Digital Computer Solutions for Excitation and Propagation of the Nerve Impulse

The Hodgkin-Huxley model of the nerve axon describes excitation and propagation of the nerve impulse by means of a nonlinear partial differential equation. This equation relates the conservation of the electric current along the cablelike structure of the axon to the active processes represented by a system of three rate equations for the transport of ions through the nerve membrane. These equations have been integrated numerically with respect to both distance and time for boundary conditions corresponding to a finite length of squid axon stimulated intracellularly at its midpoint. Computations were made for the threshold strength-duration curve and for the repetitive firing of propagated impulses in response to a maintained stimulus. These results are compared with previous solutions for the space-clamped axon. The effect of temperature on the threshold intensity for a short stimulus and for rheobase was determined for a series of values of temperature. Other computations show that a highly unstable subthreshold propagating wave is initiated in principle by a just threshold stimulus; that the stability of the subthreshold wave can be enhanced by reducing the excitability of the axon as with an anesthetic agent, perhaps to the point where it might be observed experimentally; but that with a somewhat greater degree of narcotization, the axon gives only decrementally propagated impulses.     

Bayesian image processing of data from constrained source distributions—I. non-valued,uncorrelated and correlated constraints

A series of Bayesian image processing algorithms which incorporate various classes ofa priori source information in treating data which obeys Poisson and Gaussian statistics is derived using maximum entropy considerations. The standard maximum likelihood equations are shown to be a special case of Bayesian image processing when thea priori information about a source distribution φ _j is solely that a non-vanishing probability for each element value φ _j exists only in some finite interval,a _j ≤φ _j ≤φ _j . Bayesian image processing equations for thea priori source information that all φ _j are finite -∞<φ _j <∞ and each φ _j distribution has a defined mean φ _j and a defined variance σ _j are derived. The Bayesian image processing equations are also derived when thea priori source information is that all φ _j ≥0 and that each φ _j distribution has a defined mean φ _j and a defined variance σ _j . The a priori source distribution constraint that a correlation exists among nearby elements is also considered. The results indicate improvement over standard methods.     

定量分析调控人类免疫缺陷病毒潜伏与激活的动力学机制

目的 治疗艾滋病最大的障碍在于无法根除人类免疫缺陷病毒（HIV）潜伏于人体细胞所形成的病毒存储库。构建描述病毒存储库建立分子机制的动力学模型需考虑生物体内的噪声环境和多重影响因素,本文通过一种全新的动力学结构分解方法将随机微分方程的确定性部分与随机性噪声分开,从而在仅需分析常微分方程不动点的情况下即可判断不同药物靶点的作用效果。方法 使用连续的随机微分方程构建了HIV转录过程的动力学模型,简化了描述系统所需方程的维度,增大了模型的可探索空间,在此基础上,通过计算得到的势能函数和概率分布函数直观表示病毒潜伏与激活的不同表达状态以及它们之间的关系。结果 定量分析了不同动力学参数对系统稳态和势函数的影响程度,分别得到了系统处于双稳态和单稳态时的参数范围,并将不同因素对动力系统分岔的影响程度与生物学实验结果对比,验证了本工作的理论基础。结论 本文突破了以往离散、随机的方法,可以通过常微分方程定量分析HIV转录调控的动力学机制,有利于推广到处理高维情况,进一步研究艾滋病在生物体内的发生发展,从而指导设计实验寻找临床上的治疗方案。     

Transport across the outer membrane of Escherichia coli K12 via the FhuA receptor is regulated by the TonB protein of the cytoplasmic membrane

Summary Point mutations in the “TonB box” offhuA were suppressed by point mutations in thetonB gene, suggesting both a functional and physical interaction between the FhuA receptor protein in the outer membrane and the TonB protein in the cytoplasmic membrane ofEscherichia coli K12. Mutations influA were classified into four types according to the extent by which they impaired mutant cells in their growth on ferrichrome as sole iron source and in their sensitivity to the antibiotic albomycin and to colicin M. ThetonB mutation with a glutamine to leucine replacement at position 165 was less efficient in restoring the FhuA functions than the glutamine to lysine exchange at the same position. The better the coupling between FhuA and TonB the poorer was the inhibition of phage T1 binding to FhuA by ferrichrome. A working model is proposed in which the TonB protein assumes different conformations in response to the energized state of the cytoplasmic membrane and thereby allosterically regulates the activity of the FhuA receptor. This model implies an intermembrane coupling between two proteins in adjacent membranes.     

