Reconstructing the neuron-like oscillator equations modeled by a phase-locked system with delay from scalar time series

The purpose of this work is to develop the reconstruction technique for the neuron-like oscillator equations descibed by a phase-locked system model with delay from scalar time series. Methods. We reconstruct the state vector given a scalar series of only one variable corresponding to the transmembrane potential. The second variable is obtained by numerical differentiation with smoothing by a polynomial. The third variable is obtained by numerical integration using the Simpson method. Next, the target function describing the length of the nonlinear function at a trial delay time is constructed and minimized. Results. The delay time, effective system parameters, and nonlinear function can be reconstructed using the proposed method. The method gives correct results in various periodic and chaotic regimes, including intermittency. The method works even in presence of 1% measurement noise. Conclusion. The described method is useful as a tools for reconstructing neuron models from experimental data of extracellular or intracellular recordings from the brain or in culture.