Propagation Of Electrical Signals Research Articles

Targeting the Cardiac Sodium Channel to Increase Excitability of Stem-Cell Derived Cardiomyocytes

The cardiogenic potential of stem-cell derived cardiomyocytes (iPSC-CM) and their prospective use for cardiac cell therapy crucially depends on their excitability and functional integration in myocardial tissue. Indeed, previous studies from our group have shown that cell excitability and intercellular coupling are strongly reduced in iPSC-CMs compared to primary cardiomyocytes. For clinical aspects, impaired excitability and electrical signal propagation may lead to conduction slowing and the development of arrhythmia. In this project, we focus on the idea that cardiomyocyte excitability and conduction properties are interrelated and depend on the expression of the cardiac sodium channel Nav1.5 and the major gap junction forming protein connexin-43 (Cx43). We tested the hypothesis that molecular remodeling of both proteins enhances cell excitability as reflected in Nav1.5 activity and action potential (AP) properties with the aim to approach native cardiomyocyte function. Using a combination of molecular modulation and electrophysiological evaluation in voltage and current clamp modes, our data demonstrate that enhanced Nav1.5 expression in iPSC-CMs significantly increased sodium current (INa in pA/pF: control 40.5±10.5, Nav1.5 118.3±27.2) and upstroke velocity (dv/dtmax, in V/s: 156±18 vs. 276±29, respectively) of the AP, a critical determinant of cell excitability. Typically, a fraction of iPSC-CMs also exhibited spontaneous APs with low dv/dtmax (<50 V/s) driven without Nav1.5, a hallmark of immaturity. However, after Nav1.5 overexpression, all recorded APs showed fast dv/dtmax. Furthermore, INa was also increased in Cx43-overexpressing iPSC-CMs (INa 66.2±19.8 pA/pF) suggesting that Cx43 may influence Nav1.5 expression and thereby cell excitability. This notion was further confirmed in immunostainings of Cx43-overexpressing iPSC-CMs demonstrating increased Nav1.5 expression at the plasma membrane, which suggests a common regulation pathway between both proteins. In conclusion, modulation of Nav1.5 and Cx43 expression greatly enhances the excitability of iPSC-CMs and may represent a powerful new target for improving the functional maturation of iPSC-CMs.

Toxins That Affect Voltage-Gated Sodium Channels.

Voltage-gated sodium channels (VGSCs) are critical in generation and conduction of electrical signals in multiple excitable tissues. Natural toxins, produced by animal, plant, and microorganisms, target VGSCs through diverse strategies developed over millions of years of evolutions. Studying of the diverse interaction between VGSC and VGSC-targeting toxins has been contributing to the increasing understanding of molecular structure and function, pharmacology, and drug development potential of VGSCs. This chapter aims to summarize some of the current views on the VGSC-toxin interaction based on the established receptor sites of VGSC for natural toxins.

Electrotonic signal transduction between &lt;em&gt;Aloe vera&lt;/em&gt; plants using underground pathways in soil: Experimental and analytical study

Plants communicate with other plants using different pathways: (1) volatile organic compounds’ (VOC) emission and sensing; (2) mycorrhizal networks in the soil; (3) the plants’ rhizosphere; (4) electrostatic or electromagnetic interactions; (5) roots of the same species can sometimes naturally graft. Here we show that there is an additional pathway for electrical signal transduction between neighboring plants: fast underground electrical signal propagation between roots through the soil. The mathematical model of electrical signal transduction between plants and the analytical study are supported by experimental data. The pulse train, sinusoidal and a triangular saw-shape voltage profiles were used for electrostimulation of plants and underground electrotonic signal transmission between plants. Electrostimulation of a leaf in Aloe vera by 1.5 V D-batteries or by a function generator induces electrotonic potentials propagation in the electrostimulated plants and the neighboring plants. The amplitude and sign of electrotonic potentials in both electrostimulated and neighboring Aloe vera plants depend on the amplitude, rise and fall of the applied voltage. Electrostimulation by a sinusoidal wave from a function generator induces an electrical response in leaves with a phase shift. Experimental results show cell-to-cell electrical coupling and existence of electrical differentiators in the leaves of Aloe vera. Electrostimulation is an important tool for the evaluation of mechanisms of communication between plants.

