Reconstruction of stochastic nonlinear dynamical models from trajectory measurements

Average beta burst duration profiles provide a signature of dynamical changes between the ON and OFF medication states in Parkinson's disease

Parkinson's disease motor symptoms are associated with an increase in subthalamic nucleus beta band oscillatory power. However, these oscillations are phasic, and there is a growing body of evidence suggesting that beta burst duration may be of critical importance to motor symptoms. This makes insights into the dynamics of beta bursting generation valuable, in particular to refine closed-loop deep brain stimulation in Parkinson's disease. In this study, we ask the question "Can average burst duration reveal how dynamics change between the ON and OFF medication states?". Our analysis of local field potentials from the subthalamic nucleus demonstrates using linear surrogates that the system generating beta oscillations is more likely to act in a non-linear regime OFF medication and that the change in a non-linearity measure is correlated with motor impairment. In addition, we pinpoint the simplest dynamical changes that could be responsible for changes in the temporal patterning of beta oscillations between medication states by fitting to data biologically inspired models, and simpler beta envelope models. Finally, we show that the non-linearity can be directly extracted from average burst duration profiles under the assumption of constant noise in envelope models. This reveals that average burst duration profiles provide a window into burst dynamics, which may underlie the success of burst duration as a biomarker. In summary, we demonstrate a relationship between average burst duration profiles, dynamics of the system generating beta oscillations, and motor impairment, which puts us in a better position to understand the pathology and improve therapies such as deep brain stimulation.

Detecting network structures from measurable data produced by dynamics with hidden variables

Depicting network structures from measurable data is of significance. In real-world situations, it is common that some variables of networks are unavailable or even unknown. These unavailable and unknown variables, i.e., hidden variables, will lead to much reconstruction error, even make reconstruction methods useless. In this paper, to solve hidden variable problems, we propose three reconstruction methods, respectively, based on the following conditions: statistical characteristics of hidden variables, linearizable hidden variables, and white noise injection. Among them, the method based on white noise injection is active and invasive. In our framework, theoretic analyses of these three methods are given at first, and, furthermore, the validity of theoretical derivations and the robustness of these methods are fully verified through numerical results. Our work may be, therefore, helpful for practical experiments.

Volterra-series approach to stochastic nonlinear dynamics: The Duffing oscillator driven by white noise

The Duffing oscillator is a paradigm of bistable oscillatory motion in physics, engineering, and biology. Time series of such oscillations are often observed experimentally in a nonlinear system excited by a spontaneously fluctuating force. One is then interested in estimating effective parameter values of the stochastic Duffing model from these observations-a task that has not yielded to simple means of analysis. To this end we derive theoretical formulas for the statistics of the Duffing oscillator's time series. Expanding on our analytical results, we introduce methods of statistical inference for the parameter values of the stochastic Duffing model. By applying our method to time series from stochastic simulations, we accurately reconstruct the underlying Duffing oscillator. This approach is quite straightforward-similar techniques are used with linear Langevin models-and can be applied to time series of bistable oscillations that are frequently observed in experiments.

Joint reconstruction and prediction\break of random dynamical systems under\break borrowing of strength

We propose a Bayesian nonparametric model based on Markov Chain Monte Carlo methods for the joint reconstruction and prediction of discrete time stochastic dynamical systems based on m-multiple time-series data, perturbed by additive dynamical noise. We introduce the Pairwise Dependent Geometric Stick-Breaking Reconstruction (PD-GSBR) model, which relies on the construction of an m-variate nonparametric prior over the space of densities supported over R^m. We are focusing on the case where at least one of the time-series has a sufficiently large sample size representation for an independent and accurate Geometric Stick-Breaking estimation, as defined in Merkatas et al. [Chaos 27, 063116 (2017)]. Our contention is that whenever the dynamical error processes perturbing the underlying dynamical systems share common characteristics, underrepresented data sets can benefit in terms of model estimation accuracy. The PD-GSBR estimation and prediction procedure is demonstrated specifically in the case of maps with polynomial nonlinearities of an arbitrary degree. Simulations based on synthetic time-series are presented.

