Effect of the form and anisotropy of the left ventricle on the drift of spiral waves

Three-dimensional spiral waves of electrical excitation in the myocardium are sources of dangerous cardiac arrhythmias. In this work, the dynamics of spiral waves of electrical excitation were studied in a symmetric anatomical model of the human heart left ventricle and a realistic ionic cell model of the human ventricular myocardium. Three factors that affect the drift waves in the heart were compared for the first time: the geometry of the heart wall, myocardial anisotropy, and wave chirality. Cardiac anisotropy was identified as a main factor in determining the drift of spiral waves. In the isotropic case, the dynamics were determined by the wall thickness, but did not depend on the wave chirality. In the anisotropic case, chirality was found to play a crucial role.

     

Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias

We introduce the concept of a contracting excitable medium that is capable of conducting non-linear waves of excitation that in turn initiate contraction. Furthermore, these kinematic deformations have a feedback effect on the excitation properties of the medium. Electrical characteristics resemble basic models of cardiac excitation that have been used to successfully study mechanisms of reentrant cardiac arrhythmias in electrophysiology. We present a computational framework that employs electromechanical and mechanoelectric feedback to couple a three-variable FitzHugh–Nagumo-type excitation-tension model to the non-linear stress equilibrium equations, which govern large deformation hyperelasticity. Numerically, the coupled electromechanical model combines a finite difference method approach to integrate the excitation equations, with a Galerkin finite element method to solve the equations governing tissue mechanics. We present example computations demonstrating various effects of contraction on stationary rotating spiral waves and spiral wave break. We show that tissue mechanics significantly contributes to the dynamics of electrical propagation, and that a coupled electromechanical approach should be pursued in future electrophysiological modelling studies.     

Generation of ion-acoustic and magnetoacoustic waves in an RF helicon discharge

A study is made of the generation of ion-acoustic and magnetoacoustic waves in a discharge excited in an external magnetic field by an electromagnetic wave in the whistler frequency range (ω_LH ? ω ? ω_He, where ω_LH = $\sqrt {\omega _{He} \omega _{Hi} } $ and ω_He and ω_Hi are the electron and ion gyrofrequencies, respectively). The excitation of acoustic waves is attributed to the decay of a high-frequency hybrid mode forming a plasma waveguide into low-frequency acoustic waves and new high-frequency waves that satisfy both the decay conditions and the waveguide dispersion relations. The excitation of acoustic waves is resonant in character because the conditions for the generation of waveguide modes and for the occurrence of the corresponding nonlinear wave processes should be satisfied simultaneously. An unexpected effect is the generation of magnetoacoustic waves by whistlers. A diagnostic technique is proposed that allows one to determine the thermal electron velocity by analyzing decay conditions and dispersion relations for waves in the discharge channel.     

Silicon nanoparticles synthesized using a microwave method and used as a label‐free fluorescent probe for detection of VB12

A simple and rapid detection strategy for vitamin B₁₂ (VB₁₂) was established based on label‐free silicon quantum dots (SiQDs); the detection mechanism was additionally investigated. SiQDs were synthesized using a one‐step microwave method, and their fluorescence was stronger than that synthesized using the hydrothermal method. SiQDs fluorescence was quenched using VB₁₂ due to the inner filter effect (IFE), which was demonstrated using ultraviolet (UV) absorption spectra, fluorescence lifetime, transmission electron microscopy and zeta potential analysis. Subsequently, quercetin (Que) and doxorubicin (Dox) with absorption peaks that overlapped the excitation or emission peaks of SiQDs respectively were used as control groups to investigate the quenching mechanism. Results showed that quenching efficiency was related to the level of overlap between the adsorption peak of the quencher and the excitation or emission peaks of SiQDs. A greater level of overlap caused a higher quenching efficiency. Therefore, the sensitive quenching of VB₁₂ for SiQDs was due to the synergistic effect of the synchronous overlap between the absorption peak of VB₁₂ with the excitation and emission peaks of SiQDs. Fluorescence quenching efficiency increased linearly in the 0.5 to 16.0 μmol·L^?1 VB₁₂ concentration range, and the detection limit was 158 nmol·L^?1. In addition, SiQDs were applied to determine VB₁₂ in tablets and human urine samples with satisfactory recoveries ranging from 97.7 to 101.1%.     

