An Analytic Solution of the Cable Equation Predicts Frequency Preference of a Passive Shunt-End Cylindrical Cable in Response to Extracellular Oscillating Electric Fields

Under physiological and artificial conditions, the dendrites of neurons can be exposed to electric fields. Recent experimental studies suggested that the membrane resistivity of the distal apical dendrites of cortical and hippocampal pyramidal neurons may be significantly lower than that of the proximal dendrites and the soma. To understand the behavior of dendrites in time-varying extracellular electric fields, we analytically solved cable equations for finite cylindrical cables with and without a leak conductance attached to one end by employing the Green's function method. The solution for a cable with a leak at one end for direct-current step electric fields shows a reversal in polarization at the leaky end, as has been previously shown by employing the separation of variables method and Fourier series expansion. The solution for a cable with a leak at one end for alternating-current electric fields reveals that the leaky end shows frequency preference in the response amplitude. Our results predict that a passive dendrite with low resistivity at the distal end would show frequency preference in response to sinusoidal extracellular local field potentials. The Green's function obtained in our study can be used to calculate response for any extracellular electric field.     

Cracks emanating from circular voids or elastic inclusions in PMMA near a bone-implant interface

Mechanical fracture is believed to be a primary reason for loss of fixation at the bone-cement-implant interface. In addition to the expected cracks at the bone-cement interface, cracks are also observed to be formed at voids and inclusions within the cement. An analytical solution is presented for cracks emanating from circular voids or elastic inclusions under uniaxial tension using the solution for a single dislocation as a Green's function. Stress intensity factors are calculated for arbitrary orientations of the cracks, and for varying relative stiffnesses of the inclusion and the matrix, to determine the most favorable combination of parameters for crack growth.     

On the rotational brownian motion of a bacterial idle motor. I. Theory of time-dependent fluorescence depolarization

A rotational diffusion equation and its Green's function for a spheroidal particle such as a bacterial body, to which an actively driving but idle motor is attached, are presented. As an application of the theory, general expressions for the time-dependent fluorescence depolarization caused by such a particle have been obtained. Measurement of such depolarizalion should provide a useful tool for determination of the rate of revolution of the rotating motor attached to cell bodies such as bacteria under various solution conditions, if a fluorescent (or phosphorescent) label is attached to the motor shaft.     

Transient behavior of population density with competition for resources

In this paper we derive a perturbation expansion of some nonlinear diffusion equations proposed by J. G. Skellam and M. Kimura. The solution is derived by converting the differential equations into integral equations by means of the Green's function for the diffusion equation. This research was supported in part by the Office of Naval Research under Contract Nonr-595(17).     

A new asymptotic analysis of the nth order reaction-diffusion problem: analytical and numerical studies

We present a new formulation of the steady state, isothermal, nonlinear reaction-diffusion problem involving nth order reaction kinetics for slab geometry. This results in tractable expressions for the effectiveness factor as a function of the Thiele modulus, the Thiele modulus as a function of the centerline concentration, and the concentration profiles in the slab. The expressions are valid asymptotically in the limit of large orders n. We compare these results with the exact numerical solutions obtained by transforming the nonlinear differential equation into an integral form, using Green's function methods, and solving by successive approximations. The formulation for a membrane is also given, and the nature of the asymmetrical solution discussed. The analysis is facilitated through the introduction of pseudo-reaction orders. A comparison of the asymptotic Thiele modulus obtained herein with a previously given expression shows the present theory to be an improvement.     

Further study of the two-dimensional cable theory: An electric model for a flat thin association of cells with a directional intercellular communication

Summary The two-dimensional cable theory originally presented in relation to the electrotonus along a flat tissue of the rat atrial appendage is improved by taking account of double space constants instead of a single one and an explicit boundary condition at the tip of the current injecting microelectrode. A differential equation is formulated for the membrane potential change which is produced along the tissue by the intracellular injection of a current. The solution is formally expressed in terms of the Green's function. Specific solutions corresponding to the injection of a unit current step or a linearly rising current are discussed in detail.     

Electrostatics of a simple membrane model using Green''s functions formalism.