A Bayesian nonparametric approach to reconstruction and prediction of random dynamical systems

We propose a Bayesian nonparametric mixture model for the reconstruction and prediction from observed time series data, of discretized stochastic dynamical systems, based on Markov Chain Monte Carlo methods. Our results can be used by researchers in physical modeling interested in a fast and accurate estimation of low dimensional stochastic models when the size of the observed time series is small and the noise process (perhaps) is non-Gaussian. The inference procedure is demonstrated specifically in the case of polynomial maps of an arbitrary degree and when a Geometric Stick Breaking mixture process prior over the space of densities, is applied to the additive errors. Our method is parsimonious compared to Bayesian nonparametric techniques based on Dirichlet process mixtures, flexible and general. Simulations based on synthetic time series are presented.

Reconstruction of noise-driven nonlinear networks from node outputs by using high-order correlations

Many practical systems can be described by dynamic networks, for which modern technique can measure their outputs, and accumulate extremely rich data. Nevertheless, the network structures producing these data are often deeply hidden in the data. The problem of inferring network structures by analyzing the available data, turns to be of great significance. On one hand, networks are often driven by various unknown facts, such as noises. On the other hand, network structures of practical systems are commonly nonlinear, and different nonlinearities can provide rich dynamic features and meaningful functions of realistic networks. Although many works have considered each fact in studying network reconstructions, much less papers have been found to systematically treat both difficulties together. Here we propose to use high-order correlation computations (HOCC) to treat nonlinear dynamics; use two-time correlations to decorrelate effects of network dynamics and noise driving; and use suitable basis and correlator vectors to unifiedly infer all dynamic nonlinearities, topological interaction links and noise statistical structures. All the above theoretical frameworks are constructed in a closed form and numerical simulations fully verify the validity of theoretical predictions.

Dynamical inference: where phase synchronization and generalized synchronization meet

Synchronization is a widespread phenomenon that occurs among interacting oscillatory systems. It facilitates their temporal coordination and can lead to the emergence of spontaneous order. The detection of synchronization from the time series of such systems is of great importance for the understanding and prediction of their dynamics, and several methods for doing so have been introduced. However, the common case where the interacting systems have time-variable characteristic frequencies and coupling parameters, and may also be subject to continuous external perturbation and noise, still presents a major challenge. Here we apply recent developments in dynamical Bayesian inference to tackle these problems. In particular, we discuss how to detect phase slips and the existence of deterministic coupling from measured data, and we unify the concepts of phase synchronization and general synchronization. Starting from phase or state observables, we present methods for the detection of both phase and generalized synchronization. The consistency and equivalence of phase and generalized synchronization are further demonstrated, by the analysis of time series from analog electronic simulations of coupled nonautonomous van der Pol oscillators. We demonstrate that the detection methods work equally well on numerically simulated chaotic systems. In all the cases considered, we show that dynamical Bayesian inference can clearly identify noise-induced phase slips and distinguish coherence from intrinsic coupling-induced synchronization.

Inferential framework for nonstationary dynamics. I. Theory

A general Bayesian framework is introduced for the inference of time-varying parameters in nonstationary, nonlinear, stochastic dynamical systems. Its convergence is discussed. The performance of the method is analyzed in the context of detecting signaling in a system of neurons modeled as FitzHugh-Nagumo (FHN) oscillators. It is assumed that only fast action potentials for each oscillator mixed by an unknown measurement matrix can be detected. It is shown that the proposed approach is able to reconstruct unmeasured (hidden) variables of the FHN oscillators, to determine the model parameters, to detect stepwise changes of control parameters for each oscillator, and to follow continuous evolution of the control parameters in the adiabatic limit.

Nonlinear statistical modeling and model discovery for cardiorespiratory data

We present a Bayesian dynamical inference method for characterizing cardiorespiratory (CR) dynamics in humans by inverse modeling from blood pressure time-series data. The technique is applicable to a broad range of stochastic dynamical models and can be implemented without severe computational demands. A simple nonlinear dynamical model is found that describes a measured blood pressure time series in the primary frequency band of the CR dynamics. The accuracy of the method is investigated using model-generated data with parameters close to the parameters inferred in the experiment. The connection of the inferred model to a well-known beat-to-beat model of the baroreflex is discussed.