Electrophysiological heterogeneity and stability of reentry in simulated cardiac tissue

Generation of wave break is a characteristic feature of cardiac fibrillation. In this study, we investigated how dynamic factors and fixed electrophysiological heterogeneity interact to promote wave break in simulated two-dimensional cardiac tissue, by using the Luo-Rudy (LR1) ventricular action potential model. The degree of dynamic instability of the action potential model was controlled by varying the maximal amplitude of the slow inward Ca(2+) current to produce spiral waves in homogeneous tissue that were either nearly stable, meandering, hypermeandering, or in breakup regimes. Fixed electrophysiological heterogeneity was modeled by randomly varying action potential duration over different spatial scales to create dispersion of refractoriness. We found that the degree of dispersion of refractoriness required to induce wave break decreased markedly as dynamic instability of the cardiac model increased. These findings suggest that reducing the dynamic instability of cardiac cells by interventions, such as decreasing the steepness of action potential duration restitution, may still have merit as an antifibrillatory strategy.     

Reentry in heterogeneous cardiac tissue described by the Luo-Rudy ventricular action potential model

Heterogeneity of cardiac tissue is an important factor determining the initiation and dynamics of cardiac arrhythmias. In this paper, we studied the effects of gradients of electrophysiological heterogeneity on reentrant excitation patterns using computer simulations. We investigated the dynamics of spiral waves in a two-dimensional sheet of cardiac tissue described by the Luo-Rudy phase 1 (LR1) ventricular action potential model. A gradient of action potential duration (APD) was imposed by gradually varying the local current density of K(+) current or inward rectifying K(+) current along one axis of the tissue sheet. We show that a gradient of APD resulted in spiral wave drift. This drift consisted of two components. The longitudinal (along the gradient) component was always directed toward regions of longer spiral wave period. The transverse (perpendicular to the gradient) component had a direction dependent on the direction of rotation of the spiral wave. We estimated the velocity of the drift as a function of the magnitude of the gradient and discuss its implications.     

Towards accurate numerical method for monodomain models using a realistic heart geometry

The simulation of cardiac electrophysiological waves are known to require extremely fine meshes, limiting the applicability of current numerical models to simplified geometries and ionic models. In this work, an accurate numerical method based on a time-dependent anisotropic remeshing strategy is presented for simulating three-dimensional cardiac electrophysiological waves. The proposed numerical method greatly reduces the number of elements and enhances the accuracy of the prediction of the electrical wave fronts. Illustrations of the performance and the accuracy of the proposed method are presented using a realistic heart geometry. Qualitative and quantitative results show that the proposed methodology is far superior to the uniform mesh methods commonly used in cardiac electrophysiology.     

Initiation and propagation of a neuronal intracellular calcium wave

The ability to image calcium movement within individual neurons inspires questions of functionality including whether calcium entry into the nucleus is related to genetic regulation for phenomena such as long term potentiation. Calcium waves have been initiated in hippocampal pyramidal cells with glutmatergic signals both in the presence and absence of back propagating action potentials (BPAPs). The dendritic sites of initiation of these calcium waves within about 100 μm of the soma are thought to be localized near oblique junctions. Stimulation of synapses on oblique dendrites leads to production of inositol 1,4,5-trisphosphate (IP₃) which diffuses to the apical dendrite igniting awaiting IP₃ receptors (IP₃Rs) and initiating and propagating catalytic calcium release from the endoplasmic reticulum. We construct a reduced mathematical system which accounts for calcium wave initiation and propagation due to elevated IP₃. Inhomogeneity in IP₃ distribution is responsible for calcium wave initiation versus subthreshold or spatially uniform suprathreshold activation. However, the likelihood that a calcium wave is initiated does not necessarily increase with more calcium entering from BPAPs. For low transient synaptic stimuli, timing between IP₃ generation and BPAPs is critical for calcium wave initiation. We also show that inhomogeneity in IP₃R density can account for calcium wave directionality. Simulating somatic muscarinic receptor production of IP₃, we can account for the critical difference between calcium wave entry into the soma and failure to do so.     

