Modeling experimental time series with ordinary differential equations

Recently some methods have been presented to extract ordinary differential equations (ODE) directly from an experimental time series. Here, we introduce a new method to find an ODE which models both the short time and the long time dynamics. The experimental data are represented in a state space and the corresponding flow vectors are approximated by polynomials of the state vector components. We apply these methods both to simulated data and experimental data from human limb movements, which like many other biological systems can exhibit limit cycle dynamics. In systems with only one oscillator there is excellent agreement between the limit cycling displayed by the experimental system and the reconstructed model, even if the data are very noisy. Furthermore, we study systems of two coupled limit cycle oscillators. There, a reconstruction was only successful for data with a sufficiently long transient trajectory and relatively low noise level.