84-, 100-, and 107-GBd PAM-4 Intensity-Modulation Direct-Detection Transceiver for Datacenter Interconnects

We report on an ultrahigh-speed PAM-based IM/DD optical subsystem featuring compact signal generation and tractable digital-processing complexity. The proposed transceiver is based on two main technologies: an InP DHBT selector power DAC for electrical signal generation, and a complexity-optimized receiver processing chain. We show how the complexity of the feed-forward equalizer can be considerably reduced by stacking a <3-symbol memory sequence estimator. Finally, we experimentally demonstrate successful signal recovery for 107-GBd PAM-4 in back-to-back configuration, and after 500 m and 1 km of transmission over standard single-mode fiber for 100- and 84-GBd PAM-4 configuration, respectively. A power-budget margin above 2 dB is attained in all cases.

The role of relative membrane capacitance and time delay in cerebellar Purkinje cells.

The membrane capacitance of a neuron can influence the synaptic efficacy and the speed of electrical signal propagation. Exploring the role of membrane capacitance will help facilitate a deeper understanding of the electrical properties of neurons. Thus, in this paper, we investigated the neuronal firing behaviors of a two-compartment model in Purkinje cells. We evaluated the influence of membrane capacitance under two different circumstances: in the absence of time delay and in the presence of time delay. Firstly, we separately studied the influence of somatic membrane capacitance Cs and dendritic membrane capacitance Cd on neuronal firing patterns. Through numerical simulation, we observed that they had two different types of period-adding scenarios, i.e. with and without chaotic bursting. Secondly, our results indicated that when the time delay was included in the model, periodic motions were more inclined to be destroyed, while at the same time, corresponding new chaotic motions were induced. These findings suggested that membrane capacitance and time delay play a pivotal functional role in modulating dynamical firing properties of neurons, especially aspects which lead to behaviors which result in changes to bursting patterns.

The role of electrical and jasmonate signalling in the recognition of captured prey in the carnivorous sundew plant Drosera capensis.

The carnivorous sundew plant (Drosera capensis) captures prey using sticky tentacles. We investigated the tentacle and trap reactions in response to the electrical and jasmonate signalling evoked by different stimuli to reveal how carnivorous sundews recognize digestible captured prey in their traps. We measured the electrical signals, phytohormone concentration, enzyme activities and Chla fluorescence in response to mechanical stimulation, wounding or insect feeding in local and systemic traps. Seven new proteins in the digestive fluid were identified using mass spectrometry. Mechanical stimuli and live prey induced a fast, localized tentacle-bending reaction and enzyme secretion at the place of application. By contrast, repeated wounding induced a nonlocalized convulsive tentacle movement and enzyme secretion in local but also in distant systemic traps. These differences can be explained in terms of the electrical signal propagation and jasmonate accumulation, which also had a significant impact on the photosynthesis in the traps. The electrical signals generated in response to wounding could partially mimic a mechanical stimulation of struggling prey and might trigger a false alarm, confirming that the botanical carnivory and plant defence mechanisms are related. To trigger the full enzyme activity, the traps must detect chemical stimuli from the captured prey.

Investigation of Collective Behaviour and Electrocommunication in the Weakly Electric Fish, Mormyrus rume, through a biomimetic Robotic Dummy Fish

A robotic fish has been developed to create a mixed bio-hybrid system made up of weakly electric fish and a mobile dummy fish. Weakly electric fish are capable of interacting with each other via sequences of self-generated electric signals during electrocommunication. Here we present the design of an artificial dummy fish, which is subsequently tested in behavioural experiments. The robot consists of two parts: a flexible tail that can move at different frequencies and amplitudes, performing a carangiform oscillation, and a rigid head containing the motor for the tail oscillation. The dummy fish mimics the weakly electric fish Mormyrus rume in morphology, size and electric signal generation. In order to study electrical interactions, the dummy fish is equipped with ten electrodes that record electric signals of nearby real fish and generate electric dipole fields around itself that are similar to those produced by real fish in both waveform and sequence. Behavioural experiments demonstrate that the dummy fish is able to recruit both single individuals and groups of M. rume from a shelter into an exposed area. The development of an artificial dummy fish may help to understand fundamental aspects of collective behaviour in weakly electric fish and the properties necessary to initiate and sustain it in closed-loop feedback experiments based on electrocommunication.