Damage induced arrhythmias: mechanisms and implications

Little is known about the role played by non-uniform myocardial stress and strain distributions and by non-uniform excitation contraction coupling in mechanisms underlying the premature beats that initiate an arrhythmia. We will review the evidence in support of a mechanism in which both non-uniform contraction and increased Ca2+ load of cells adjacent to acutely damaged cells are essential in the "spontaneous" generation of Ca2+ transients during the relaxation phase of the electrically driven twitch. The putative mechanism of initiation of the propagating Ca2+ waves involves feedback of rapid length (or force) changes to dissociation of Ca2+ from the contractile filaments. A novel aspect of this concept is that these mechanically elicited Ca2+ transients induce propagating Ca2+ waves that travel into the adjacent normal myocardium and cause after-depolarizations, which, in turn, may cause premature action potentials. These premature action potentials will further load the cells with Ca2+, which promotes the subsequent generation of propagating Ca2+ transients and leads to triggered arrhythmias. The damage-induced premature beats may also initiate re-entry arrhythmias in non-uniform myocardium. These observations strongly support the concept that abnormal cellular Ca2+ transport plays a crucial role in the initiation of arrhythmias in damaged and non-uniform myocardium.     

Mechanisms of ventricular arrhythmias elicited by coexistence of multiple electrophysiological remodeling in ischemia: A simulation study

Myocardial ischemia, injury and infarction (MI) are the three stages of acute coronary syndrome (ACS). In the past two decades, a great number of studies focused on myocardial ischemia and MI individually, and showed that the occurrence of reentrant arrhythmias is often associated with myocardial ischemia or MI. However, arrhythmogenic mechanisms in the tissue with various degrees of remodeling in the ischemic heart have not been fully understood. In this study, biophysical detailed single-cell models of ischemia 1a, 1b, and MI were developed to mimic the electrophysiological remodeling at different stages of ACS. 2D tissue models with different distributions of ischemia and MI areas were constructed to investigate the mechanisms of the initiation of reentrant waves during the progression of ischemia. Simulation results in 2D tissues showed that the vulnerable windows (VWs) in simultaneous presence of multiple ischemic conditions were associated with the dynamics of wave propagation in the tissues with each single pathological condition. In the tissue with multiple pathological conditions, reentrant waves were mainly induced by two different mechanisms: one is the heterogeneity along the excitation wavefront, especially the abrupt variation in conduction velocity (CV) across the border of ischemia 1b and MI, and the other is the decreased safe factor (SF) for conduction at the edge of the tissue in MI region which is attributed to the increased excitation threshold of MI region. Finally, the reentrant wave was observed in a 3D model with a scar reconstructed from MRI images of a MI patient. These comprehensive findings provide novel insights for understanding the arrhythmic risk during the progression of myocardial ischemia and highlight the importance of the multiple pathological stages in designing medical therapies for arrhythmias in ischemia.     

Theoretical analysis of the magnetocardiographic pattern for reentry wave propagation in a three-dimensional human heart model

We present a computational study of reentry wave propagation using electrophysiological models of human cardiac cells and the associated magnetic field map of a human heart. We examined the details of magnetic field variation and related physiological parameters for reentry waves in two-dimensional (2-D) human atrial tissue and a three-dimensional (3-D) human ventricle model. A 3-D mesh system representing the human ventricle was reconstructed from the surface geometry of a human heart. We used existing human cardiac cell models to simulate action potential (AP) propagation in atrial tissue and 3-D ventricular geometry, and a finite element method and the Galerkin approximation to discretize the 3-D domain spatially. The reentry wave was generated using an S1-S2 protocol. The calculations of the magnetic field pattern assumed a horizontally layered conductor for reentry wave propagation in the 3-D ventricle. We also compared the AP and magnetocardiograph (MCG) magnitudes during reentry wave propagation to those during normal wave propagation. The temporal changes in the reentry wave motion and magnetic field map patterns were also analyzed using two well-known MCG parameters: the current dipole direction and strength. The current vector in a reentry wave forms a rotating spiral. We delineated the magnetic field using the changes in the vector angle during a reentry wave, demonstrating that the MCG pattern can be helpful for theoretical analysis of reentry waves.     

