Solutions to axon equations

The solutions to a general class of axon partial differential equations proposed by FitzHugh which includes the Hodgkin-Huxley equations are studied. It is shown that solutions to the partial differential equations are exactly the solutions to a related set of integral equations. An iterative procedure for constructing the solutions based on standard methods for ordinary differential equations is given and each set of initial values is shown to lead to a unique solution. Continuous dependence of the solutions on the initial values is established and solutions with initial values in a restricted (physiological) range are shown to remain in that range for all time. The iterative procedure is not suggested as the basis for numerical integration.     

Equations and calculations of product yields and preferred pathways for butanediol and mixed-acid fermentations

Using the available information of fermentation biochemistry, fermentation (stoichiometric) equations are derived for anaerobic saccharolytic fermentations of butanediol and mixed acids. The equations describe the interrelations among the fermentation products, biomass, and consumed substrate (glucose). The validity of the equations is tested using a variety of batch data from the literature. The validity of the equations is expected to extend to steady-state and transient fermentations, as well. Uses, improvements, and extensions of the equations are also discussed in detail. Among others, it is shown that the equations are useful for checking the consistency of experimental data, for calculating maximal yields and selectivities for the fermentation products, and calculating the extent of utilization of the Embden-Meyerhof-Parnas pathway versus the Hexose Monophosphate pathway of glucose utilization.     

Neural rate equations for bursting dynamics derived from conductance-based equations

A method of obtaining rate equations from conductance-based equations is developed and applied to fast-spiking and bursting neocortical neurons. It involves splitting systems of conductance-based equations into fast and slow subsystems, and averaging the effects of fast terms that drive the slowly varying quantities by showing that their average is closely proportional to the firing rate. The dependence of the firing rate on the injected current is then approximated in the analysis. The resulting behavior of the slow variables is then substituted back into the fast equations, with the further approximation of replacing the fast voltages in these terms by effective values. For bursting neurons the method yields two coupled limit-cycle oscillators: a self-exciting oscillator for the slow variables that commences limit-cycle oscillations at a critical current and modulates a fast spike-generating oscillator, thereby leading to slowly modulated bursts with a group of spikes in each burst. The dynamics of these coupled oscillators are then verified against those of the conductance-based equations. Finally, it is shown how to place the results in a form suitable for use in mean-field equations for neural population dynamics.     

Spatial reconstruction of the human motion based on images of a single camera

The inverse dynamic analysis procedures used in the study of the human gait require that the kinematics of the supporting biomechanical model is known beforehand. The first step to obtain the kinematic data is the reconstruction of human spatial motion, i.e., the evaluation of the anatomic points positions that enables to uniquely define the position of all anatomical segments. In photogrammetry, the projection of each anatomical point is described by two linear equations relating its three spatial coordinates with the two coordinates of the projected point. The need for the image of two cameras arises from the fact that three equations are necessary to find the original spatial position of the anatomical point. It is shown here that the kinematic constraint equations associated with a biomechanical model can be used as the extra set of equations required for the reconstruction process, instead of the equations associated with the second camera. With this methodology, the system of equations arising from the point projections and biomechanical model kinematic constraints are solved simultaneously. Since the system of equations has multiple solutions for each image, a strategy based on the minimization of the cost function associated to the smoothness of the reconstructed motion is devised, leading to an automated computer procedure enabling a unique reconstruction.     

The appropriate use of regression equations for the estimation of segmental inertia parameters

Linear regression equations are commonly used in conjunction with experimental data to provide linear relationships between quantities which are dimensionally distinct. In many cases theoretical relationships between such quantities are known and can be used as a basis for non-linear regression equations. This study compares linear and non-linear approaches for estimating the segmental moments of inertia from anthropometric measurements using the data of Chandler et al. [Chandler et al. (1975) Investigation of inertial properties of the human body. AMRL Technical Report 74-137, Wright Patterson Air Force Base. OH.] Right limb data were used to derive the equations while left limb data were used as a cross-validation sample to evaluate the inertia estimates calculated from the equations. For the limb segments the standard error estimates had average values of 21% for the linear equations and 13% for the non-linear equations. Data on a 10 yr-old boy was used to compare the two approaches outside the sample range. The mean percentage residuals were 286% for the linear equations and 20% for the non-linear equations. A set of non-linear equations is provided.     