Cardiac sodium channel, its mutations and their spectrum of arrhythmia phenotypes

The mechanisms of cellular excitability and propagation of electrical signals in the cardiac muscle are very important functionally and pathologically. The heart is constituted by three types of muscle: atrial, ventricular, and specialized excitatory and conducting fi bers. From a physiological and pathophysiological point of view, the conformational states of the sodium channel during heart function constitute a signifi cant aspect for the diagnosis and treatment of heart diseases. Functional states of the sodium channel (closed, open, and inactivated) and their structure help to understand the cardiac regulation processes. There are areas in the cardiac muscle with anatomical and functional differentiation that present automatism, thus subjecting the rest of the fi bers to their own rhythm. The rate of these (pacemaker) areas could be altered by modifi cations in ions, temperature and especially, the autonomic system. Excitability is a property of the myocardium to react when stimulated. Another electrical property is conductivity, which is characterized by a conduction and activation process, where the action potential, by the all-or-nothing law, travels throughout the heart. Heart relaxation also stands out as an active process, dependent on the energetic output and on specificion and enzymatic actions, with the role of sodium channel being outstanding in the functional process. In the gene mutation aspects that encode the rapid sodium channel (SCN5A gene), this channel is responsible for several phenotypes, such as Brugada syndrome, idiopathic ventricular fibrillation, dilated cardiomyopathy, early repolarization syndrome, familial atrial fibrillation, variant 3 of long QT syndrome, multifocal ectopic ventricular contractions originating in Purkinje arborizations, progressive cardiac conduction defect (Lenègre disease), sudden infant death syndrome, sick sinus syndrome, sudden unexplained nocturnal death syndrome, among other sodium channel alterations with clinical overlapping. Finally, it seems appropriate to consider the “sodium channel syndrome” (mutations in the gene of the α subunit of the sodium channel, SCN5A gene) as a single clinical entity that may manifest in a wide range of phenotypes, to thus have a better insight on these cardiac syndromes and potential outcomes for their clinical treatment.

Recent advances in animal and human pluripotent stem cell modeling of cardiac laminopathy.

Laminopathy is a disease closely related to deficiency of the nuclear matrix protein lamin A/C or failure in prelamin A processing, and leads to accumulation of the misfold protein causing progeria. The resultant disrupted lamin function is highly associated with abnormal nuclear architecture, cell senescence, apoptosis, and unstable genome integrity. To date, the effects of loss in nuclear integrity on the susceptible organ, striated muscle, have been commonly associated with muscular dystrophy, dilated cardiac myopathy (DCM), and conduction defeats, but have not been studied intensively. In this review, we aim to summarize recent breakthroughs in an in vivo laminopathy model and in vitro study using patient-specific human induced pluripotent stem cells (iPSCs) that reproduce the pathophysiological phenotype for further drug screening. We describe several in-vivo transgenic mouse models to elucidate the effects of Lmna H222P, N195K mutations, and LMNA knockout on cardiac function, in terms of hemodynamic and electrical signal propagation; certain strategies targeted on stress-related MAPK are mentioned. We will also discuss human iPSC cardiomyocytes serving as a platform to reveal the underlying mechanisms, such as the altered mechanical sensation in electrical coupling of the heart conduction system and ion channel alternation in relation to altered nuclear architecture, and furthermore to enable screening of drugs that can attenuate this cardiac premature aging phenotype by inhibition of prelamin misfolding and oxidative stress, and also enhancement of autophagy protein clearance and cardiac-protective microRNA.

