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PubMed2003

Strategy and model building in the fourth dimension: a null model for genotype x age interaction as a Gaussian stationary stochastic process.

BACKGROUND: Using univariate and multivariate variance components linkage analysis methods, we studied possible genotype x age interaction in cardiovascular phenotypes related to the aging process from the Framingham Heart Study. RESULTS: We found evidence for genotype x age interaction for fasting glucose and systolic blood pressure. CONCLUSIONS: There is polygenic genotype x age interaction for fasting glucose and systolic blood pressure and quantitative trait locus x age interaction for a linkage signal for systolic blood pressure phenotypes located on chromosome 17 at 67 cM.

Adult↗
PubMed2001

Numerical studies of backscattering enhancement of electromagnetic waves from two-dimensional random rough surfaces with the forward-backward/novel spectral acceleration method.

The forward-backward method with a novel spectral acceleration algorithm (FB/NSA) has been shown to be a highly efficient O(Ntot) iterative method of moments, where Ntot is the total number of unknowns to be solved, for the computation of electromagnetic (EM) wave scattering from both one-dimensional and two-dimensional (2-D) rough surfaces. The efficiency of the method makes studies of backscattering enhancement from moderately rough impedance surfaces at large incident angles tractable. Variations in the characteristics of backscattering enhancement with incident angle, surface impedance, polarization, and surface statistics are investigated by use of the 2-D FB/NSA method combined with parallel computing techniques. The surfaces considered are Gaussian random processes with an isotropic Gaussian spectrum and root-mean-square surface heights and slopes ranging from 0.5 lambda to lambda and from 0.5 to 1.0, respectively, where lambda is the EM wavelength in free space. Incident angles ranging from normal incidence up to 70 degrees are considered in this study. It is found that backscattering enhancement depends strongly on all parameters of interest. America

Journal Article↗
PubMed2006

Apparent diffusion coefficients from high angular resolution diffusion imaging: estimation and applications.

High angular resolution diffusion imaging has recently been of great interest in characterizing non-Gaussian diffusion processes. One important goal is to obtain more accurate fits of the apparent diffusion processes in these non-Gaussian regions, thus overcoming the limitations of classical diffusion tensor imaging. This paper presents an extensive study of high-order models for apparent diffusion coefficient estimation and illustrates some of their applications. Using a meaningful modified spherical harmonics basis to capture the physical constraints of the problem, a new regularization algorithm is proposed. The new smoothing term is based on the Laplace-Beltrami operator and its closed form implementation is used in the fitting procedure. Next, the linear transformation between the coefficients of a spherical harmonic series of order l and independent elements of a rank-l high-order diffusion tensor is explicitly derived. This relation allows comparison of the state-of-the-art anisotropy measures computed from spherical harmonics and tensor coefficients. Published results are reproduced accurately and it is also possible to recover voxels with isotropic, single fiber anisotropic, and multiple fiber anisotropic diffusion. Validation is performed on apparent diffusion coefficients from synthetic data, from a biological phantom, and from a human brain dataset.

Algorithms↗
PubMed1992

A numerical method for electron transport calculations.

A numerical algorithm for calculating the penetration of electrons in dense media is presented. The numerical algorithm is intended for future application to radiotherapy dose calculations. The method is generic in the sense that it may be used with different theoretical models describing the angular scattering of electrons with depth. It is also general enough that it may be applied to electron dose calculations in heterogeneous as well as homogeneous media. The assumptions used in the algorithm are examined and equations describing the evolution of the distribution of electrons with depth are presented. Calculations have been performed for 10 MeV broad beams and pencil beams incident on water. It is shown that the Fermi-Eyges analytical solutions are recovered if the angular scattering process is assumed to be a Gaussian Markov process and the cumulative angle of electron travel remains small. In the case where the small angle approximation is not imposed, the numerical method qualitatively reproduces, at large depths, the wide angle scattering 'tails' seen in Monte Carlo generated profiles.