Stochastic Hierarchical Systems: Excitable Dynamics

We present a discrete model of stochastic excitability by a low-dimensional set of delayed integral equations governing the probability in the rest state, the excited state, and the refractory state. The process is a random walk with discrete states and nonexponential waiting time distributions, which lead to the incorporation of memory kernels in the integral equations. We extend the equations of a single unit to the system of equations for an ensemble of globally coupled oscillators, derive the mean field equations, and investigate bifurcations of steady states. Conditions of destabilization are found, which imply oscillations of the mean fields in the stochastic ensemble. The relation between the mean field equations and the paradigmatic Kuramoto model is shown.     

Boundary element solution of macromolecular electrostatics: interaction energy between two proteins.

The boundary element technique is implemented to solve for the electrostatic potential of macromolecules in an ionic solution. This technique entails solving surface integral equations that are equivalent to the Poisson and the Poisson-Boltzmann equations governing the electrostatic potential inside the macromolecules and and in the solvent. A simple but robust method is described for discretizing the macromolecular surfaces in order to approximate the integral equations by linear algebraic equations. Particular attention is paid to the interaction energy between two macromolecules, and an iterative procedure is devised to make the calculation more efficient. This iterative procedure is illustrated in the electron transfer system of cytochrome c and cytochrome c peroxidase.     

Calculation of tilt angles for crystal specimen orientation adjustment using double-tilt and tilt-rotate holders

In this communication we wish to present a group of new equations which can be used to calculate the tilt angle for crystal specimen orientation adjustment in the transmission electron microscope. The experiments were concerned with double-tilt and tilt-rotate holders and the new equations deduced using matrix geometry. The specimen orientation adjustment using the tilt angles calculated by these equations is considered to be more convenient and less time-consuming than following the Kikuchi map method. Our method avoids the difficulties associated with orientation adjustment of severely strained and small grain size specimens using the Kikuchi map procedure. The algorithms for deducing the new equations, together with an experimental example using the equations, are described.     

Analysis of the transient behavior of kidney models

Non-steady-state equations for kidney models are stated. General conservation relations for these equations are derived. Transient equations for the central core model of the renal medulla are developed. Solution of the equations by Laplace transform methods for time invariant volume flows is discussed. The general theory of solving models with time dependent flows by finite difference methods is developed.     

种间竞争的一个新的数学模型—对经典的Lotka—Vol—terra竞争方程的扩充

本文根据营养动力学理论,建立了一类种间竞争的新的数学模型:它是单种群增长的Cui-Lawson模型,在种间竞争上的推广。新的种间竞争模型克服了经典的种间竞争的Lotka-Volteira方程的局限与不足,具有更广泛和复杂的行为,并在特殊条件下以Lotka-Volterra竞争方程为其特例。因此,新的种间竞争的数学模型是更一般的解释性模型,是对经典的Lotka-Voterra竞争方程的扩充。     

Estimating equations for parameters in means and covariances of multivariate discrete and continuous responses.

Generalized estimating equations are introduced in an ad hoc fashion for the covariance matrix of a multivariate response. These equations are to be solved jointly with score equations from a generalized linear model for mean parameters. A class of quadratic exponential models is used to develop joint estimating equations for mean and covariance parameters in a more systematic fashion, and proposals for the use of such equations are developed. Comments on the relative merits of the ad hoc and model-based approaches to estimation are given and a regression illustration with a bivariate response is provided.     

A conversational program for analytical modelling of action potentials in nerve and muscle.

A conversational program is described which is being used for exploratory modelling of digestive tract electrical activity. The structure of the mathematical model is based on the Hodgkin-Huxley equations for nerve axons and the Noble equations for Purkinje fibres of the heart. The parameters of the models are changed conversationally to investigate the effects of frequency, amplitude and waveshape. In this way it is intended to extend the models to include nervous control. The program is also being used to correlate the behaviour of H-H type equations with the simpler dynamics of Van der Pol's equation which is the basis of an oscillator-array model for the stomach and small intestines.     

Reduced MHD equations with allowance for slow magnetosonic waves

The so-called reduced magnetohydrodynamics, which deals with the motion of incompressible fluids and is usually applied to describe plasma flows in a strong toroidal magnetic field, has a number of drawbacks and, in some cases, fails to produce correct results. The equations proposed here are simpler than the original MHD equations and are free of these drawbacks. These equations, like reduced MHD equations, make it possible to remove from consideration fast magnetosonic waves and to introduce the vector potential for the poloidal magnetic field. However, our equations differ from the reduced MHD equations in that they completely incorporate slow magnetosonic waves, the specific features of the toroidal geometry, and the effects of the toroidal velocity.     