Piecewise-linear Models of Genetic Regulatory Networks: Equilibria and their Stability

A formalism based on piecewise-linear (PL) differential equations, originally due to Glass and Kauffman, has been shown to be well-suited to modelling genetic regulatory networks. However, the discontinuous vector field inherent in the PL models raises some mathematical problems in defining solutions on the surfaces of discontinuity. To overcome these difficulties we use the approach of Filippov, which extends the vector field to a differential inclusion. We study the stability of equilibria (called singular equilibrium sets) that lie on the surfaces of discontinuity. We prove several theorems that characterize the stability of these singular equilibria directly from the state transition graph, which is a qualitative representation of the dynamics of the system. We also formulate a stronger conjecture on the stability of these singular equilibrium sets.     

Possibility of electron tunneling through a nerve membrane

From a quantum mechanical standpoint, electron tunneling may occur in a nerve axon because the nerve membrane (60 to 100 angstroms) is thin enough for the tunnel effect and because the external solution and the axoplasma are redox systems which can provide electrons. Calculations and diagrams of the density-of-states are given to predict theN-shaped current-voltage characteristics which have been observed in the studies of squid giant axons, of the reconstituted liquid membranes and of the marine algae.     

Response properties of the glossopharyngeal taste system of the mud puppy (Necturus maculosus)

Summary The responses of individual glossopharyngeal neurons of the mud puppy,Necturus maculosus, were examined over an extended concentration series of NaCl, HCl, quinine hydrochloride (QHCl) and sucrose solutions. Neuron isolation was evaluated by a computer program that sorted neural impulses according to amplitude (Fig. 1). When sufficient isolation existed, a second program counted the impulses in each test period as well as in pre- and post-stimulus periods. Response latencies were calculated independently.The response to taste stimulation took one of three forms: 1) increased activity, 2) decreased activity, or 3) increased activity to the water rinse. For each concentration series the magnitude (SR) and latency functions of the responses were determined. These varied among stimuli and among nerve fibers (Fig. 4). However, the SR and latency functions were found in specific combinations, the most unique being one in which both functions remained constant over an entire concentration series (Fig. 4E, F).Most neurons responded to more than one of the stimuli. Many, however, responded to at least one of the stimuli with a particular form of response and combination of SR and latency functions (Fig. 6). In this sense they may be considered chemospecific as well as multiply sensitive. Despite the many types of response, the sum of the individual SR functions closely resembled the SR functions of the whole nerve (Fig. 7) and the summed latencies produced a temporal pattern with a phasic component similar to that of the whole nerve response (Fig. 8).Abbreviations SR stimulus concentration vs response magnitude - L latency of response - +,0, -,I (SR orL) magnitude or latency functions with positive, zero, negative or indeterminate slope - QHCl quinine hydrochloride Supported in part by NIH Grant NS09168The authors wish to thank Mr. Marc Schneider for the computer programs and his expert assistance and Mrs. G. Chapman of Florida State University for photography. We also thank Dr. David V. Smith for his critical comments. This work was submitted in partial fulfillment of the requirements for the Ph. D. degree to Michigan State University by the senior author.     

Reduced transmitter release conferred by mutations in theslowpoke-encoded Ca2+-activated K+ channel gene ofDrosophila

Potassium channels control the repolarization of nerve terminals and thus play important roles in the control of synaptic transmission. Here we describe the effects of mutations in theslowpoke gene, which is the structural gene for a calcium activated potassium channel, on transmitter release at the neuromuscular junction inDrosophila melanogaster. Surprisingly, we find that theslowpoke mutant exhibits reduced transmitter release compared to normal. Similarly, theslowpoke mutation significantly suppresses the increased transmitter release conferred either by a mutation inShaker or by application of 4-aminopyridine, which blocks theShaker-encoded potassium channel at theDrosophila nerve terminal. Furthermore, theslowpoke mutation suppresses the striking increase in transmitter release that occurs following application of 4-aminopyridine to theether a go-go mutant. This suppression is most likely the result of a reduction of Ca²⁺ influx into the nerve terminal in theslowpoke mutant. We hypothesize that the effects of theslowpoke mutation are indirect, perhaps resulting from increased Ca²⁺ channel inactivation, decreased Na⁺ or Ca²⁺ channel localization or gene expression, or by increases in the expression or activity of potassium channels distinct fromslowpoke.     

