Methods of data assimilation are established in physical sciences and engineering for the merging of observed data with dynamical models. When the model is nonlinear, methods such as the ensemble Kalman filter have been developed for this purpose. At the other end of the spectrum, when a model is not known, the delay coordinate method introduced by Takens has been used to reconstruct nonlinear dynamics. In this article, we merge these two important lines of research. A model-free filter is introduced based on the filtering equations of Kalman and the data-driven modeling of Takens. This procedure replaces the model with dynamics reconstructed from delay coordinates, while using the Kalman update formulation to reconcile new observations. We find that this combination of approaches results in comparable efficiency to parametric methods in identifying underlying dynamics, and may actually be superior in cases of model error.

Scientific analysis often relies on the ability to make accurate predictions of a system’s dynamics. Mechanistic models, parameterized by a number of unknown parameters, are often used for this purpose. Accurate estimation of the model state and parameters prior to prediction is necessary, but may be complicated by issues such as noisy data and uncertainty in parameters and initial conditions. At the other end of the spectrum exist nonparametric methods, which rely solely on data to build their predictions. While these nonparametric methods do not require a model of the system, their performance is strongly influenced by the amount and noisiness of the data. In this article, we consider a hybrid approach to modeling and prediction which merges recent advancements in nonparametric analysis with standard parametric methods. The general idea is to replace a subset of a mechanistic model’s equations with their corresponding nonparametric representations, resulting in a hybrid modeling and prediction scheme. Overall, we find that this hybrid approach allows for more robust parameter estimation and improved short-term prediction in situations where there is a large uncertainty in model parameters. We demonstrate these advantages in the classical Lorenz-63 chaotic system and in networks of Hindmarsh-Rose neurons before application to experimentally collected structured population data.

A nonlinear data assimilation technique is applied to determine and track effective connections between ensembles of cultured spinal cord neurons measured with multielectrode arrays. The method is statistical, depending only on confidence intervals, and requiring no form of arbitrary thresholding. In addition, the method updates connection strengths sequentially, enabling real-time tracking of nonstationary networks. The ensemble Kalman filter is used with a generic spiking neuron model to estimate connection strengths as well as other system parameters to deal with model mismatch. The method is validated on noisy synthetic data from Hodgkin-Huxley model neurons before being used to find network connections in the neural culture recordings.

No abstract

Methods for forecasting time series are a critical aspect of the understanding and control of complex networks. When the model of the network is unknown, nonparametric methods for prediction have been developed, based on concepts of attractor reconstruction pioneered by Takens and others. In this Rapid Communication we consider how to make use of a subset of the system equations, if they are known, to improve the predictive capability of forecasting methods. A counterintuitive implication of the results is that knowledge of the evolution equation of even one variable, if known, can improve forecasting of all variables. The method is illustrated on data from the Lorenz attractor and from a small network with chaotic dynamics.