An improved saddlepoint approximation

Given a set of third- or higher-order moments, not only is the saddlepoint approximation the only realistic 'family-free' technique available for constructing an associated probability distribution, but it is 'optimal' in the sense that it is based on the highly efficient numerical method of steepest descents. However, it suffers from the problem of not always yielding full support, and whilst [S. Wang, General saddlepoint approximations in the bootstrap, Prob. Stat. Lett. 27 (1992) 61.] neat scaling approach provides a solution to this hurdle, it leads to potentially inaccurate and aberrant results. We therefore propose several new ways of surmounting such difficulties, including: extending the inversion of the cumulant generating function to second-order; selecting an appropriate probability structure for higher-order cumulants (the standard moment closure procedure takes them to be zero); and, making subtle changes to the target cumulants and then optimising via the simplex algorithm.     

A Derivative Matching Approach to Moment Closure for the Stochastic Logistic Model

Continuous-time birth-death Markov processes serve as useful models in population biology. When the birth-death rates are nonlinear, the time evolution of the first n order moments of the population is not closed, in the sense that it depends on moments of order higher than n. For analysis purposes, the time evolution of the first n order moments is often made to be closed by approximating these higher order moments as a nonlinear function of moments up to order n, which we refer to as the moment closure function. In this paper, a systematic procedure for constructing moment closure functions of arbitrary order is presented for the stochastic logistic model. We obtain the moment closure function by first assuming a certain separable form for it, and then matching time derivatives of the exact (not closed) moment equations with that of the approximate (closed) equations for some initial time and set of initial conditions. The separable structure ensures that the steady-state solutions for the approximate equations are unique, real and positive, while the derivative matching guarantees a good approximation, at least locally in time. Explicit formulas to construct these moment closure functions for arbitrary order of truncation n are provided with higher values of n leading to better approximations of the actual moment dynamics. A host of other moment closure functions previously proposed in the literature are also investigated. Among these we show that only the ones that achieve derivative matching provide a close approximation to the exact solution. Moreover, we improve the accuracy of several previously proposed moment closure functions by forcing derivative matching.     

Moments,cumulants, and polygenic dynamics

A new approach for describing the evolution of polygenic traits subject to selection and mutation is presented. Differential equations for the change of cumulants of the allelic frequency distribution at a particular locus and for the cumulants of the distributions of genotypic and phenotypic values are derived. The derivation is based on the assumptions of random mating, no sex differences, absence of random drift, additive gene action, linkage equilibrium, and Hardy-Weinberg proportions. Cumulants are a set of parameters that, like moments, describe the shape of a probability density. Compared with moments, however, they have properties that make them a much more convenient tool for investigating polygenic traits. Applications to directional and stabilizing selection are given.     

A well-posed, embedded constraint representation of joint moments from kinesiological measurements

Joint moment estimation using the traditional inverse dynamics analysis presents two challenging problems, which limit its reliability. First, the quality of the computed moments depends directly on unreliable estimates of the segment accelerations obtained numerically by differentiating noisy marker measurements. Second, the representation of joint moments from combined video and force plate measurements belongs to a class of ill-posed problems, which does not possess a unique solution. This paper presents a well-posed representation derived from an embedded constraint equation. The proposed method, referred to as the embedded constraint representation (ECR), provides unique moment estimates, which satisfy all measurement constraints and boundary conditions and require fewer acceleration components than the traditional inverse dynamics method. Specifically, for an n-segment open chain planar system, the ECR requires n-3 acceleration components as compared to 3(n-1) components required by the traditional (from ground up) inverse dynamics analysis. Based on a simulated experiment using a simple three-segment model, the precision of the ECR is evaluated at different noise levels and compared to the traditional inverse dynamics technique. At the lowest noise levels, the inverse dynamics method is up to 50 percent more accurate while at the highest noise levels the ECR method is up to 100 percent more accurate. The ECR results over the entire range of noise levels reveals an average improvement on the order 20 percent in estimating the moments distal to the force plate and no significant improvement in estimating moments proximal to the force plate. The new method is particularly advantageous in a combined video, force plate, and accelerometery sensing strategy.     

