Propagation of action potentials between parallel chains of cardiac muscle cells in PSpice simulation

Propagation of action potentials between parallel chains of cardiac muscle cells was simulated using the PSpice program. Excitation was transmitted from cell to cell along a strand of three or four cells not connected by low-resistance tunnels (gap-junction connexons) in parallel with one or two similar strands. Thus, two models were used: a 2 x 3 model (two parallel chains of three cells each) and a 3 x 4 model (three parallel chains of four cells each). The entire surface membrane of each cell fired nearly simultaneously, and nearly all the propagation time was spent at the cell junctions, thus giving a staircase-shaped propagation profile. The junctional delay time between contiguous cells in a chain was about 0.2-0.5 ms. A significant negative cleft potential develops in the narrow junctional clefts, whose magnitude depends on several factors, including the radial cleft resistance (Rjc). The cleft potential (Vjc) depolarizes the postjunctional membrane to threshold by a patch-clamp action. Therefore, one mechanism for the transfer of excitation from one cell to the next is by the electric field (EF) that is generated in the junctional cleft when the prejunctional membrane fires. Propagation velocity increased with elevation of Rjc. With electrical stimulation of the first cell of the first strand (cell A1), propagation rapidly spread down that chain and then jumped to the second strand (B chain), followed by jumping to the third strand (C chain) when present. The rapidity by which the parallel chains became activated depended on the longitudinal resistance of the narrow extracellular cleft between the parallel strands (Rol2). The higher the Rol2 resistance, the faster the propagation (lower propagation time) over the cardiac muscle sheet (2-dimensional). The transverse resistance of the cleft had no effect. When the first cell of the second strand (cell B1) was stimulated, propagation spread down the B chain and jumped to the other two strands (A and C) nearly simultaneously. When cell C1 was stimulated, propagation traveled down the C chain and jumped to the B chain, followed by excitation of the A chain. Thus, there was transverse propagation of excitation as longitudinal propagation was occurring. Therefore, transmission of excitation by the EF mechanism can occur between myocardial cells lying closely parallel to one another without the requirement of a specialized junction.     

Effects of bath resistance on action potentials in the squid giant axon: myocardial implications.

This study presents a simplified version of the quasi-one-dimensional theory (Wu, J., E. A. Johnson, and J. M. Kootsey. 1996. A quasi-one-dimensional theory for anisotropic propagation of excitation in cardiac muscle. Biophys. J. 71:2427-2439) with two components of the extracellular current, along and perpendicular to the axis, and a simulation and its experimental confirmation for the giant axon of the squid. By extending the one-dimensional core conductor cable equations, this theory predicts, as confirmed by the experiment, that the shapes of the intracellular and the extracellular action potentials are related to the resistance of the bath. Such a result was previously only expected by the field theories. The correlation between the shapes of the intracellular and the extracellular potentials of the giant axon of the squid resembles that observed during the anisotropic propagation of excitation in cardiac muscle. Therefore, this study not only develops a quasi-one-dimensional theory for a squid axon, but also provides one possible factor contributing to the anisotropic propagation of action potentials in cardiac muscle.     

Finite element modeling of impulsive excitation and shear wave propagation in an incompressible,transversely isotropic medium

Elastic properties of materials can be measured by observing shear wave propagation following localized, impulsive excitations and relating the propagation velocity to a model of the material. However, characterization of anisotropic materials is difficult because of the number of elasticity constants in the material model and the complex dependence of propagation velocity relative to the excitation axis, material symmetries, and propagation directions. In this study, we develop a model of wave propagation following impulsive excitation in an incompressible, transversely isotropic (TI) material such as muscle. Wave motion is described in terms of three propagation modes identified by their polarization relative to the material symmetry axis and propagation direction. Phase velocities for these propagation modes are expressed in terms of five elasticity constants needed to describe a general TI material, and also in terms of three constants after the application of two constraints that hold in the limit of an incompressible material. Group propagation velocities are derived from the phase velocities to describe the propagation of wave packets away from the excitation region following localized excitation. The theoretical model is compared to the results of finite element (FE) simulations performed using a nearly incompressible material model with the five elasticity constants chosen to preserve the essential properties of the material in the incompressible limit. Propagation velocities calculated from the FE displacement data show complex structure that agrees quantitatively with the theoretical model and demonstrates the possibility of measuring all three elasticity constants needed to characterize an incompressible, TI material.     

Electrophysiological interaction through the interstitial space between adjacent unmyelinated parallel fibers.