Algorithms↗
PubMed1996

Variability of extracellular spike waveforms of cortical neurons.

1. Here we study the variability in extracellular records of action potentials. Our work is motivated, in part, by the need to construct effective algorithms to classify single-unit waveforms from multiunit recordings. 2. We used microwire electrode pairs (stereotrodes) to record from primary somatosensory cortex of awake, behaving rat. Our data consist of continuous records of extracellular activity and segmented records of extracellular spikes. Spectral and principal component techniques are used to analyze mean single-unit wave-forms, the variability between different instances of a single-unit waveform, and the underlying background activity. 3. The spectrum of the variability between different instances of a single-unit waveforms is not white, and falls off above 1 kHz with a frequency dependence of roughly f-2. This spectrum is different from that of the mean spike waveforms, which falls off roughly as f-4, but is essentially identical with the spectrum of background activity. The spatial coherence of the variability on the 10-micron scale also falls off at high frequencies. 4. The variability between different instances of a single-unit waveform is dominated by a relatively small number of principal components. As a consequence, there is a large anisotropy in the cluster of the spike waveforms. 5. The background noise cannot be represented as a stationary Gaussian random process. In particular, we observed that the spectrum changes significantly between successive 20-ms intervals. Furthermore, the total power in the background activity exhibits larger fluctuations than is consistent with a stationary Gaussian random process. 6. Roughly half of the single-unit spike waveforms exhibit systematic changes as a function of the interspike interval. Although this results in a non-Gaussian distribution in the space of waveforms, the distribution can be modeled by a scalar function of the interspike interval. 7. We use a set of 44 mean single-unit waveforms to define the space of differences between spike waveforms. This characterization, together with that of the background activity, is used to construct a filter that optimizes the detection of differences between single-unit waveforms. Further, an information theoretic measure is defined that characterizes the detectability.

Action Potentials↗
PubMed2004

When is approximation by Gaussian networks necessarily a linear process?

Let s > or = 1 be an integer. A Gaussian network is a function on Rs of the form [Formula: see text]. The minimal separation among the centers, defined by (1/2) min(1 < or = j not = k < or = N) [Formula: see text], is an important characteristic of the network that determines the stability of interpolation by Gaussian networks, the degree of approximation by such networks, etc. Let (within this abstract only) the set of all Gaussian networks with minimal separation exceeding 1/m be denoted by Gm. We prove that for functions [Formula: see text] such that [Formula: see text], if the degree of L2(nonlinear) approximation of [Formula: see text] from Gm is [Formula: see text] then necessarily the degree of approximation of [Formula: see text] by (rectangular) partial sums of degree m2 of the Hermite expansion of [Formula: see text] is also [Formula: see text]. Moreover, Gaussian networks in Gm having fixed centers in a ball of radius [Formula: see text] and coefficients being linear functionals of [Formula: see text] can be constructed to yield the same degree of approximation. Similar results are proved for the Lp norms, 1 < or = p < or =[Formula: see text] but with the condition that the number of neurons N, should satisfy logN = [Formula: see text](m2).

Humans↗
PubMed2002

Persistence of a continuous stochastic process with discrete-time sampling: non-Markov processes.

We consider the problem of "discrete-time persistence," which deals with the zero crossings of a continuous stochastic process X(T) measured at discrete times T=nDeltaT. For a Gaussian stationary process the persistence (no crossing) probability decays as exp(-theta(D)T)=[rho(a)](n) for large n, where a=exp(-DeltaT/2) and the discrete persistence exponent theta(D) is given by theta(D)=(ln rho)/(2 ln a). Using the "independent interval approximation," we show how theta(D) varies with DeltaT for small DeltaT and conclude that experimental measurements of persistence for smooth processes, such as diffusion, are less sensitive to the effects of discrete sampling than measurements of a randomly accelerated particle or random walker. We extend the matrix method developed by us previously [Phys. Rev. E 64, 015101(R) (2001)] to determine rho(a) for a two-dimensional random walk and the one-dimensional random-acceleration problem. We also consider "alternating persistence," which corresponds to a<0, and calculate rho(a) for this case.

