ABSTRACT
The last decade showed an increased interest in Langevin equations for modeling time series recorded from complex dynamical systems. These equations allow to discriminate between deterministic (drift) and stochastic (diffusion) components of the recorded time series. In practice, the estimation of drift and diffusion is often based on approximations of the models' dynamics that are valid only for high sampling frequencies. Also, model assessment is not or only indirectly performed, potentially leading to false claims. In this study we compare the performance of an asymptotically unbiased estimation method with a generally used approximate method, demonstrating the necessity of using (asymptotically) unbiased estimators. Furthermore, we describe how confidence intervals for the unknown parameters can be constructed and how model assessment can be carried out. We apply the methodology to local field potentials recorded in vitro from mouse hippocampus from eight genetically different strains. The recorded field potentials turn out to be well described by linearly damped Langevin equations with parabolic diffusion. The modeling enables a dynamical interpretation of the spectral power of the field potentials. It reveals that observed spectral power differences in the field potentials across hippocampal regions are associated with differences in the deterministic component of the system, and it reveals transiently active current dipoles, which are not detectable by conventional methods. Also, all estimated parameters have significant heritabilities, which suggests that the Langevin equations capture biological relevant aspects of electrical hippocampal activity.
Subject(s)
Hippocampus/physiology , Models, Biological , Animals , Diffusion , Electrophysiological Phenomena , Linear Models , Mice , Time FactorsABSTRACT
We present a statistical method, and its accompanying algorithms, for the selection of a mathematical model of the gating mechanism of an ion channel and for the estimation of the parameters of this model. The method assumes a hidden Markov model that incorporates filtering, colored noise and state-dependent white excess noise for the recorded data. The model selection and parameter estimation are performed via a Bayesian approach using Markov chain Monte Carlo. The method is illustrated by its application to single-channel recordings of the K(+) outward-rectifier in barley leaf.
Subject(s)
Hordeum/metabolism , Models, Biological , Potassium Channels/metabolism , Algorithms , Biometry , Markov Chains , Mathematics , Models, Statistical , Monte Carlo Method , Plant Leaves/metabolismABSTRACT
OBJECTIVE: To determine the magnitude of the growth retardation in Dutch monozygotic and dizygotic twins during infancy in comparison with the Dutch reference growth charts for general population infants from 1997 and to construct reference growth charts for twins. DESIGN: Descriptive. METHOD: The growth of twins was studied using longitudinal data on over 4000 Dutch twin pairs from birth until 2.5 years of age. The LMS method was used to obtain growth charts for height, weight and body-mass index (BMI) for twin pairs during infancy. Centiles were estimated by the Box-Cox power curve (L), the median curve (M) and the coefficient of variation curve (S). RESULTS: From birth until the age of half a year, the average height and weight of twin pairs were at about the 10th percentile of the Dutch reference population. One year later this difference had decreased to about the 25th percentile, and when the twin pairs were between 1.5 and 2.5 years of age the difference was further decreased to the 35th percentile. The BMI deviated less from that of the reference population: during the first half a year the BMI of twin pairs was at about the 25th percentile. Subsequently, the BMI improved, but remained slightly below the median of the reference population at the age of about two years. Approximately half (50% for height, 58% for weight) of the growth retardation from birth until 1.5 years was attributable to gestational age. Between 1.5 years and 2.5 years of age, this difference was reduced to one third: 33% for both height and weight. Thus, a substantial part of the growth difference could not be explained by gestational age. CONCLUSION: Correcting for gestational age alone is not sufficient to make possible a comparison of the growth of twin pairs with the growth of general population infants. The development of twins can, however, be followed by means of the reference growth charts designed by the authors.