The electrostatics of a simple membrane model picturing a lipid bilayer as a low dielectric constant slab immersed in a homogeneous medium of high dielectric constant (water) can be accurately computed using the exact Green's functions obtainable for this geometry. We present an extensive discussion of the analysis and numerical aspects of the problem and apply the formalism and algorithms developed to the computation of the energy profiles of a test charge (e.g., ion) across the bilayer and a molecular model of the acetylcholine receptor channel embedded in it. The Green's function approach is a very convenient tool for the computer simulation of ionic transport across membrane channels and other membrane problems where a good and computationally efficient first-order treatment of dielectric polarization effects is crucial.     

The contribution of phosphate-phosphate repulsions to the free energy of DNA bending

DNA bending is important for the packaging of genetic material, regulation of gene expression and interaction of nucleic acids with proteins. Consequently, it is of considerable interest to quantify the energetic factors that must be overcome to induce bending of DNA, such as base stacking and phosphate–phosphate repulsions. In the present work, the electrostatic contribution of phosphate–phosphate repulsions to the free energy of bending DNA is examined for 71 bp linear and bent-form model structures. The bent DNA model was based on the crystallographic structure of a full turn of DNA in a nucleosome core particle. A Green's function approach based on a linear-scaling smooth conductor-like screening model was applied to ascertain the contribution of individual phosphate–phosphate repulsions and overall electrostatic stabilization in aqueous solution. The effect of charge neutralization by site-bound ions was considered using Monte Carlo simulation to characterize the distribution of ion occupations and contribution of phosphate repulsions to the free energy of bending as a function of counterion load. The calculations predict that the phosphate–phosphate repulsions account for ~30% of the total free energy required to bend DNA from canonical linear B-form into the conformation found in the nucleosome core particle.     

Intrinsic time scaling in survival analysis: Application to biological populations

A method of dimensionless time-scaling based on extrinsic expectation of life at birth but intrinsic to a system generating a survival distribution is introduced. Such scaling allows the survival fraction function and its associated mortality function to serve as Green's functions for their generalized equivalents. i.e. a “population” function and a “death” function. The analytical mechanics of utilizing these concepts are formulated, applied to the classical Gompertz and Weibull survival models, and discussed with respect to biological relevance.     

A method of neuronal spike train study based on autoregressive analysis

The method of autoregressive (AR) analysis for neuronal spike trains (NST) is proposed in the paper. The AR model and the Green's function as well as the power spectral density function are used to process and analyse the neuronal interspike interval (ISI) sequences of cat's first somatosensory area of cortex (SI area) under various situations. With these methods the characteristics of the ISI sequence such as the AR order and parameters, memory property, correlativity and periodicity etc. can be extracted.     

The interspike interval of a cable model neuron with white noise input

The firing time of a cable model neuron in response to white noise current injection is investigated with various methods. The Fourier decomposition of the depolarization leads to partial differential equations for the moments of the firing time. These are solved by perturbation and numerical methods, and the results obtained are in excellent agreement with those obtained by Monte Carlo simulation. The convergence of the random Fourier series is found to be very slow for small times so that when the firing time is small it is more efficient to simulate the solution of the stochastic cable equation directly using the two different representations of the Green's function, one which converges rapidly for small times and the other which converges rapidly for large times. The shape of the interspike interval density is found to depend strongly on input position. The various shapes obtained for different input positions resemble those for real neurons. The coefficient of variation of the interspike interval decreases monotonically as the distance between the input and trigger zone increases. A diffusion approximation for a nerve cell receiving Poisson input is considered and input/output frequency relations obtained for different input sites. The cases of multiple trigger zones and multiple input sites are briefly discussed.     

Analytical study on bioheat transfer problems with spatial or transient heating on skin surface or inside biological bodies

Several closed form analytical solutions to the bioheat transfer problems with space or transient heating on skin surface or inside biological bodies were obtained using Green's function method. The solutions were applied to study several selected typical bioheat transfer processes, which are often encountered in cancer hyperthermia, laser surgery, thermal comfort analysis, and tissue thermal parameter estimation. Thus a straightforward way to quantitatively interpret the temperature behavior of living tissues subject to constant, sinusoidal, step, point or stochastic heatings etc. both in volume and on boundary were established. Further solution to the three-dimensional bioheat transfer problems was also given to illustrate the versatility of the present method. Implementations of this study to the practical problems were addressed.     

Green's functions in biological potential problems

Electrical potential problems encountered in biology differ from those usually considered in electrical theory first, because the membranes of tissues satisfy a non-linear relation between current flow and polarization, and second, because the interior of the tissues are not equipotentials. A Green's function suitable for discussing such problems is defined, and a cylindrical illustration of such a function is discussed.     

