One way traffic of pulses in a neuron

In this paper we investigate the effect of a change in geometry of a nerve axon on the propagation of potential waves along the axon. In particular we show that potential waves are stopped at a sudden large increase of cross-section area such as increase of diameter or branching. Some special examples are treated. The results do also apply to problems in population genetics and chemical reaction theory.     

Remark on some correspondences between the characteristics of the nerve impulse and those of ferroelectric polarization

Ferroelectric polarization currents have characteristics which may be of use in the study of the nerve impulse either directly or indirectly through the suggestion of parallel experiments in the two domains. For example, the “all or none” aspect of the excitation of the nerve impulse can be considered to correspond in some sense to the coercive force required for the reversal of a ferroelectric polarization. The refractory period then corresponds to the lack of a relaxation process in ferroelectric polarization; for a second application of voltage in a given direction does not produce a ferroelectric current; it is necessary that the polarization be reversed by a reversal of voltage. The “heat block” in nerve conduction may correspond to a ferroelectric Curie point at about 40°C. Velocities of propagation of ferroelectric polarization are of the order of magnitude of the velocities of sound and therefore in a range suitable for the interpretation of the observed velocities of nerve impulses. These and other parallels between ferroelectric behavior and the characteristics of the nerve impulse suggest that there may be a useful degree of similarity between the molecular processes responsible for the nerve impulse and those responsible for ferroelectric polarization. *** DIRECT SUPPORT *** A01E2080 00002     

ELECTRIC IMPEDANCE OF THE SQUID GIANT AXON DURING ACTIVITY

Alternating current impedance measurements have been made over a wide frequency range on the giant axon from the stellar nerve of the squid, Loligo pealii, during the passage of a nerve impulse. The transverse impedance was measured between narrow electrodes on either side of the axon with a Wheatstone bridge having an amplifier and cathode ray oscillograph for detector. When the bridge was balanced, the resting axon gave a narrow line on the oscillograph screen as a sweep circuit moved the spot across. As an impulse passed between impedance electrodes after the axon had been stimulated at one end, the oscillograph line first broadened into a band, indicating a bridge unbalance, and then narrowed down to balance during recovery. From measurements made during the passage of the impulse and appropriate analysis, it was found that the membrane phase angle was unchanged, the membrane capacity decreased about 2 per cent, while the membrane conductance fell from a resting value of 1000 ohm cm.² to an average of 25 ohm cm.² The onset of the resistance change occurs somewhat after the start of the monophasic action potential, but coincides quite closely with the point of inflection on the rising phase, where the membrane current reverses in direction, corresponding to a decrease in the membrane electromotive force. This E.M.F. and the conductance are closely associated properties of the membrane, and their sudden changes constitute, or are due to, the activity which is responsible for the all-or-none law and the initiation and propagation of the nerve impulse. These results correspond to those previously found for Nitella and lead us to expect similar phenomena in other nerve fibers.     

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.     

The nerve impulse in the axon — a new theory

The Classical Theory of function in the nervous system postulates that the nerve impulse is the result of a sequential reversal of the membrane potential due to an increased permeability of the membrane, first to sodium ions, then to potassium ions. The new theory presents a bio-physical model which depicts the nerve impulse as an event involving the motions of electrons and waves, and their interactions with sodium and potassium atoms and ions. The velocity of the nerve impulse (the most important parameter of nerve function) is determined by the product of two constants: c = the speed of light, which is a constant for all nerves; k =a constant for each nerve and is believed to be a specific property of nerve matter related in some way to the atomic process. The theory proposes that the nerve impulse in the axon is dualistic in nature (particles and waves play equally significant roles). The dualistic nature accounts for the three most fundamental characteristics of conduction of the nerve impulse: periodicity (conduction of a nerve impulse over long distances with constant velocity and form); non-summing (two nerve impulses cannot be in the same place at the same time); quantum nature of each nerve impulse — i.e., the unit message of the nerve impulse is an indivisible unit.     