Dynamic arm swinging in human walking

Humans tend to swing their arms when they walk, a curious behaviour since the arms play no obvious role in bipedal gait. It might be costly to use muscles to swing the arms, and it is unclear whether potential benefits elsewhere in the body would justify such costs. To examine these costs and benefits, we developed a passive dynamic walking model with free-swinging arms. Even with no torques driving the arms or legs, the model produced walking gaits with arm swinging similar to humans. Passive gaits with arm phasing opposite to normal were also found, but these induced a much greater reaction moment from the ground, which could require muscular effort in humans. We therefore hypothesized that the reduction of this moment may explain the physiological benefit of arm swinging. Experimental measurements of humans (n = 10) showed that normal arm swinging required minimal shoulder torque, while volitionally holding the arms still required 12 per cent more metabolic energy. Among measures of gait mechanics, vertical ground reaction moment was most affected by arm swinging and increased by 63 per cent without it. Walking with opposite-to-normal arm phasing required minimal shoulder effort but magnified the ground reaction moment, causing metabolic rate to increase by 26 per cent. Passive dynamics appear to make arm swinging easy, while indirect benefits from reduced vertical moments make it worthwhile overall.     

CuBIC: cumulant based inference of higher-order correlations in massively parallel spike trains

Recent developments in electrophysiological and optical recording techniques enable the simultaneous observation of large numbers of neurons. A meaningful interpretation of the resulting multivariate data, however, presents a serious challenge. In particular, the estimation of higher-order correlations that characterize the cooperative dynamics of groups of neurons is impeded by the combinatorial explosion of the parameter space. The resulting requirements with respect to sample size and recording time has rendered the detection of coordinated neuronal groups exceedingly difficult. Here we describe a novel approach to infer higher-order correlations in massively parallel spike trains that is less susceptible to these problems. Based on the superimposed activity of all recorded neurons, the cumulant-based inference of higher-order correlations (CuBIC) presented here exploits the fact that the absence of higher-order correlations imposes also strong constraints on correlations of lower order. Thus, estimates of only few lower-order cumulants suffice to infer higher-order correlations in the population. As a consequence, CuBIC is much better compatible with the constraints of in vivo recordings than previous approaches, which is shown by a systematic analysis of its parameter dependence.     

A new method to infer higher-order spike correlations from membrane potentials

What is the role of higher-order spike correlations for neuronal information processing? Common data analysis methods to address this question are devised for the application to spike recordings from multiple single neurons. Here, we present a new method which evaluates the subthreshold membrane potential fluctuations of one neuron, and infers higher-order correlations among the neurons that constitute its presynaptic population. This has two important advantages: Very large populations of up to several thousands of neurons can be studied, and the spike sorting is obsolete. Moreover, this new approach truly emphasizes the functional aspects of higher-order statistics, since we infer exactly those correlations which are seen by a neuron. Our approach is to represent the subthreshold membrane potential fluctuations as presynaptic activity filtered with a fixed kernel, as it would be the case for a leaky integrator neuron model. This allows us to adapt the recently proposed method CuBIC (cumulant based inference of higher-order correlations from the population spike count; Staude et al., J Comput Neurosci 29(1–2):327–350, 2010c) with which the maximal order of correlation can be inferred. By numerical simulation we show that our new method is reasonably sensitive to weak higher-order correlations, and that only short stretches of membrane potential are required for their reliable inference. Finally, we demonstrate its remarkable robustness against violations of the simplifying assumptions made for its construction, and discuss how it can be employed to analyze in vivo intracellular recordings of membrane potentials.     

Higher-Order Spike Triggered Analysis of Neural Oscillators

For the purpose of elucidating the neural coding process based on the neural excitability mechanism, researchers have recently investigated the relationship between neural dynamics and the spike triggered stimulus ensemble (STE). Ermentrout et al. analytically derived the relational equation between the phase response curve (PRC) and the spike triggered average (STA). The STA is the first cumulant of the STE. However, in order to understand the neural function as the encoder more explicitly, it is necessary to elucidate the relationship between the PRC and higher-order cumulants of the STE. In this paper, we give a general formulation to relate the PRC and the nth moment of the STE. By using this formulation, we derive a relational equation between the PRC and the spike triggered covariance (STC), which is the covariance of the STE. We show the effectiveness of the relational equation through numerical simulations and use the equation to identify the feature space of the rat hippocampal CA1 pyramidal neurons from their PRCs. Our result suggests that the hippocampal CA1 pyramidal neurons oscillating in the theta frequency range are commonly sensitive to inputs composed of theta and gamma frequency components.     