Brain stiffness increases with myelin content

Brain stiffness plays an important role in neuronal development and disease, but reported stiffness values vary significantly for different species, for different brains, and even for different regions within the same brain. Despite extensive research throughout the past decade, the mechanistic origin of these stiffness variations remains elusive. Here we show that brain tissue stiffness is correlated to the underlying tissue microstructure and directly proportional to the local myelin content. In 116 indentation tests of six freshly harvested bovine brains, we found that the cerebral stiffnesses of 1.33±0.63kPa in white matter and 0.68±0.20kPa in gray matter were significantly different (p<0.01). Strikingly, while the inter-specimen variation was rather moderate, the minimum and maximum cerebral white matter stiffnesses of 0.59±0.19 kPa and 2.36±0.64kPa in each brain varied by a factor of four on average. To provide a mechanistic interpretation for this variation, we performed a histological characterization of the tested brain regions. We stained the samples with hematoxylin and eosin and luxol fast blue and quantified the local myelin content using image analysis. Interestingly, we found that the cerebral white matter stiffness increased with increasing myelin content, from 0.72kPa at a myelin content of 64–2.45kPa at a myelin content of 89%, with a Pearson correlation coefficient of ρ=0.91 (p<0.01). This direct correlation could have significant neurological implications. During development, our results could help explain why immature, incompletely myelinated brains are softer than mature, myelinated brains and more vulnerable to mechanical insult as evident, for example, in shaken baby syndrome. During demyelinating disease, our findings suggest to use stiffness alterations as clinical markers for demyelination to quantify the onset of disease progression, for example, in multiple sclerosis. Taken together, our study indicates that myelin might play a more important function than previously thought: It not only insulates signal propagation and improves electrical function of single axons, it also provides structural support and mechanical stiffness to the brain as a whole. Statement of SignificanceIncreasing evidence suggests that the mechanical environment of the brain plays an important role in neuronal development and disease. Reported stiffness values vary significantly, but the origin of these variations remains unknown. Here we show that stiffness of our brain is correlated to the underlying tissue microstructure and directly proportional to the local myelin content. Myelin has been discovered in 1854 as an insulating layer around nerve cells to improve electric signal propagation. Our study now shows that it also plays an important mechanical role: Using a combined mechanical characterization and histological characterization, we found that the white matter stiffness increases linearly with increasing myelin content, from 0.5kPa at a myelin content of 63–2.5kPa at 92%.

Changes in H(+)-ATP Synthase Activity, Proton Electrochemical Gradient, and pH in Pea Chloroplast Can Be Connected with Variation Potential.

Local stimulation induces generation and propagation of electrical signals, including the variation potential (VP) and action potential, in plants. Burning-induced VP changes the physiological state of plants; specifically, it inactivates photosynthesis. However, the mechanisms that decrease photosynthesis are poorly understood. We investigated these mechanisms by measuring VP-connected systemic changes in CO2 assimilation, parameters of light reactions of photosynthesis, electrochromic pigment absorbance shifts, and light scattering. We reveal that inactivation of photosynthesis in the pea, including inactivation of dark and light reactions, was connected with the VP. Inactivation of dark reactions decreased the rate constant of the fast relaxation of the electrochromic pigment absorbance shift, which reflected a decrease in the H+-ATP synthase activity. This decrease likely contributed to the acidification of the chloroplast lumen, which developed after VP induction. However, VP-connected decrease of the proton motive force across the thylakoid membrane, possibly, reflected a decreased pH in the stroma. This decrease may be another mechanism of chloroplast lumen acidification. Overall, stroma acidification can decrease electron flow through photosystem I, and lumen acidification induces growth of fluorescence non-photochemical quenching and decreases electron flow through photosystem II, i.e., pH decreases in the stroma and lumen, possibly, contribute to the VP-induced inactivation of light reactions of photosynthesis.

Modeling of Kidney Hemodynamics: Probability-Based Topology of an Arterial Network.

Through regulation of the extracellular fluid volume, the kidneys provide important long-term regulation of blood pressure. At the level of the individual functional unit (the nephron), pressure and flow control involves two different mechanisms that both produce oscillations. The nephrons are arranged in a complex branching structure that delivers blood to each nephron and, at the same time, provides a basis for an interaction between adjacent nephrons. The functional consequences of this interaction are not understood, and at present it is not possible to address this question experimentally. We provide experimental data and a new modeling approach to clarify this problem. To resolve details of microvascular structure, we collected 3D data from more than 150 afferent arterioles in an optically cleared rat kidney. Using these results together with published micro-computed tomography (μCT) data we develop an algorithm for generating the renal arterial network. We then introduce a mathematical model describing blood flow dynamics and nephron to nephron interaction in the network. The model includes an implementation of electrical signal propagation along a vascular wall. Simulation results show that the renal arterial architecture plays an important role in maintaining adequate pressure levels and the self-sustained dynamics of nephrons.