Nonlinear propagation of agonist-induced cytoplasmic calcium waves in single astrocytes

In astrocytes in primary culture, activation of neurotransmitter receptors results in intracellular calcium signals that propagate as waves across the cell. Similar agonist-induced calcium waves have been observed in astrocytes in organotypic cultures in response to synaptic activation. By using primary cultured astrocytes grown on glass coverslips, in conjunction with fluorescence microscopy we have analyzed agonist-induced Ca²⁺ wave initiation and propagation in individual cells. Both norepinephrine and glutamate elicited Ca²⁺ signals which were initiated focally and discretely in one region of the cell, from where the signals spread as waves along the entire length of the cell. Analysis of the wave propagation and the waveform revealed that the propagation was nonlinear with one or more focal loci in the cytoplasm where the wave was regeneratively amplified. These individual loci appear as discrete focal areas 7–15 μm in diameter and having intrinsic oscillatory properties that differ from each other. The wave initiation locus and the different amplification loci remained invariant in space during the course of the experiment and supported an identical spatiotemporal pattern of signalling in any given cell in response to multiple agonist applications and when stimulated with different agonists which are coupled via InsP₃. Cytoplasmic Ca²⁺ concentration at rest was consistently higher (17 ± 4nM, mean ± S.E.M.) in the wave initiation locus compared with the rest of the cytoplam. The nonlinear propagation results from significant changes in signal rise times, amplitudes, and wave velocity in cellular regions of active loci. Analysis of serial slices across the cell revealed that the rise times and amplitudes of local signals were as much as three- to fourfold higher in the loci of amplification. A phenomenon of hierarchy in local amplitudes of the signal in the amplification loci was observed with the wave initiation locus having the smallest and the most distal locus having largest amplitude. By this mechanism locally very high concentrations of Ca²⁺ are achieved in strategic locations in the cell in response to receptor activation. While the average wave velocity calculated over the length of the cell was 10–15 μm/s, in the active loci rates as high as 40 μm/s were measured. Wave velocity was fivefold lower in regions of the cell separating active loci. The differences in the intrinsic oscillatory periods give rise to local Ca²⁺ waves that show the properties of collision and annihilation. It is hypothesized that the wave front provokes regenerative Ca²⁺ release from specialized areas in the cell where the endoplasmic reticulum is endowed with higher density of InsP₃ receptor channels. Thus wave propagation is achieved by a process of diffusion and regenerative Ca²⁺ release in multiple cellular loci provoked by the advancing wave front; in this way, wave propagation is nonlinear and saltatory. Regenerative Ca²⁺ wave propagation from distal atrocytic processes to the cell body and neighboring cells is likely to provide an important signalling mechanism in the nervous system. 1994 John Wiley & Sons, Inc.     

An image-based model of calcium waves in differentiated neuroblastoma cells

Calcium waves produced by bradykinin-induced inositol-1,4, 5-trisphosphate (InsP(3))-mediated release from endoplasmic reticulum (ER) have been imaged in N1E-115 neuroblastoma cells. A model of this process was built using the "virtual cell," a general computational system for integrating experimental image, biochemical, and electrophysiological data. The model geometry was based on a cell for which the calcium wave had been experimentally recorded. The distributions of the relevant cellular components [InsP(3) receptor (InsP(3)R)], sarcoplasmic/endoplasmic reticulum calcium ATPase (SERCA) pumps, bradykinin receptors, and ER] were based on 3D confocal immunofluorescence images. Wherever possible, known biochemical and electrophysiological data were used to constrain the model. The simulation closely matched the spatial and temporal characteristics of the experimental calcium wave. Predictions on different patterns of calcium signals after InsP(3) uncaging or for different cell geometries were confirmed experimentally, thus helping to validate the model. Models in which the spatial distributions of key components are altered suggest that initiation of the wave in the center of the neurite derives from an interplay of soma-biased ER distribution and InsP(3) generation biased toward the neurite. Simulations demonstrate that mobile buffers (like the indicator fura-2) significantly delay initiation and lower the amplitude of the wave. Analysis of the role played by calcium diffusion indicated that the speed of the wave is only slightly dependent on the ability of calcium to diffuse to and activate neighboring InsP(3) receptor sites.     