The influence of interstitial or extracellular potentials on propagation usually has been ignored, often through assuming these potentials to be insignificantly different from zero, presumably because both measurements and calculations become much more complex when interstitial interactions are included. This study arose primarily from an interest in cardiac muscle, where it has been well established that substantial interstitial potentials occur in tightly packed structures, e.g., tens of millivolts within the ventricular wall. We analyzed the electrophysiological interaction between two adjacent unmyelinated fibers within a restricted extracellular space. Numerical evaluations made use of two linked core-conductor models and Hodgkin-Huxley membrane properties. Changes in transmembrane potentials induced in the second fiber ranged from nonexistent with large intervening volumes to large enough to initiate excitation when fibers were coupled by interstitial currents through a small interstitial space. With equal interstitial and intracellular longitudinal conductivities and close coupling, the interaction was large enough (induced Vm approximately 20 mV peak-to-peak) that action potentials from one fiber initiated excitation in the other, for the 40-microns radius evaluated. With close coupling but no change in structure, propagation velocity in the first fiber varied from 1.66 mm/ms (when both fibers were simultaneously stimulated) to 2.84 mm/ms (when the second fiber remained passive). Although normal propagation through interstitial interaction is unlikely, the magnitudes of the electrotonic interactions were large and may have a substantial modulating effect on function.     

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.     

Scale of dispersal in varying environments and its implications for life histories of marine invertebrates

Summary We present several models concerning the short term consequences of spreading offspring in varying environments. Our goal is to determine what patterns of spatial and temporal variation yield an advantage to increasing scale of dispersal. Of necessity, the models are somewhat artificial but we feel they are a reasonable approximation of and hence generalizable to natural systems. With these models we examine consequences of dispersal arising from environmental variation: increased environmental variance, different degrees of spatial and temporal correlation, some arbitrary spatial patterns of favorability and finally some patterns derived from long-term, large-scale weather data collected along a contiguous stretch of coastline from southern Oregon to northern Washington (USA). We examine the costs and benefits of increasing sclae of dispersal in both density dependent and density independent models.Several conclusions may be drawn from the results of these models. In the absence of any spatial or temporal order to favorability (where favorability is directly proportional to either fitness or carrying capacity) increasing scale of spread produces a higher tate of population increase. At larger scales, though, an asymptote of maximum relative advantage is approached, so each added increment of spread has a smaller contribution to fitness. This asymptote is higher and the approach to it relatively slower with increasing environmental variance. For a given environmental variance, increasing spatial correlation results in a slower approach to the same asymptote. In density independent models, increasing temporal correlation of fitness selects against increased dispersal if expected differences between sites are sufficiently great relative to variation within sites; but in this instance, density dependence yields a somewhat different result: dispersers have a refuge at sites of low carrying capacity or sites lacking non-dispersers. Finally, optimum intermediate scales of dispersal can occur where differences in expected fitness increase with increasing distance from the parental site, such as in a gradient, but where the environmental variation at a given site is fairly large relative to differences in expected fitness between adjacent sites.The foregoing results are extended for the following predictions. When greater longevity in a resistant phase of the life cycle reduces temporal variation in survival and fecundity, increased generation time should decrease the benefits of spreading offspring in an environment that would otherwise favor spread and could either increase or decrease the costs of spreading offspring in an environment selecting against spread. We speculate that if large scale patterns of varying survival and fecundity are similar to the variation in the physical environment which we examined with weather data, there should be little or no short term advantage to large scale spread of offspring (on the order of 50 kilometers or more) because expected differences increase and seldom if ever decrease with increasing distance between sites.This suggests that feeding larvae of benthic invertebrates with their concomitant long planktonic period, receive little if any advantage from increased scale of dispersal, and consequently that the advantages to planktotrophy over lecithotrophy must lie in other life history aspects, such as the ability to produce a greater number of smaller eggs.Order of authorship alphabetical and by increasing age and height     

Use of the correct heat conduction-convection equation as basis for heat-pulse sap flow methods in anisotropic wood

Heat-pulse methods to determine sap flux density in trees are founded on the theory of heat conduction and heat convection in an isotropic medium. However, sapwood is clearly anisotropic, implying a difference in thermal conductivity along and across the grain, and hence necessitates the theory for an anisotropic medium. This difference in thermal conductivities, which can be up to 50%, is, however, not taken into account in the key equation leading to the currently available heat-pulse methods. Despite this major flaw, the methods remain theoretically correct as they are based on derivations of the key equation, ruling out any anisotropic aspects. The importance of specifying the thermal characteristics of the sapwood according to axial, tangential or radial direction is revealed as well as referring to and using the proper anisotropic theory in order to avoid confusion and misinterpretation of thermal properties when dealing with sap flux density measurements or erroneous results when modelling heat transport in sapwood.     