Effect of vascular volume and edema on wave propagation in canine lungs

The velocities of longitudinal and transverse stress waves transmitted through inflated lung parenchyma depend on the lung stiffness, as defined by the bulk and shear moduli, and the lung density. We examined the relationship between stress wave velocities and lung density. A saline-filled reservoir was connected to the vessels of caudal dog lobes held inflated at 5 cmH2O transpulmonary pressure, and vascular volume and extravascular lung water were increased in steps by increasing vascular pressure. At each step, we measured the transmitted signals at locations 2 and 7 cm from an impulse surface distortion by means of microphones embedded in the lung surface. Longitudinal and transverse wave velocities were computed by using cross-correlation analysis of microphone signal pairs. Both wave velocities decreased as lung density increased: as a first approximation, wave velocities were inversely proportional to the square root of lung density. This behavior is consistent with the propagation of small-amplitude stress waves through an elastic continuum. Estimated bulk and shear moduli were 26 and 3.5 cmH2O, respectively, and were consistent with results from quasi-static deformation tests.     

The Creation of Proton Hopping from a Drug?Receptor Encounter

We extend recent modeling studies of proton hopping, used to describe the functioning of membrane channels and axon nerve conduction, to offer an explanation of the initiation of the nerve impulse at an effector? ligand encounter. This encounter is proposed to create a hydronium ion in the vicinity of the effector and ligand, which leads to a continuous flow of protons, called proton hopping, through water adjacent to this encounter. This proton hopping is proposed to be the message carried from the encounter to the axon of a particular nerve system associated with that particular effector? ligand system.     

Longitudinal elastic wave propagation in pulmonary parenchyma

Consideration of the lung as an elastic continuum led us to investigate the possible propagation of elastic waves. Here the relevant stiffness and density are given by the Lamé constants and density of the parenchyma. To test this hypothesis, we measured propagation velocities (c) in dog lobes by recording transit times of a velocity impulse on one side of the lobe and the subsequent arrival on the other side. We compared our measured values of c with elastic longitudinal wave velocities (c long) predicted by values of elastic moduli given by Lai-Fook et al. (J. Appl. Physiol. 40: 508-513, 1976) as a function of translobar pressure (PL) and our measured densities. Good agreement was found between c and c long. Typical values of c ranged from 250-1,500 cm/s as PL ranged from 2-20 cmH2O. No systematic difference in the c-c long relation was found between inflation and deflation, suggesting that the elastic moduli of lungs are essentially a function of pressure. No significant effect was observed by changing the physical properties of the gas within the lobe [air vs. He vs. sulfur hexafluoride (SF6)], suggesting that indeed we were observing waves associated with the coupling of parenchymal density to parenchymal stiffness.     

Conduction in demyelinated axons—A simplified model

This paper is concerned with conduction of the nervous impulse in a myelinated axon and the effect of demyelination on conduction characteristics. A model of nerve conduction called the “gunpowder fuse” model is presented which accurately predicts conduction velocities in both myelinated and unmyelinated nerves. The effect on conduction velocity in this model by reducing myelin thickness is examined by utilizing basic data and building and equivalent circuit. The result is a curve relating reduced conduction velocity to reduced myelin thickness. A similar analysis and resultant curve is derived from a saltatory conduction model. Supported in part by National Multiple Sclerosis Society Research Grant No. 516 and Air Force Grant AFOSR 669-67.     

Anisotropic propagation of Ca2+ waves in isolated cardiomyocytes.

Digital imaging microscopy of fluor-3 fluorescence was used to study the propagation of intracellular Ca2+ waves in isolated adult rat cardiomyocytes from 17 to 37 degrees C. Ca2+ waves spread in both transverse and longitudinal direction of a myocyte. Transverse propagation was pronounced in waves starting from a focus at the edge of a myocyte and in waves following an irregular, curved path (spiral waves). For the former type of waves, propagation velocities were determined. Both transverse and longitudinal wave components propagated at constant velocity ranging from 30 to 125 micron/s. Myocytes were anisotropic with respect to wave propagation: waves propagated faster in the longitudinal than in the transverse direction. The ratio between longitudinal and transverse velocity increased from 1.30 at 17 degrees C to 1.55 at 37 degrees C. Apparent activation energies for transverse and longitudinal wave propagation were estimated to be -20 kJ/mol, suggesting that these processes are limited by diffusion of Ca2+. Direction-dependent propagation velocities are interpreted to result from the highly ordered structure of the myocytes, especially from the anisotropic arrangement of diffusion obstacles such as myofilaments and mitochondria.     