Reentrant waves in a ring of embryonic chick ventricular cells imaged with a Ca2+ sensitive dye

According to the classic model initially formulated by Mines, reentrant cardiac arrhythmias may be associated with waves circulating in a ring geometry. This study was designed to study the dynamics of reentry in a ring geometry of cardiac tissue culture. Reentrant calcium waves in rings of cultured embryonic chick cardiac myocytes were imaged using a macroscope to monitor the fluorescence of intracellular Calcium Green-1 dye. The rings displayed a variety of stable rhythms including pacemaker activity and spontaneous reentry. Waves originating from a localized pacemaker could lead to reentry as a consequence of unidirectional block. In addition, more complex patterns were observed due to the interactions between reentrant and pacemaker rhythms. These rhythms included instances in which pacemakers accelerated the reentrant rhythm, and instances in which the excitation was blocked in the vicinity of pacemakers. During reentrant activity an appropriately timed electrical stimulus could induce resetting of activity or cause complete annihilation of the propagating waves. This experimental preparation reveals many spontaneously occuring complex rhythms. These complex rhythms are hypothesized to reflect interactions between spontaneous pacemakers, wave propagation, refractory period, and overdrive suppression. This preparation may serve as a useful model system to further investigate complex dynamics arising during reentrant rhythms in cardiac tissue.     

Spiral waves of spreading depression in the isolated chicken retina

Existence of the theoretically predicted spiral waves of excitation in intact two-dimensional networks of excitable elements has been experimentally confirmed in the isolated chicken retina. The preparation supports the waves of Leão's spreading depression (SD) the concentric propagation of which from the point of origin can be directly observed as a change of the optical properties of the retinal tissue. The propagation rate of 3.7 mm/min (35°C) decreased to 1.5 mm/min for SD waves elicited during relative refractory period. When a several-mm long segment of the SD wave had been blocked by anodal polarization, the laterally opened ends of the wavefront started to spread after termination of polarization into the previously blocked tissue, gradually turning around and penetrating into the region recovering from the original SD. One or two simultaneously generated spiral waves of SD continued to rotate for several cycles. Spiral SD could also be elicited by punctiform cathodal polarization (1 mA) applied to the SD wave-rear. Since the new SD wave could only spread into the recovering tissue it formed a laterally open wavefront, the free ends of which eventually turned around and started spiral SD. With continued reverberation the nucleus of the spiral SD wave gradually migrated across the retina until it approached an obstacle (e.g., pecten) which stopped further spiral propagation. Spiral SD waves were elicited in 31 retinal preparations and lasted for 4.5 cycles on the average. Average cycle duration was 4.7 min. Spontaneous spiral SD waves were observed in preparations incubated in Mg²⁺-free media. The spiral SD waves in retina are compared with mathematical models of analogous phenomena. It is argued that spiral SD waves probably exist in the cerebral cortex of rats and account for generation of repetitive SD waves sometimes elicited by overlapping stimulation of two cortical regions.     

Action potential duration dispersion and alternans in simulated heterogeneous cardiac tissue with a structural barrier

Structural barriers to wave propagation in cardiac tissue are associated with a decreased threshold for repolarization alternans both experimentally and clinically. Using computer simulations, we investigated the effects of a structural barrier on the onset of spatially concordant and discordant alternans. We used two-dimensional tissue geometry with heterogeneity in selected potassium conductances to mimic known apex-base gradients. Although we found that the actual onset of alternans was similar with and without the structural barrier, the increase in alternans magnitude with faster pacing was steeper with the barrier--giving the appearance of an earlier alternans onset in its presence. This is consistent with both experimental structural barrier findings and the clinical observation of T-wave alternans occurring at slower pacing rates in patients with structural heart disease. In ionically homogeneous tissue, discordant alternans induced by the presence of the structural barrier arose at intermediate pacing rates due to a source-sink mismatch behind the barrier. In heterogeneous tissue, discordant alternans occurred during fast pacing due to a barrier-induced decoupling of tissue with different restitution properties. Our results demonstrate a causal relationship between the presence of a structural barrier and increased alternans magnitude and action potential duration dispersion, which may contribute to why patients with structural heart disease are at higher risk for ventricular tachyarrhythmias.     