Parameter estimation in Hodgkin-Huxley-Type equations for membrane action potentials in nerve and heart muscle

A method is presented which renders parameter estimation possible in systems of non-linear differential equations where normally no solution exists in terms of analytic functions and which have to be solved numerically. The method uses the concept of sensitivity equations. Two examples are given, taking mathematical models for membrane action potentials in nerve and heart muscle by Hodgkin and Huxley and by Beeler and Reuter. The model equations together with the corresponding system of sensitivity equations are given, which are necessary to estimate maximum conductivity coefficients defining the interactions of different ionic current components. A computer program is described and results of action potential numerical analysis are presented using simulated data. It can be seen, that even with superimposed simulated noise the real parameter values are estimated in an excellent manner. The method can be used to interpret observed changes in action potential time courses under physiological and pharmacological conditions.     

A numerical solution of the mechanical bidomain model

Introduction: The mechanical bidomain model predicts forces on integrin proteins in the membrane. It has been solved analytically for idealized examples, but a numerical algorithm is needed to address realistic problems. Methods: The bidomain equations are approximated using finite differences. An ischemic region is modeled as a circular area having no active tension, surrounded by normal tissue. Results: The membrane force is large in the ischemic border zone, but is small elsewhere. Strain is distributed widely throughout the ischemic region and surrounding tissue. Conclusion: This calculation provides a testable prediction for the mechanism of mechanotransduction and remodeling in cardiac tissue.     

A Lyapunov function for piecewise-independent differential equations: stability of the ideal free distribution in two patch environments

In this article we construct Lyapunov functions for models described by piecewise-continuous and independent differential equations. Because these models are described by discontinuous differential equations, the theory of Lyapunov functions for smooth dynamical systems is not applicable. Instead, we use a geometrical approach to construct a Lyapunov function. Then we apply the general approach to analyze population dynamics describing exploitative competition of two species in a two-patch environment. We prove that for any biologically meaningful parameter combination the model has a globally stable equilibrium and we analyze this equilibrium with respect to parameters.      

Asymptotics of the total net direct pharmacological effect for large drug doses

The direct pharmacological effect E is described by the E _max model relating E to the drug plasma concentration C _p . The area under the effect vs. time curve (AUC _E ) is used as the measurement of the total net pharmacological effect. The drug plasma concentrations are solutions of compartmental systems of ordinary differential equations with the input terminated after a finite time and controlled in a proportional manner by a single dose-like parameter. The asymptotics of the time derivative of C _p for large doses are derived and used as conditions which have to be satisfied by functions for which the asymptotics of the integral defining AUC _E are derived. The AUC _E is proportional to the time T _C>EC50 for which the drug concentration stays above the threshold level EC ₅₀. The threshold EC ₅₀ denotes the drug plasma concentration which elicits 50% of the maximum effect. The parameter T _C>EC50 is proportional to the logarithm of drug dose for large doses and its asymptotics is calculated up to the order o(1) as dose increases to infinity. The results are applied to basic pharmacokinetic systems. Received: 7 December 1999 / Revised version: 23 May 2000 / Published online: 23 October 2000     

Non-smooth plant disease models with economic thresholds

In order to control plant diseases and eventually maintain the number of infected plants below an economic threshold, a specific management strategy called the threshold policy is proposed, resulting in Filippov systems. These are a class of piecewise smooth systems of differential equations with a discontinuous right-hand side. The aim of this work is to investigate the global dynamic behavior including sliding dynamics of one Filippov plant disease model with cultural control strategy. We examine a Lotka–Volterra Filippov plant disease model with proportional planting rate, which is globally studied in terms of five types of equilibria. For one type of equilibrium, the global structure is discussed by the iterative equations for initial numbers of plants. For the other four types of equilibria, the bounded global attractor of each type is obtained by constructing appropriate Lyapunov functions. The ideas of constructing Lyapunov functions for Filippov systems, the methods of analyzing such systems and the main results presented here provide scientific support for completing control regimens on plant diseases in integrated disease management.     

