of optimization problem the nonlinear least-squares routine is used. The method does not exploit any parameterization for approximation of nonlinear functions of individual nodes. In addition, an original two-step algorithm for the removal of spurious couplings is proposed based on the clusterization of coefficients of the reconstructed coupling functions and the analysis of their variation. The method efficiency is shown for periodic and chaotic vector time series for ensembles of different size that contain from 8 to 32 oscillators. These oscillators have a cubic nonlinearity and sigmoid is considered as a coupling function. The effect of measurement noise on the results of coupling architecture reconstruction is studied in detail and the method is shown to be effective for relatively high noise (signal to noise ratio equal to eight).