Journal Article↗
PubMed2001

Persistence of a continuous stochastic process with discrete-time sampling.

We introduce the concept of "discrete-time persistence," which deals with zero-crossings of a continuous stochastic process, X(T), measured at discrete times, T=n Delta T. For a Gaussian Markov process with relaxation rate mu, we show that the persistence (no crossing) probability decays as [rho(a)](n) for large n, where a = exp(-mu Delta T), and we compute rho(a) to high precision. We also define the concept of "alternating persistence," which corresponds to a<0. For a>1, corresponding to motion in an unstable potential (mu<0), there is a nonzero probability of having no zero-crossings in infinite time, and we show how to calculate it.

Journal Article↗
PubMed2001

The asymptotic distribution of canonical correlations and vectors in higher-order cointegrated models.

The study of the large-sample distribution of the canonical correlations and variates in cointegrated models is extended from the first-order autoregression model to autoregression of any (finite) order. The cointegrated process considered here is nonstationary in some dimensions and stationary in some other directions, but the first difference (the "error-correction form") is stationary. The asymptotic distribution of the canonical correlations between the first differences and the predictor variables as well as the corresponding canonical variables is obtained under the assumption that the process is Gaussian. The method of analysis is similar to that used for the first-order process.

Journal Article↗
PubMed2002

Does the EEG during isoflurane/alfentanil anesthesia differ from linear random data?

OBJECTIVE: Bispectral analysis of the electroencephalogram (EEG) has been used to monitor depth of anesthesia. In the majority of publications this has been done using the so called Bispectral (BIS) Index. The exact relation of this index to bispectral quantities like the bispectrum and its normalized version the bicoherence has not yet been published. In case the EEG is a linear random process the bicoherence is trivial. It is a mere constant independent of the EEG frequency. If the signal is a linear Gaussian random process this constant is zero. In this case both the bispectrum and bicoherence are zero. The aim of this study was to determine the proportion of EEG epochs with non-trivial bicoherence during anesthesia with isoflurane/nitrous oxide. METHODS: We reanalyzed 26.4 hr of EEG signal recorded in 8 patients during anesthesia for general abdominal surgery which were stored in digitized form on CD-Rom. The test developed by Hinich for Gaussianity and linearity was applied to these data. The test was validated with various kinds of surrogate data; especially the phase randomized (pr) EEG, synthetic Gaussian random data and the z-component of the Lorenz attractor and its pr version. RESULTS: The proportion of epochs for which a non-trivial bicoherence was detected by the test was as follows: Lorenz data 95%, pr Lorenz data 5%, synthetic Gaussian data 13.8%, pr EEG 5.4%, original EEG 6.2%. CONCLUSION: As expected the test procedure correctly identified for the Lorenz data for 95% of all epochs a non-trivial bicoherence. For the original EEG data we could not find a significant greater percentage of epochs with non-trivial bicoherence than for the pr data and the synthetic Gaussian data. We conclude that the EEG during anesthesia with isoflurane/alfentanil appears to be largely a linear random process.

Alfentanil↗
PubMed2005

Universal non-Gaussian velocity distribution in violent gravitational processes.

We study the velocity distribution in spherical collapses and cluster-pair collisions by use of N -body simulations. Reflecting the violent gravitational processes, the velocity distribution of the resultant quasistationary state generally becomes non-Gaussian. Through the strong mixing of the violent process, there appears a universal non-Gaussian velocity distribution, which is a democratic (equal-weighted) superposition of many Gaussian distributions (DT distribution). This is deeply related with the local virial equilibrium and the linear mass-temperature relation which characterize the system. We show the robustness of this distribution function against various initial conditions which leads to the violent gravitational process. The DT distribution has a positive correlation with the energy fluctuation of the system. On the other hand, the coherent motion such as the radial motion in the spherical collapse and the rotation with the angular momentum suppress the appearance of the DT distribution.