Dimensional analysis of nerve models

General equations for (i) a uniform patch of nerve membrane, (ii) a continuous (unmyelinated) axon and (iii) a noded (myelinated) axon are analyzed using dimensional analysis. The original dimensioned equations are transformed to dimensionless equations. These equations contain dimensionless constants called similarity parameters which are functions of the physical constants or parameters of the system. (The similarity parameters are analogous to such quantities as the Reynolds number and Mach number used in fluid dynamics.) There is one similarity parameter for each of cases (i) and (ii), and four for case (iii). All dimensioned systems having the same values of all the similarity parameters form a similarity class.Once a quantity such as threshold stimulus or conduction velocity is computed for one member of any similarity class, the same quantity can be easily computed for any other member of the same class, by a simple formula containing the physical constants of the system, called a generating equation, or by an even simpler expression of proportionality, called a scaling relation.     

Method for evaluating metabolic functions of drugs in bioartificial liver

Lidocaine and galactose loading tests were performed on a bioartificial liver (BAL), an extracorporeal medical device incorporating living hepatocytes in a cartridge without a transport barrier across the membranes. The concentration changes were analyzed using pharmacokinetic equations to evaluate the efficacy and limitation of the proposed method. Lidocaine and galactose were found to be suitable drugs for a quantitative evaluation of the BAL functions, as they did not interact with the plasma proteins or blood vessels, making their concentrations easy to determine. The drug concentration changes after drug loading were easily analyzed using pharmacokinetic equations, and the BAL functions quantitatively expressed by pharmacokinetic parameters, such as the clearance (CL) and galactose elimination capacity (GEC). In addition, these two drugs have already been used in clinical tests to evaluate human liver functions over long periods, and lidocaineCL values andGEC values reported for a normal human liver. Thus, a comparison of theCL andGEC values for theBAL and a natural liver revealed what proportion of normal liver functions could be replaced by the BAL.     

A methodological basis for description and analysis of systems with complex switch-like interactions

 A wide range of complex systems appear to have switch-like interactions, i.e. below (or above) a certain threshold x has no or little influence on y, while above (or below) this threshold the effect of x on y saturates rapidly to a constant level. Switching functions are frequently described by sigmoid functions or combinations of these. Within the context of ordinary differential equations we present a very general methodological basis for designing and analysing models involving complicated switching functions together with any other non-linearities. A procedure to determine position and stability properties of all stationary points lying close to a threshold for one or several variables, so-called singular stationary points, is developed. Such points may represent homeostatic states in models, and are therefore of considerable interest. The analysis provides a profound insight into the generic effects of steep sigmoid interactions on the dynamics around homeostatic points. It leads to qualitative as well as quantitative predictions without using advanced mathematical methods. Thus, it may have an important heuristic function in connection with numerical simulations aimed at unfolding the predictive potential of realistic models. Received 25 January 1996; received in revised form 29 June 1997     

Extracellular potentials of a single myelinated nerve fiber in an unbounded volume conductor

The extracellular potentials of a single myelinated nerve fiber in an unbounded volume conductor were studied. The spatial distribution of the transmembrane potential was obtained by integrating the system of partial differential equations characterizing the electric processes in the active myelinated nerve fiber. The spatial distribution of the extracellular potentials at various radial distances in the volume conductor were calculated using the line source model. Up to a certain radial distance (500 m) the discontinuity of the action potential propagation is reflected in the extracellular potentials, while further in the volume conductor the potentials are smooth. The effect of the fiber diameter and the internodal distance on the volume conductor potentials as well as the changes in the magnitude of the extracellular potential (in the time domain) between two adjacent nodes at various radial distances were studied. The radial decline of the peak-to-peak amplitude of the extracellular potential depends on the radial coordinater of the field point and increases with the increase ofr.     

A mathematical note on the fahraeuslindqvist effect in power law fluid

An axisymmetric flow of a power law fluid through circular tubes under constant pressure gradient with the flow parameters varying radially is analyzed theoretically. The main finding is that for the Fahraeus-Lindqvist (F-L) effect to occur, it is necessary to have at least one of the parametersK (consistency) andn (index) as a discontinuous function ofr in the absence of wall slip; and with slip condition the parameters could be continuous functions ofr under specific conditions. In both the cases the existence of more than one discontinuity cannot be ruled out. The results obtained are consistent with experimental findings of blood flow through narrow tubes.