Dispersal and competition models for plants

New models for seed dispersal and competition between plant species are formulated and analyzed. The models are integrodifference equations, discrete in time and continuous in space, and have applications to annual and perennial species. The spread or invasion of a single plant species into a geographic region is investigated by studying the travelling wave solutions of these equations. Travelling wave solutions are shown to exist in the single-species models and are compared numerically. The asymptotic wave speed is calculated for various parameter values. The single-species integrodifference equations are extended to a model for two competing annual plants. Competition in the two-species model is based on a difference equation model developed by Pakes and Maller [26]. The two-species model with competition and dispersal yields a system of integrodifference equations. The effects of competition on the travelling wave solutions of invading plant species is investigated numerically.     

New percentage body fat prediction equations for Japanese females

Anthropometry is a simple and cost-efficient method for the assessment of body composition. However prediction equations to estimate body composition using anthropometry should be 'population-specific'. Most popular body composition prediction equations for Japanese females were proposed more than 40 years ago and there is some concern regarding their usefulness in Japanese females living today. The aim of this study was to compare percentage body fat (%BF) estimated from anthropometry and dual energy x-ray absorptiometry (DXA) to examine the applicability of commonly used prediction equations in young Japanese females. Body composition of 139 Japanese females aged between 18 and 27 years of age (BMI range: 15.1-29.1 kg/m(2)) was measured using whole-body DXA (Lunar DPX-LIQ) scans. From anthropometric measurements %BF was estimated using four equations developed from Japanese females. The results showed that the traditionally employed prediction equations for anthropometry significantly (p<0.01) underestimate %BF of young Japanese females and therefore are not valid for the precise estimation of body composition. New %BF prediction equations were proposed from the DXA and anthropometry results. Application of the proposed equations may assist in more accurate assessment of body fatness in Japanese females living today.     

State-dependent neutral delay equations from population dynamics

A novel class of state-dependent delay equations is derived from the balance laws of age-structured population dynamics, assuming that birth rates and death rates, as functions of age, are piece-wise constant and that the length of the juvenile phase depends on the total adult population size. The resulting class of equations includes also neutral delay equations. All these equations are very different from the standard delay equations with state-dependent delay since the balance laws require non-linear correction factors. These equations can be written as systems for two variables consisting of an ordinary differential equation (ODE) and a generalized shift, a form suitable for numerical calculations. It is shown that the neutral equation (and the corresponding ODE—shift system) is a limiting case of a system of two standard delay equations.     

Membrane permeability equations and their solutions for red cells

Summary The mathematical equations for the transport of nonelectrolytes across cell membranes are critically examined and cast in forms suitable for solution which involve fewer approximations than has heretofore been commonly done. For the case of red cells, the equations are developed to include the effect of the variation in apparent nonosmotic water owing to the variation in hemoglobin concentration as the cell swells or shrinks. Two methods of solution of the equations are developed and studied and sample calculations are provided. It is shown that the solutions to the linearized equations commonly found in the literature are insufficiently accurate for some purposes and this inaccuracy is avoided by the methods given here. The importance of retaining the effects of variations in apparent nonosmotic water and in solute volume in the cell is demonstrated.     

Oscillations in a model of repression with external control

A mathematical model for control by repression by an extracellular substance is developed, including diffusion and time delays. The model examines how active transport of a nutrient can produce either oscillatory or stable responses depending on a variety of parameters, such as diffusivity, cell size, or nutrient concentration. The system of equations for the mathematical model is reduced to a system of delay differential equations and linear Volterra equations. After linearizing these equations and forming the limiting Volterra equations, the resulting linear system no longer has any spatial dependence. Local stability analysis of the radially symmetric model shows that the system of equations can undergo Hopf bifurcations for certain parameter values, while other ranges of the parameters guarantee asymptotic stability. One numerical study shows that the model can exhibit intracellular biochemical oscillations with increasing extracellular concentrations of the nutrient, which suggests a possible trigger mechanism for morphogenesis.The work of this author was supported in part by NSF grants DMS-8603787 and DMS-8807360The work of this author was supported under the REU program of NSF by grant DMS-8807360The work of this author was supported under the REU program of NSF by grant DMS-8807360     

Numerical modeling of molecular-motor-assisted transport of adenoviral vectors in a spherical cell

Viral gene delivery in a spherical cell is investigated numerically. The model of intracellular trafficking of adenoviruses is based on molecular-motor-assisted transport equations suggested by Smith and Simmons. These equations are presented in spherical coordinates and extended by accounting for the random component of motion of viral particles bound to filaments. This random component is associated with the stochastic nature of molecular motors responsible for locomotion of viral particles bound to filaments. The equations are solved numerically to simulate viral transport between the cell membrane and cell nucleus during initial stages of viral infection.