Journal Article↗
PubMed1999

Manifestations of slow site exchange processes in solution NMR: a continuous Gaussian exchange model.

The effects of site exchange due to slow conformational changes in rapidly rotating molecules in solution are examined in detail. Significant gaps in the currently available theory are filled. The effects of site exchange on the lineshape, decay of a simple spin-echo, decay of the even echoes in a Carr-Purcell-Meiboom-Gill (CPMG) pulse-sequence, and decay of the transverse magnetization in a resonant spin-locking field are investigated. Both trajectory and stochastic operator approaches are formulated and shown to be completely equivalent whenever the dynamics of population transfers among the inequivalent sites is governed by either a stationary or a nonstationary Markov process. A nonstationary Markov process may result from Brownian dynamics (a stationary Markov process) in a larger conformational space that contains the subspace of inequivalent sites. A continuous Gaussian exchange model is formulated in which a nucleus undergoes continuous one-dimensional motion in a harmonic potential well that is located in a linear chemical shift gradient. The effects of this Gaussian exchange model on the lineshape, simple spin-echo decay, and decay of the even echoes of a CPMG pulse train are treated rigorously via the trajectory approach. Compact analytical expressions are obtained for the relevant correlation functions in each case. The relevant decays are found to be exponential in the very short time and long time limits, which are not necessarily experimentally significant in any given case. In the fast exchange limit the relevant decays are exponential at all times, and explicit formulas are given for their decay rates. In the long time limit, all discrete multisite models with the same intrinsic Ro2 at every site are shown to be completely equivalent to a continuous Gaussian model with appropriate relaxation time and variance of the Larmor frequency. The effects of this Gaussian exchange model on the decay of the transverse magnetization in a resonant spin-locking field are treated heuristically by a trajectory approach. The intrinsic contribution (Ro1rho) of rapid rotations and dipole-dipole interactions to relax the transverse magnetizations of two nuclei of the same kind in the presence of a (nearly) resonant spin-locking field is also derived and found to be practically the same as the intrinsic contribution, Ro2, of those same rotations to the simple and CPMG spin-echo decay rates and linewidth. Literature data for the linewidth, decay rate of the CPMG even spin-echoes, and R(1rho) decay rate for the A9-H2 protons of adenines at the central TpA step in the sequence, 5'-GCAGGTTTAAACCTCG-3', are analyzed using the Gaussian exchange model to assess the time-scale and variance of the site exchange process as well as the intrinsic Ro2 rate. Although a single Gaussian exchange process with appropriate parameters can fit these three A9-H2 data rather well, this particular "solution" cannot be reconciled with NMR relaxation data on other protons in the same DNA molecule. Rather good agreement with all of the observations is obtained by using a model of two concurrent Gaussian exchange processes, whose relaxation times, tau = 7 and 460 micros, differ in time-scale by a factor of 65. The insensitivity of R1rho in the presence of a fast site exchange process to much slower concurrent site exchange processes is explicitly demonstrated. Protocols for detecting and characterizing a second slow site exchange process are suggested.

Adenine↗
PubMed1989

Applications of the expectation-maximization algorithm to quantal analysis of postsynaptic potentials.

The expectation-maximization (EM) algorithm is a robust method for maximum likelihood estimation of the parameters of an incompletely sampled distribution. It has been used to resolve the trial-to-trial amplitude fluctuations of postsynaptic potentials, when these are recorded in the presence of noise. Its use has however been limited by the need for different recursion equations for each set of conditions defined by the signal and noise processes. These equations are derived for the following conditions which arise in studies of synaptic transmission: non-gaussian noise process; quantal fluctuation; quantal variability. In addition, a constraint can be incorporated to accommodate simple and compound binomial models of transmitter release. Some advantages of these methods are illustrated by Monte Carlo simulations.

Algorithms↗