Effect of Selection on a Quantitative Characteristic

In this paper we study the effect of selection procedures on certain parameters of the distribution function (d.f.) (such as mean, percentile etc.) of a quantitative characteristic X, in successive generations when the d.f. is governed by a single locus. We study the changes in gene frequencies under the truncation and genotype selection procedures by obtaining approximations to the gene frequencies, since exact expressions are not available. Using these approximations for the gene frequencies, we compute the selection differentials of X for different values of n, the number of generations. We also obtain the limiting distributions as n → ∞ and compute the number of generations required for the above parameter(s) of the d.f. to reach a value close enough to the limiting value.     

Food material properties and mandibular load resistance abilities in large-bodied hominoids

Numerous comparative studies have sought to demonstrate a functional link between feeding behavior, diet, and mandibular form in primates. In lieu of data on the material properties of foods ingested and masticated, many investigators have relied on qualitative dietary classifications such as "folivore" or "frugivore." Here we provide the first analysis of the relationship between jaw form, dietary profiles, and food material properties in large-bodied hominoids. We employed ratios of area moments of inertia and condylar area to estimate moments imposed on the mandible in order to evaluate and compare the relative ability to counter mandibular loads among central Bornean orangutans (Pongo pygmaeus wurmbii), Virunga mountain gorillas (Gorilla beringei beringei), and east African chimpanzees (Pan troglodytes schweinfurthii). We used data on elastic modulus (E) of fruit, fracture toughness (R) of fruit, leaves, and non-fruit, non-leaf vegetation, and derived fragmentation indices ( radicalR/E and radicalER), as proxies for bite force. We generated bending and twisting moments (forcexmoment arm) for various mandibular loading behaviors using food material properties to estimate minimally required bite forces. Based on E and R of foods ingested and masticated, we hypothesized improved resistance to mandibular loads in Pongo p. wurmbii compared to the African apes, and in G. b. beringei compared to Pan t. schweinfurthii. Results reveal that our predictions are borne out only when bite forces are estimated from maximum R of non-fruit, non-leaf vegetation. For all other tissues and material properties results were contrary to our predictions. Importantly, as food material properties change, the moments imposed on the mandible change; this, in turn, alters the entire ratio of relative load resistance to moment. The net effect is that species appear over- or under-designed for the moments imposed on the mandible. Our hypothesis, therefore, is supported only if we accept that maximum R of these vegetative tissues represents the relevant mechanical property influencing the magnitude of neuromuscular activity, food fragmentation, and mandibular morphology. A general implication is that reliable estimates of average and maximum bite forces from food material properties require that the full range of tissues masticated be tested. Synthesizing data on ingestive and masticatory behaviors, the number of chewing cycles associated with a given food, and food mechanical properties, should inform the broader question of which foods and feeding behaviors are most influential on the mandibular loading environment.     

Effects of low-pass filter combinations on lower extremity joint moments in distance running

Inverse dynamics is a standard tool in biomechanics, which requires low-pass filtering of external force and kinematic signals. Unmatched filtering procedures are reported to affect joint moment amplitudes in high impact movements, like landing or cutting, but are also common in the analysis of distance running. We analyzed the effects of cut-off frequencies in 94 rearfoot runners at a speed of 3.5 m/s. Additionally, we investigated whether the evaluation of footwear interventions is affected by the choice of cut-off frequencies. We performed 3D inverse dynamics for the hip, knee and ankle joints using different low-pass filter cut-off frequency combinations for a recursive fourth-order Butterworth filter. We observed fluctuations of joint moment curves in the first half of stance, which were most pronounced for the most unmatched cut-off frequency combination (kinematics: 10 Hz; ground reaction forces (GRFs): 100 Hz) and for more proximal joints. Peak sagittal plane hip joint moments were altered by 94% on average. We observed a change in the ranking of subjects based on joint moment amplitude. We found significant (p < 0.001) footwear by cut-off frequency combination interaction effects for most peak joint moments. These findings highlight the importance of cut-off frequency choice in the analysis of joint moments and the assessment of footwear interventions in distance running. Based on our results, we propose to use matched cut-off frequencies around 20 Hz in order to avoid large artificial fluctuations in joint moment curves while at the same time avoiding a severe removal of physiological high-frequency signal content from the GRF signals.     