Cracks emanating from slipping inclusions near a bone-implant interface.

It has been hypothesized that mechanical fracture at the bone-cement-implant interface is the initial cause for loosening of orthopedic implants. Previous investigators have observed cracks to emanate from methacrylate beads, apparently acting as inclusions within the cement. It is believed that the bond between these inclusions and the surrounding matrix breaks prior to emanation of radial cracks from the inclusion. An analytical model is developed for radial cracks emanating from circular inclusions that allow slip along their interface. The solution to the interaction of a single dislocation and a slipping inclusion is used as a Green's Function to model the crack. The Mode I stress intensity factors are calculated for arbitrary orientations of the crack and for varying relative stiffness of the matrix and the inclusion to test feasibility of crack growth.     

Biochemical oscillations and the development of organization

Systems of linear inhomogeneous rate equations for reaction-diffusion kinetics are solved utilizing a Green's integral operator technique. The general theory for the eigenvalues and eigenfunctions of n-component linear systems undergoing diffusion and reaction is derived. The time development of temporal oscillations of the system's reactants is studied. Systemic homogeneity is related to a quantity defined as redundancy. The latter is a function of the system's entropy and a correlation is made between redundancy and noise. It is hypothesized that this correlation may provide a tie between the time development of noise and the time development of pathology in living systems.     

Frequency distributions from birth,death, and creation processes

The time-dependent frequency distribution of groups of individuals versus group size was investigated within a continuum approximation, assuming a simplified individual growth, death and creation model. The analogy of the system to a physical fluid exhibiting both convection and diffusion was exploited in obtaining various solutions to the distribution equation. A general solution was approximated through the application of a Green's function. More specific exact solutions were also found to be useful. The solutions were continually checked against the continuum approximation through extensive simulation of the discrete system. Over limited ranges of group size, the frequency distributions were shown to closely exhibit a power-law dependence on group size, as found in many realizations of this type of system, ranging from colonies of mutated bacteria to the distribution of surnames in a given population. As an example, the modeled distributions were successfully fit to the distribution of surnames in several countries by adjusting the parameters specifying growth, death and creation rates.     

Through-bond electron-transfer interaction in proteins

A model for the through-bond electronic interaction between electron donor and acceptor in proteins is developed. We use a one-electron Hamiltonian, write the Dyson's equation in site representation and solve it by using a Green's function formalism with some renormalization ideas. An expression for Tab which describes the exponential decay with distance bond per bond is obtained. Covalent, non-covalent and convergent pathways are considered and no periodic approximation is needed.     

On the optimal control of fed-batch reactors with substrate-inhibited kinetics

The optimal feed rate profiles, for fed-batch fermentation that maximizes the biomass production and accounts for time, are analyzed. The solution can be found only if the final arc of the optimal control is a batch arc, since in this case the final concentrations of substrate and biomass can be determined by ulterior conditions on the mass balance and on the final growth rate of biomass and thus it is possible to solve the resulting time optimal problem by using Green's theorem. This evidences the "turnpike property" of the solution, which tries to spend the maximum time on or at least near the singular arc along which the substrate concentration is maintained constant. The optimality of the final batch arc is related to the time operational cost in the performance index. The sequence of the control depends on the initial conditions for which six different regions, with the respective patterns, have been identified, in case the performance index allows the control sequence to have a final batch.     

Fröhlich's model of bose condensation in biological systems

Fröhlich's model of Bose-Einstein condensation in biological systems is analyzed using finite temperature Green's function techniques. The thermal average of the rate of change of vibrational mode quanta in the biological system is calculated and compared with Fröhlich's postulated rate equations. The appropriate phonon lifetime is also calculated and represents the minimum time needed to produce the condensed phase. The possible importance of this lifetime and its relation to recent microwave absorption experiments is discussed.     

Vibrational fluctuations of hydrogen bonds in a DNA double helix with nonuniform base pairs.

The Green's function technique is applied to a study of breathing modes in a DNA double helix which contains a region of different base pairs from the rest of the double helix. The calculation is performed on a G-C helix in the B conformation with four consecutive base pairs replaced by A-T. The average stretch in hydrogen bonds is found amplified around the A-T base pair region compared with that of poly(dG)-poly(dC). This is likely related to the A-T regions lower stability against hydrogen bond melting. The A-T region may be considered to be the initiation site for melting in such a helix.