Analysis of parametric estimation of head tissue conductivities using Electrical Impedance Tomography

We study the theoretical performance of using Electrical Impedance Tomography (EIT) to measure the conductivity of the main tissues of the head. The governing equations are solved using the Finite Element Method for realistically shaped head models with isotropic and anisotropic electrical conductivities. We focus on the Electroencephalography (EEG) signal frequency range since EEG source localization is the assumed application. We obtain the Cramér-Rao Lower Bound (CRLB) to find the minimum conductivity estimation error expected with EIT measurements. The more convenient electrode pairs selected for current injection from a typical EEG array are determined from the CRLB. Moreover, using simulated data, the Maximum Likelihood Estimator of the conductivity parameters is shown to be close to the CRLB for a relatively low number of measurements. The results support the idea of using EIT as a low-cost and practical tool for individually measure the conductivity of the head tissues, and to use them when solving the EEG source localization. Even when the conductivity of the soft tissues of the head is available from Diffusion Tensor Imaging, EIT can complement the electrical model with the estimation of the skull and scalp conductivities.     

Non-Uniform Dispersion of the Source-Sink Relationship Alters Wavefront Curvature

The distribution of cellular source-sink relationships plays an important role in cardiac propagation. It can lead to conduction slowing and block as well as wave fractionation. It is of great interest to unravel the mechanisms underlying evolution in wavefront geometry. Our goal is to investigate the role of the source-sink relationship on wavefront geometry using computer simulations. We analyzed the role of variability in the microscopic source-sink relationship in driving changes in wavefront geometry. The electrophysiological activity of a homogeneous isotropic tissue was simulated using the ten Tusscher and Panfilov 2006 action potential model and the source-sink relationship was characterized using an improved version of the Romero et al. safety factor formulation (SF_m2). Our simulations reveal that non-uniform dispersion of the cellular source-sink relationship (dispersion along the wavefront) leads to alterations in curvature. To better understand the role of the source-sink relationship in the process of wave formation, the electrophysiological activity at the initiation of excitation waves in a 1D strand was examined and the source-sink relationship was characterized using the two recently updated safety factor formulations: the SF_m2 and the Boyle-Vigmond (SF_VB) definitions. The electrophysiological activity at the initiation of excitation waves was intimately related to the SF_m2 profiles, while the SF_VB led to several counterintuitive observations. Importantly, with the SF_m2 characterization, a critical source-sink relationship for initiation of excitation waves was identified, which was independent of the size of the electrode of excitation, membrane excitability, or tissue conductivity. In conclusion, our work suggests that non-uniform dispersion of the source-sink relationship alters wavefront curvature and a critical source-sink relationship profile separates wave expansion from collapse. Our study reinforces the idea that the safety factor represents a powerful tool to study the mechanisms of cardiac propagation in silico, providing a better understanding of cardiac arrhythmias and their therapy.     

Computational Study of Subdural Cortical Stimulation: Effects of Simulating Anisotropic Conductivity on Activation of Cortical Neurons

Subdural cortical stimulation (SuCS) is an appealing method in the treatment of neurological disorders, and computational modeling studies of SuCS have been applied to determine the optimal design for electrotherapy. To achieve a better understanding of computational modeling on the stimulation effects of SuCS, the influence of anisotropic white matter conductivity on the activation of cortical neurons was investigated in a realistic head model. In this paper, we constructed pyramidal neuronal models (layers 3 and 5) that showed primary excitation of the corticospinal tract, and an anatomically realistic head model reflecting complex brain geometry. The anisotropic information was acquired from diffusion tensor magnetic resonance imaging (DT-MRI) and then applied to the white matter at various ratios of anisotropic conductivity. First, we compared the isotropic and anisotropic models; compared to the isotropic model, the anisotropic model showed that neurons were activated in the deeper bank during cathodal stimulation and in the wider crown during anodal stimulation. Second, several popular anisotropic principles were adapted to investigate the effects of variations in anisotropic information. We observed that excitation thresholds varied with anisotropic principles, especially with anodal stimulation. Overall, incorporating anisotropic conductivity into the anatomically realistic head model is critical for accurate estimation of neuronal responses; however, caution should be used in the selection of anisotropic information.     