Calculation and measurement of the dipole moment of small proteins: Use of protein data base

The dipole moments of small protein molecules were determined experimentally in order to validate the calculated dipole moments by previous investigators. We found that the agreements are satisfactory for some proteins. There are, however, many proteins for which the agreement is less than satisfactory. In order to find the cause of the disagreement, the dipole moments of these proteins were recalculated using the Brookhaven Protein Data Bank. We calculated the dipole moment due to fixed surface charges and the bond moments of all the carbonyl groups in main chain and side chains. The calculation consists of the mean moments and their mean square fluctuations. In addition, we investigated the effect of electrostatic interactions between charged sites for several proteins. These results show that incorporation of the interactions does not affect substantially the calculated dipole moments. The rms fluctuation of the dipole moment is found to be small but not negligible. In conclusion, recalculated dipole moments are in good agreement with the observed values. © 1993 John Wiley & Sons, Inc.     

Aerodynamic yawing moment characteristics of bird wings

The aerodynamic yawing moments due to sideslip are considered for wings of birds. Reference is made to the experience with aircraft wings in order to identify features which are significant for the yawing moment characteristics. Thus, it can be shown that wing sweep, aspect ratio and lift coefficient have a great impact. Focus of the paper is on wing sweep which can considerably increase the yawing moment due to sideslip when compared with unswept wings. There are many birds the wings of which employ sweep. To show the effect of sweep for birds, the aerodynamic characteristics of a gull wing which is considered as a representative example are treated in detail. For this purpose, a sophisticated aerodynamic method is used to compute results of high precision. The yawing moments of the gull wing with respect to the sideslip angle and the lift coefficient are determined. They show a significant level of yaw stability which strongly increases with the lift coefficient. It is particularly high in the lift coefficient region of best gliding flight conditions. In order to make the effect of sweep more perspicuous, a modification of the gull wing employing no sweep is considered for comparison. It turns out that the unswept wing yields yawing moments which are substantially smaller than those of the original gull wing with sweep. Another feature significant for the yawing moment characteristics concerns the fact that sweep is at the outer part of bird wings. By considering the underlying physical mechanism, it is shown that this feature is most important for the efficiency of wing sweep. To sum up, wing sweep provides a primary contribution to the yawing moments. It may be concluded that this is an essential reason why there is sweep in bird wings.     

Definition and Estimation of Higher-Order Gene Fixation Indices

Fixation indices summarize the associations between genes that arise from the joint effects of inbreeding and selection. In this paper, fixation indices are derived for pairs, triplets and quadruplets of genes at a single multiallelic locus. The fixation indices are obtained by dividing cumulants by constants; the cumulants describe the statistical distribution of alleles and the constants are functions of gene frequency. The use of cumulants instead of moments is necessary only for four-gene indices, when the fourth cumulant is used. A second type of four-gene index is also required, and this index is based upon the covariation of second-order cumulants. At multiallelic loci, a large number of indices is possible. If alleles are selectively neutral, the number of indices is reduced and the relationship between gene identity and gene cumulants is shown.--Two-gene indices can always be estimated from genotypic frequency data at a single polymorphic locus. Three-gene indices are also estimable except when allele frequency equals one-half. Four-gene indices are not estimable unless selection is assumed to have an equal effect upon each allele (such as under selective neutrality) and the locus contains at least three alleles of unequal frequency. For diallelic or selected loci, an alternative four-gene fixation index is proposed. This index incorporates both types of four-gene associations but cannot be related to gene identity.     