An alternative ultrasonic method for measuring the elastic properties of cortical bone

We studied the elastic properties of bone to analyze its mechanical behavior. The basic principles of ultrasonic methods are now well established for varying isotropic media, particularly in the field of biomedical engineering. However, little progress has been made in its application to anisotropic materials. This is largely due to the complex nature of wave propagation in these media. In the present study, the theory of elastic waves is essential because it relates the elastic moduli of a material to the velocity of propagation of these waves along arbitrary directions in a solid. Transducers are generally placed in contact with the samples which are often cubes with parallel faces that are difficult to prepare. The ultrasonic method used here is original, a rough preparation of the bone is sufficient and the sample is rotated. Moreover, to analyze heterogeneity of the structure we measure velocities in different points on the sample. The aim of the present study was to determine in vitro the anisotropic elastic properties of cortical bones. For this purpose, our method allowed measurement of longitudinal and transverse velocities (C(L) and C(T)) in longitudinal (fiber direction) and the radial directions (orthogonal to the fiber direction) of compact bones. Young's modulus E and Poisson's ratio nu, were then deduced from the velocities measured considering the compact bone as transversely isotropic or orthotropic. The results are in line with those of other methods.     

Measurements of the elasticity of monolayers consisting of lipids from nerve membranes

For measurement of viscoelastic properties of monolayer-covered interfaces a longitudinal wave is generated in the plane of the interface, using a horizontal oscillating barrier. The wave propagation depends on the values of the viscoelastic parameters of the monolayer. The technique is applied here to study the surface elasticity of layers consisting of lipids extracted from nerve membranes. It is concluded that mechanical disturbances are propagated as longitudinal waves. The possibility that longitudinal waves occur in nerve membranes and the role they might play in the transmission of information in biological membranes is discussed.     

ON THE THEORY OF NERVOUS CONDUCTION

Assuming that the propagation of the nervous impulse consists in the excitation of adjacent regions of the nerve by the action current of the already excited region, exact equations for the velocity of such a propagation are established and integrated. The result depends on the assumptions which we make about the laws of excitation. If Hoorweg''s law is accepted, it is found that the velocity of propagation decreases exponentially with time, and that there is a limiting distance which the impulse will travel and which cannot be exceeded. If however a set of equations proposed by L. Lapique is assumed to govern the process of excitation, we find that the velocity of propagation asymptotically reaches a constant value.     

Generation of the nervous impulse and periodic oscillations

Usually the models for the excitation and propagation of the nervous impulse are studied either in the space-clamp situation or on a model axon extended on both sides to infinity. Following Fitzhugh in the present paper the release of an impulse train at the axon hillock is studied within the scope of Fitzhugh's BVP model. The existence and stability of periodic oscillations are studied by direct methods, also the relation to Liénard's equation. The exact correspondence between the BVP model and the socalled Nagumo-equation is established. For typical examples the solutions are computed by numerical methods.     

Propagation of stress waves in inflated sheep lungs

If the lung is an elastic continuum, both longitudinal and transverse stress waves should be propagated in the medium with distinct velocities. In five isolated sheep lungs, we investigated the propagation of stress waves. The lungs were degassed and then inflated to a constant transpulmonary pressure (Ptp). We measured signals transmitted at locations approximately 1.5, 6, and 11 cm from an impulse surface distortion with the use of small microphones embedded in the pleural surface. Two transit times were computed from the first two significant peaks of the cross-correlation of microphone signal pairs. The "fast" wave velocities averaged 301 +/- 92, 445 +/- 80, and 577 +/- 211 (SD) cm/s for Ptp values of 5, 10, and 15 cmH2O, respectively. Corresponding "slow" wave velocities averaged 139 +/- 22, 217 +/- 36, and 255 +/- 89 cm/s. The fast waves were consistent with longitudinal waves of velocity [(K + 4G/3)/p]1/2, where bulk modulus K = 4 Ptp and shear modulus G = 0.7 Ptp. The slow waves were consistent with transverse (and/or Rayleigh) waves of velocity (G/p)1/2, with a G value of 0.9 Ptp. Measured values of K were 5 Ptp and values of G measured by indentation tests were 0.7 Ptp. Thus, stress wave velocities measured on pleural surface of isolated lungs correlated well with elastic moduli of lung parenchyma.     