96-04: An MEA-Based Stem Cell-Cardiomyocyte Coculture Model for Studying Electrical Signal Propagation after Stem Cell Transplantation

96-04: An MEA-Based Stem Cell-Cardiomyocyte Coculture Model for Studying Electrical Signal Propagation after Stem Cell Transplantation

Analysis of horn type waveguide with impulse excitation

The aim of this research is theoretical investigation of dynamics of the horn type waveguides with impact excitation. For this purpose, a scheme of system with piezoelectric transducer for burst-type electrical signal generation is presented. Such a system could be used in generation of burst-type electric signals or direct control for some kind of stepper motors, e.g. piezo motors. In this paper horn (waveguide), which could be used in a system, for burst type electrical signal generation, is theoretically investigated in different way – end of the horn with smaller cross sectional area is excited with the impulse excitation and displacement is obtained at the end with greater cross sectional area.

Electrical signals in higher plants: Mechanisms of generation and propagation

Local stimulation induces generation and propagation of electric signals in higher plants. Noninvasive stimulus induces an action potential and damaging influences lead to the variation potential. The mechanism of the generation of an action potential is rather complex in nature and is associated with both activation of ion channels (Ca2+, Cl–, and K+) and transient change in the activity of the plasma membrane H+-ATPase. Generation of the variation potential, the duration of which is considerably longer than that of the action potential, is based on transient inactivation of the electrogenic pump; however, passive ion fluxes also contribute to such process, which causes qualitative similarity of the mechanisms of action potential and variation potential generation. Propagation of electrical signals mainly occurs in conducting bundles; thus, transfer of an action potential is associated with vascular parenchyma and sieve elements, while the variation potential is connected to the xylem vessels. The mechanism of the distribution the action potential is similar to nerve impulse transmission, while generation of the variation potential is induced by transfer of a chemical substance, whose propagation is accelerated by a hydraulic wave.

Different Profile of mRNA Expression in Sinoatrial Node from Streptozotocin-Induced Diabetic Rat.

BackgroundExperiments in isolated perfused heart have shown that heart rate is lower and sinoatrial node (SAN) action potential duration is longer in streptozotocin (STZ)–induced diabetic rat compared to controls. In sino-atrial preparations the pacemaker cycle length and sino-atrial conduction time are prolonged in STZ heart. To further clarify the molecular basis of electrical disturbances in the diabetic heart the profile of mRNA encoding a wide variety of proteins associated with the generation and transmission of electrical activity has been evaluated in the SAN of STZ-induced diabetic rat heart.Methodology/Principal FindingsHeart rate was measured in isolated perfused heart with an extracellular suction electrode. Expression of mRNA encoding a variety of intercellular proteins, intracellular Ca2+-transport and regulatory proteins, cell membrane transport proteins and calcium, sodium and potassium channel proteins were measured in SAN and right atrial (RA) biopsies using real-time reverse transcription polymerase chain reaction techniques. Heart rate was lower in STZ (203±7 bpm) compared to control (239±11 bpm) rat. Among many differences in the profile of mRNA there are some worthy of particular emphasis. Expression of genes encoding some proteins were significantly downregulated in STZ-SAN: calcium channel, Cacng4 (7-fold); potassium channel, Kcnd2 whilst genes encoding some other proteins were significantly upregulated in STZ-SAN: gap junction, Gjc1; cell membrane transport, Slc8a1, Trpc1, Trpc6 (4-fold); intracellular Ca2+-transport, Ryr3; calcium channel Cacna1g, Cacna1h, Cacnb3; potassium channels, Kcnj5, Kcnk3 and natriuretic peptides, Nppa (5-fold) and Nppb (7-fold).Conclusions/SignificanceCollectively, this study has demonstrated differences in the profile of mRNA encoding a variety of proteins that are associated with the generation, conduction and regulation of electrical signals in the SAN of STZ-induced diabetic rat heart. Data from this study will provide a basis for a substantial range of future studies to investigate whether these changes in mRNA translate into changes in electrophysiological function.