Electric dipole theory and thermodynamics of actomyosin molecular motor in muscle contraction

Movements in muscles are generated by the myosins which interact with the actin filaments. In this paper we present an electric theory to describe how the chemical energy is first stored in electrostatic form in the myosin system and how it is then released and transformed into work. Due to the longitudinal polarized molecular structure with the negative phosphate group tail, the ATP molecule possesses a large electric dipole moment (p(0)), which makes it an ideal energy source for the electric dipole motor of the actomyosin system. The myosin head contains a large number of strongly restrained water molecules, which makes the ATP-driven electric dipole motor possible. The strongly restrained water molecules can store the chemical energy released by ATP binding and hydrolysis processes in the electric form due to their myosin structure fixed electric dipole moments (p(i)). The decrease in the electric energy is transformed into mechanical work by the rotational movement of the myosin head, which follows from the interaction of the dipoles p(i) with the potential field V(0) of ATP and with the potential field Psi of the actin. The electrical meaning of the hydrolysis reaction is to reduce the dipole moment p(0)-the remaining dipole moment of the adenosine diphosphate (ADP) is appropriately smaller to return the low negative value of the electric energy nearly back to its initial value, enabling the removal of ADP from the myosin head so that the cycling process can be repeated. We derive for the electric energy of the myosin system a general equation, which contains the potential field V(0) with the dipole moment p(0), the dipole moments p(i) and the potential field psi. Using the previously published experimental data for the electric dipole of ATP (p(0) congruent with 230 debye) and for the amount of strongly restrained water molecules (N congruent with 720) in the myosin subfragment (S1), we show that the Gibbs free energy changes of the ATP binding and hydrolysis reaction steps can be converted into the form of electric energy. The mechanical action between myosin and actin is investigated by the principle of virtual work. An electric torque always appears, i.e. a moment of electric forces between dipoles p(0) and p(i)(/M/ > or = 16 pN nm) that causes the myosin head to function like a scissors-shaped electric dipole motor. The theory as a whole is illustrated by several numerical examples and the results are compared with experimental results.     

First passage time for a supercritical Galton-Watson process restricted to the non-extinction set

In the present paper we obtain the probability distribution and the first two moments, of the number of generations required by a supercritical branching process to cross for the first time a given population size, when it is restricted to the non-extinction set.     

Combinatorial properties of RNA secondary structures.

The secondary structure of an RNA molecule is of great importance and possesses influence, e.g., on the interaction of tRNA molecules with proteins or on the stabilization of mRNA molecules. The classification of secondary structures by means of their order proved useful with respect to numerous applications. In 1978, Waterman, who gave the first precise formal framework for the topic, suggested to determine the number a(n,p) of secondary structures of size n and given order p. Since then, no satisfactory result has been found. Based on an observation due to Viennot et al., we will derive generating functions for the secondary structures of order p from generating functions for binary tree structures with Horton-Strahler number p. These generating functions enable us to compute a precise asymptotic equivalent for a(n,p). Furthermore, we will determine the related number of structures when the number of unpaired bases shows up as an additional parameter. Our approach proves to be general enough to compute the average order of a secondary structure together with all the r-th moments and to enumerate substructures such as hairpins or bulges in dependence on the order of the secondary structures considered.     

Primary stability of threaded cups in THR--an experimental study.

Four threaded cups were tested up to their lever-out moments, torque-in moments and their resistance to failure. A was a parabolic-shaped, B was a spherical, C was a spherical-shaped too, and D was a conical shaped cup. Cup A and D represent cups which have proven themselves in clinical applications, but not cup B. The threads were determined and showed different constructive features. The cups were torqued into precise cavities in PVC foam cubes, after that they were levered out in a testing machine. The lever-out moments of all the cups showed significant differences; the results were: A: 78.4 Nm, B: 88.7 Nm, C: 117.5 Nm, D; 136.6 Nm. In the case of the torque-in moments there were no significant differences between A and B, neither between C and D. The differences in stiffness between B and C were not significant, but they were between the others. The primary stability against lever-out and the torque-in moment of threaded cups for artificial hip replacement can be basically influenced by different constructive features. Hence lever-out moment and torque-in moment should be understood and tested as independent variables.     

Moment closure and the stochastic logistic model

The quasi-stationary distribution of the stochastic logistic model is studied in the parameter region where its body is approximately normal. Improved asymptotic approximations of its first three cumulants are derived. It is shown that the same results can be derived with the aid of the moment closure method. This indicates that the moment closure method leads to expressions for the cumulants that are asymptotic approximations of the cumulants of the quasi-stationary distribution.