The “Lillie Transition”: models of the onset of saltatory conduction in myelinating axons

Almost 90 years ago, Lillie reported that rapid saltatory conduction arose in an iron wire model of nerve impulse propagation when he covered the wire with insulating sections of glass tubing equivalent to myelinated internodes. This led to his suggestion of a similar mechanism explaining rapid conduction in myelinated nerve. In both their evolution and their development, myelinating axons must make a similar transition between continuous and saltatory conduction. Achieving a smooth transition is a potential challenge that we examined in computer models simulating a segmented insulating sheath surrounding an axon having Hodgkin-Huxley squid parameters. With a wide gap under the sheath, conduction was continuous. As the gap was reduced, conduction initially slowed, owing to the increased extra-axonal resistance, then increased (the “rise”) up to several times that of the unmyelinated fiber, as saltatory conduction set in. The conduction velocity slowdown was little affected by the number of myelin layers or modest changes in the size of the “node,” but strongly affected by the size of the “internode” and axon diameter. The steepness of the rise of rapid conduction was greatly affected by the number of myelin layers and axon diameter, variably affected by internode length and little affected by node length. The transition to saltatory conduction occurred at surprisingly wide gaps and the improvement in conduction speed persisted to surprisingly small gaps. The study demonstrates that the specialized paranodal seals between myelin and axon, and indeed even the clustering of sodium channels at the nodes, are not necessary for saltatory conduction.     

Electromechanical coupling of waves in nerve fibres

The propagation of an action potential (AP) in a nerve fibre is accompanied by mechanical and thermal effects. In this paper, an attempt is made to build up a mathematical model which couples the AP with a possible pressure wave (PW) in the axoplasm and waves in the nerve fibre wall (longitudinal—LW and transverse—TW) made of a lipid bilayer (biomembrane). A system of differential equations includes the governing equations of single waves with coupling forces between them. The single equations are kept as simple as possible in order to carry out the proof of concept. An assumption based on earlier studies is made that the coupling forces depend on changes (the gradient, time derivative) of the voltage. In addition, it is assumed that the transverse displacement of the biomembrane can be calculated from the gradient of the LW in the biomembrane. The computational simulation is focused to determining the influence of possible coupling forces on the emergence of mechanical waves from the AP. As a result, an ensemble of waves (AP, PW, LW, TW) emerges. The further experiments should verify assumptions about coupling forces.     

On the velocity of conduction in nerve fibers with saltatory transmission

Using an electrical model to represent certain features of a nerve fiber together with a one-factor theory of excitation, an expression is obtained for the velocity of propagation of a nerve impulse along a nerve fiber exhibiting saltatory transmission. The velocity shows a maximum with respect to internodal length. A critical internodal length, a critical radius, and a critical value for the resistance of the external medium are required in order to make transmission of the impulse possible.     

The velocity of conduction in nerve fiber and its electric characteristics

The theory developed in this paper shows that the propagation of spike potential along a nerve fiber and the conduction of an electric wave along an inert inorganic conductor follow a common quantitative relationship. This result gives further support to the belief that propagation of excitation is an electrical process. The basic idea of the theory is derived from the consideration that velocity has, by its mathematical definition, a local meaning; conduction in a nerve is completely determined by the local characteristics of the latter, as well as those of the wave. The final formula derived does not make use of any other field of science beyond the fundamental principles of electricity. It gives the conduction velocity in terms of the electric characteristics of the fiber and of the duration of the spike potential. The formula is in agreement with the known dependence of the conduction velocity on various parameters characterizing the axon. The computed velocity agrees with the measured ones on the squid giant axon, crab nerve axon, frog muscle fiber and Nitella cell. The membrane inductance appears as a velocity controling agent which prevents also a possible distortion of the spike potential during conduction. The structural meaning of the electric characteristics of the axon membrane is discussed from the viewpoint of the diffusion theory. A formula for the velocity of spread of the electrotonus is also derived.     

神经元轴突生长的力生物学

神经系统作为一个复杂的体系,在其发育过程中轴突需要延伸较长的距离才能与下一级神经元或靶细胞形成突触。在这个复杂的移动过程中,神经元轴突在空间分布上形成了精确有序的结构。过去认为这种有序结构的形成主要由形态发生素的化学浓度梯度来指导,而最近的研究发现力学因素对调控轴突的延伸速度与方向发挥着重要的作用。因此,轴突的延伸本质上是一个力化学耦合过程。本文将结合自己过去的工作论述力学因素对轴突延伸的调控机制及相关的信号转导。这一领域的研究将为认识对神经系统疾病的发生以及神经再生提供重要的参考。