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Identifying nonlinear dynamical systems via generative recurrent neural networks with applications to fMRI

Fig 4

Evaluation of training protocol and KL measure on dynamical systems benchmarks.

A. True trajectory from chaotic Lorenz attractor (with parameters s = 10, r = 28, b = 8/3). B. Distribution of (Eq 9) across all samples, binned at .05, for PLRNN-SSM (black) and LDS-SSM (red). For the PLRNN-SSM, around 26% of these samples (grey shaded area, pooled across different numbers of latent states M) captured the butterfly structure of the Lorenz attractor well (see also D). Unsurprisingly, the LDS completely failed to reconstruct the Lorenz attractor. C. Estimated Lyapunov exponents for reconstructed Lorenz systems for PLRNN-SSM (black) and LDS-SSM (red) (estimated exponent for true Lorenz system ≈.9, cyan line). A significant positive correlation between the absolute deviation in Lyapunov exponents for true and reconstructed systems with (r = .27, p < .001) further supports that the latter measures salient aspects of the nonlinear dynamics in the PLRNN-SSM (for the LDS-SSM, all of these empirically determined Lyapunov exponents were either < 0, as indicative of convergence to a fixed point, or at least very close to 0, light-gray line). D. Samples of PLRNN-generated trajectories for different values. The grey shaded area indicates successful estimates. E. True van der Pol system trajectories (with μ = 2 and ω = 1). F. Same as in B but for van der Pol system. G. Correlation of the spectral density between true and reconstructed van der Pol systems for the PLRNN-SSM (black) and LDS-SSM (red). A significant negative correlation for the PLRNN-SSM between the agreement in the power spectrum (high values on y-axis) and again supports that the normalized KL divergence defined across state space (Eq 9) captures the dynamics (we note that measuring the correlation between power spectra comes with its own problems, however). For the LDS-SSM, in contrast, all power-spectra correlations and measures were poor. H. Same as in D for van der Pol system. Note that even reconstructed systems with high values may capture the limit cycle behavior and thus the basic topological structure of the underlying true system (in general, the 2-dimensional vdP system is likely easier to reconstruct than the chaotic Lorenz system; vice versa, low values do not ascertain that the reconstructed system exhibits the same frequencies).

doi: https://doi.org/10.1371/journal.pcbi.1007263.g004