The bone tissue of the canine mandible is elastically isotropic

This paper reports experimental measurements which show that canine mandibular bone tissue is elastically isotropic. Earlier work has established that human, canine and bovine cortical bone tissue of the femur, tibia and skull are elastically anisotropic and therefore the reported isotropy of mandibular tissue was unexpected. The isotropic elastic moduli of the canine mandible are represented by a Young's modulus of 7.5 GPa and a Poisson's ratio of 0.4. Earlier work gave the three orthotropic Young's moduli of the cortical one of the canine femur as 12.8 GPa, 15.6 GPa and 20.1 GPa. The experimental technique employed is elastic wave propagation at ultrasonic frequencies.     

Diffusion in a Fluid Membrane with a Flexible Cortical Cytoskeleton

We calculate the influence of a flexible network of long-chain proteins, which is anchored to a fluid membrane, on protein diffusion in this membrane. This is a model for the cortical cytoskeleton and the lipid bilayer of the red blood cell, which we apply to predict the influence of the cytoskeleton on the diffusion coefficient of a mobile band 3 protein. Using the pressure field that the cytoskeleton exerts on the membrane, from the steric repulsion between the diffusing protein and the cytoskeletal filaments, we define a potential landscape for the diffusion within the bilayer. We study the changes to the diffusion coefficient on removal of one type of anchor proteins, e.g., in several hemolytic anemias, as well as for isotropic and anisotropic stretching of the cytoskeleton. We predict an overall increase of the diffusion for a smaller number of anchor proteins and increased diffusion for anisotropic stretching in the direction of the stretch, because of the decrease in the spatial frequency as well as in the height of the potential barriers.     

Hypoxia influences generation and propagation of electrical activity in embryonic cardiomyocyte clusters

The influence of tissue hypoxia on the generation and propagation of excitation was studied in spontaneously beating embryonic cardiomyocyte clusters grown in eight 9-12 days old embryoid bodies. Within the embryoid bodies one to three separately active clusters of cardiomyocytes were found, each having its own pacemaker cell. Lowering of tissue PO(2) caused bradycardia as well as arrhythmia in all embryoid bodies investigated. The mean frequency of the extracellularly recorded action potentials decreased under conditions of pronounced hypoxia from a mean of 1.4-1.8 Hz to below 0.8 Hz. In three embryoid bodies hypoxia-sensitive as well as hypoxia-tolerant cardiomyocyte clusters were found. The hypoxia-insensitive cardiomyocytes showed a low frequency of spontaneous activity. In addition to the observed changes in the generation of excitation, tissue hypoxia caused an approximately 60% reduction in the velocity of conduction within the cardiomyocyte clusters. Moreover, in at least one of the eight experiments propagation failure with an incomplete block in spread of excitation was observed. All hypoxia-induced effects on generation and propagation of embryonic cardiomyocyte excitation were completely reversible after reoxygenation.     

Effect of a Single Excitation Stimulus on Photosynthetic Activity and Light-dependent pH Banding in <Emphasis Type="Italic">Chara</Emphasis> Cells

Using pH microelectrodes and a Micro-scopy PAM (pulse-amplitude modulated) chlorophyll fluorometer, it is shown that a propagation of an action potential in Chara corallina leads to transient suppression of spatially periodic pH profiles along the illuminated cell. The suppression was manifested as a large pH decrease in the alkaline zones and a slight pH increase in the acid zones. The propagating action potential diminished the maximum yield of chlorophyll fluorescence (F_m′) in the alkaline cell regions, as well as the quantum yield of photosystem II photochemistry, without affecting F_m′ in the acid cell regions. The results indicate an interference of membrane excitation in the mechanisms responsible for pH banding patterns in Characean algae. Apparently, the electrical excitation of the plasma membrane in the alkaline cell regions initiates a pathway that can modulate membrane events at the thylakoid membrane.     