Agonist‐evoked Ca2+ wave progression requires Ca2+ and IP3

Smooth muscle responds to IP₃‐generating agonists by producing Ca²⁺ waves. Here, the mechanism of wave progression has been investigated in voltage‐clamped single smooth muscle cells using localized photolysis of caged IP₃ and the caged Ca²⁺ buffer diazo‐2. Waves, evoked by the IP₃‐generating agonist carbachol (CCh), initiated as a uniform rise in cytoplasmic Ca²⁺ concentration ([Ca²⁺]_c) over a single though substantial length (～30 µm) of the cell. During regenerative propagation, the wave‐front was about 1/3 the length (～9 µm) of the initiation site. The wave‐front progressed at a relatively constant velocity although amplitude varied through the cell; differences in sensitivity to IP₃ may explain the amplitude changes. Ca²⁺ was required for IP₃‐mediated wave progression to occur. Increasing the Ca²⁺ buffer capacity in a small (2 µm) region immediately in front of a CCh‐evoked Ca²⁺ wave halted progression at the site. However, the wave front does not progress by Ca²⁺‐dependent positive feedback alone. In support, colliding [Ca²⁺]_c increases from locally released IP₃ did not annihilate but approximately doubled in amplitude. This result suggests that local IP₃‐evoked [Ca²⁺]_c increases diffused passively. Failure of local increases in IP₃ to evoke waves appears to arise from the restricted nature of the IP₃ increase. When IP₃ was elevated throughout the cell, a localized increase in Ca²⁺ now propagated as a wave. Together, these results suggest that waves initiate over a surprisingly large length of the cell and that both IP₃ and Ca²⁺ are required for active propagation of the wave front to occur. J. Cell. Physiol. 224: 334–344, 2010. © 2010 Wiley‐Liss, Inc.     

Enhancement of daptomycin production in <Emphasis Type="Italic">Streptomyces roseosporus</Emphasis> LC-51 by manipulation of cofactors concentration in the fermentation culture

The effects of eight cofactors of enzymes on daptomycin production were investigated in this work, which included nicotinic acid (VPP), riboflavin (VB₂), heme, thiamine (VB₁), biotin (VH), cyanocobalamin (VB₁₂), tetrahydrofolic acid (THF) and pyridoxal 5-phosphate (VB₆). The dry cell weight (DCW), consumption of glucose, and daptomycin production were obviously improved when proper amount of exogenous cofactors were supplemented in the medium. The effects of heme, THF, VB₁₂ and VB₆ on daptomycin production were especially notable. The daptomycin yield enhanced 363, 104, 53 and 46%, respectively, when optimized amount of these four cofactors were supplemented in the broth. Moreover, the daptomycin yield further increased to 632 mg/l, which was over 4.5-fold higher than that of the control (without cofactors), at 132 h in a 7.5-l fermenter, by supplementation all of the eight cofactors at optimized concentrations (VPP 4 mg/l, VB₂ 0.5 mg/l, heme 9 mg/l, VB₁ 0.4 mg/l, VH 0.1 mg/l, VB₁₂ 0.04 mg/l, THF 6 mg/l and VB₆ 0.4 mg/l). Further, the effects of cofactors on the corresponding key enzymes and important intracellular metabolites were studied in order to elucidate the mechanism of enhancement of daptomycin production by manipulation of cofactors concentration in the fermentation culture. It is suggested that this strategy for increasing the daptomycin production in Streptomyces roseosporus LC-51 by manipulation of cofactors concentration in the fermentation culture may provide an alternative approach to enhance the production of metabolites in other Streptomyces.     

Slow Conduction in Cardiac Muscle: A Biophysical Model

Mechanisms of slow conduction in cardiac muscle are categorized and the most likely identified. Propagating action potentials were obtained experimentally from a synthetically grown strand of cardiac muscle (around 50 μm by 30 mm) and theoretically from a one-dimensional cable model that incorporated varying axial resistance and membrane properties along its length. Action potentials propagated at about 0.3 m/s, but in some synthetic strands there were regions (approximately 100 μm in length) where the velocity decreased to 0.002 m/s. The electrophysiological behavior associated with this slow conduction was similar to that associated with slow conduction in naturally occurring cardiac muscle (notches, Wenckebach phenomena, and block). Theoretically, reasonable changes in specific membrane capacitance, membrane activity, and various changes in geometry were insufficient to account for the observed slow conduction velocities. Conduction velocities as low as 0.009 m/s, however, could be obtained by increasing the resistance (r_i) of connections between the cells in the cable; velocities as low as 0.0005 m/s could be obtained by a further increase in r_i made possible by a reduction in membrane activity by one-fourth, which in itself decreased conduction velocity by only a factor of 1/1.4. As a result of these findings, several of the mechanisms that have been postulated, previously, are shown to be incapable of accounting for delays such as those which occur in the synthetic strand as well as in the atrioventricular (VA) node.