Nanometer analysis of cell spreading on matrix-coated surfaces reveals two distinct cell states and STEPs

When mouse embryonic fibroblasts in suspension contact a matrix-coated surface, they rapidly adhere and spread. Using total internal reflection fluorescence microscopy of dye-loaded fibroblasts to quantify cell-substrate contact, we found that increasing the surface matrix density resulted in faster spreading initiation whereas lamellipodial dynamics during spreading were unaltered. After spreading initiation, most cells spread in an anisotropic manner through stochastic, transient extension periods (STEPs) with approximately 30 STEPs over 10 min to reach an area of 1300 micro m(2) +/- 300 micro m(2). A second mode of spreading, increased in serum-deprived cells, lacked STEPs and spread in a rapid, isotropic manner for 1-4 min. This isotropic mode was characterized by a high rate of area increase, 340 micro m(2)/min with 78% of the cell edge extending. Anisotropic cells spread slower via STEPs, 126 micro m(2)/min with 34% of the edge extending. During the initial 2-4 min of fast, isotropic spreading, centripetal flow of actin was low (0.8 micro m/min) whereas in anisotropic cells it was high from early times (4.7 micro m/min). After initial isotropic spreading, rearward actin movement increased and isotropic cells displayed STEPs similar to anisotropic cells. Thus, the two cell states display dramatically different spreading whereas long-term motility is based on STEPs.     

A simple theory of peptide interactions on a membrane surface: excluded volume and entropic order

A simple theory of the interactions of peptides bound onto a lipid membrane is developed, modeling the peptides as rods on a surface. At low peptide surface-concentration, excluded volume dominates the peptide-peptide interactions and the orientation of the peptides is random, resulting in an isotropic configuration. However, at high peptide density on the membrane, the peptides become orientationally ordered, resulting in an anisotropic configuration. This effect is entirely entropic in origin, and simply reflects the fact that peptides can be exchanged more easily on the surface if they are equally aligned, resulting in a larger number of possible configurations. In three dimensions, this phenomenon corresponds to the well-known isotropic-nematic phase transition. In two dimensions, the problem is not as well understood. The theoretical treatment presented here yields a simple, manageable expression which can be compared with experimental data. Two-dimensional ordering results in an increase in the apparent binding constant of peptides to membranes at high concentration of peptides relative to what is expected from the effect of excluded volume alone. The possible implications of side-by-side alignment for several biological processes, such as peptide translocation across membranes and plaque formation in Alzheimer's disease, are discussed.     

Conformational changes of colicin Ia channel-forming domain upon membrane binding: a solid-state NMR study

Channel-forming colicins are bactericidal proteins that spontaneously insert into hydrophobic lipid bilayers. We have used magic-angle spinning solid-state nuclear magnetic resonance spectroscopy to examine the conformational differences between the water-soluble and the membrane-bound states of colicin Ia channel domain, and to study the effect of bound colicin on lipid bilayer structure and dynamics. We detected (13)C and (15)N isotropic chemical shift differences between the two forms of the protein, which indicate structural changes of the protein due to membrane binding. The Val C(alpha) signal, unambiguously assigned by double-quantum experiments, gave a 0.6 ppm downfield shift in the isotropic position and a 4 ppm reduction in the anisotropic chemical shift span after membrane binding. These suggest that the alpha-helices in the membrane-bound colicin adopt more ideal helical torsion angles as they spread onto the membrane. Colicin binding significantly reduced the lipid chain order, as manifested by (2)H quadrupolar couplings. These results are consistent with the model that colicin Ia channel domain forms an extended helical array at the membrane-water interface upon membrane binding.     

Membrane-Mediated Interaction between Strongly Anisotropic Protein Scaffolds

Specialized proteins serve as scaffolds sculpting strongly curved membranes of intracellular organelles. Effective membrane shaping requires segregation of these proteins into domains and is, therefore, critically dependent on the protein-protein interaction. Interactions mediated by membrane elastic deformations have been extensively analyzed within approximations of large inter-protein distances, small extents of the protein-mediated membrane bending and small deviations of the protein shapes from isotropic spherical segments. At the same time, important classes of the realistic membrane-shaping proteins have strongly elongated shapes with large and highly anisotropic curvature. Here we investigated, computationally, the membrane mediated interaction between proteins or protein oligomers representing membrane scaffolds with strongly anisotropic curvature, and addressed, quantitatively, a specific case of the scaffold geometrical parameters characterizing BAR domains, which are crucial for membrane shaping in endocytosis. In addition to the previously analyzed contributions to the interaction, we considered a repulsive force stemming from the entropy of the scaffold orientation. We computed this interaction to be of the same order of magnitude as the well-known attractive force related to the entropy of membrane undulations. We demonstrated the scaffold shape anisotropy to cause a mutual aligning of the scaffolds and to generate a strong attractive interaction bringing the scaffolds close to each other to equilibrium distances much smaller than the scaffold size. We computed the energy of interaction between scaffolds of a realistic geometry to constitute tens of k_BT, which guarantees a robust segregation of the scaffolds into domains.