New roles for the GLUTAMATE RECEPTOR-LIKE 3.3, 3.5, and 3.6 genes as on/off switches of wound-induced systemic electrical signals

ABSTRACTWounding induces systemic potentials in Arabidopsis thaliana that can be abolished by concomitant suppression of the GLUTAMATE RECEPTOR-LIKE GLR3.3 and GLR3.6 genes. However, the roles of specific GLR channels to these potentials remain unclear. Here I applied the Electrical Penetration Graph (EPG) to study the contribution of three GLR channels to wound-induced, systemically propagated electrical potentials in Arabidopsis thaliana. In contrast to recordings made with conventional rigs for whole-plant electrophysiology, the EPG allows for the unambiguous distinction of the phloem-propagated action potential (AP) from the electrical activity outside of the phloem. The data reported here suggest that: (a) the transmission of wound-induced, phloem-propagated AP to neighbor leaves, requires expression of GLR3.3 or GLR3.6, whereas GLR3.5 prevents its transmission to non-neighbor leaves; (b) the generation of wound-induced electrical signals outside the phloem network depends on GLR3.6 expression; and (c) wound-induced systemic potentials initiated in the shoot are transmitted to the root in the adult plant, which suggests a role for these electrical signals in coordinating the plant defenses in the shoot and in the root. Here, I propose a model for wound-induced systemic electrical signals at the molecular, cellular and anatomical level. In this model, GLR3.3 and GLR3.6 function as on switches for the propagation of wound-induced potentials beyond the wounded leaf, while GLR3.5 functions as an off switch that prevents the propagation of wound-induced electrical potentials to distal, non-neighbor leaves.

Voltage-Sensitive Fluorescence of Indocyanine Green in the Heart

So far, the optical mapping of cardiac electrical signals using voltage-sensitive fluorescent dyes has only been performed in experimental studies because these dyes are not yet approved for clinical use. It was recently reported that the well-known and widely used fluorescent dye indocyanine green (ICG), which has FDA approval, exhibits voltage sensitivity in various tissues, thus raising hopes that electrical activity could be optically mapped in the clinic. The aim of this study was to explore the possibility of using ICG to monitor cardiac electrical activity. Optical mapping experiments were performed on Langendorff rabbit hearts stained with ICG and perfused with electromechanical uncouplers. The residual contraction force and electrical action potentials were recorded simultaneously. Our research confirms that ICG is a voltage-sensitive dye with a dual-component (fast and slow) response to membrane potential changes. The fast component of the optical signal (OS) can have opposite polarities in different parts of the fluorescence spectrum. In contrast, the polarity of the slow component remains the same throughout the entire spectrum. Separating the OS into these components revealed two different voltage-sensitivity mechanisms for ICG. The fast component of the OS appears to be electrochromic in nature, whereas the slow component may arise from the redistribution of the dye molecules within or around the membrane. Both components quite accurately track the time of electrical signal propagation, but only the fast component is suitable for estimating the shape and duration of action potentials. Because ICG has voltage-sensitive properties in the entire heart, we suggest that it can be used to monitor cardiac electrical behavior in the clinic.

Exact traveling wave solutions for nonlinear PDEs in mathematical physics using the generalized Kudryashov method

The generalized Kudryashov method is applied in this article for finding the exact solutions of nonlinear partial differential equations (PDEs) in mathematical physics. Solitons and other solutions are given. To illustrate the validity of this method, we apply it to three nonlinear PDEs, namely, the diffusive predator-prey system, the nonlinear Bogoyavlenskii equations and the nonlinear telegraph equation. These equations are related to signal analysis for transmission and propagation of electrical signals. As a result, many analytical exact solutions of these equations are obtained including symmetrical Fibonacci function solutions and hyperbolic function solutions. Physical explanations for some solutions of the given three nonlinear PDEs are obtained. Comparison our new results with the well-known results are given.

Ion channels in mechanosensing and electrical signaling in plants

Plants as animals have cell membrane equipped with ion channels. These channels embedded in the membrane allowed ion to move in or out the cell. Therefore these proteins are involved in several physiological processes such as nutrition, sensing (of environmental factors) and long-distance communication. In this “slide presentation” we report the possible role of ion channels in sensing oscillation and in long distance signalling in plant.The first example concerns the mechanosensitive channels of the MSL (Mechanosensitive channel Small conductance-Like) family and illustrate how it possibly behave as an oscillation sensor for the plant. The function of such a channel is consistent with the capacity of terrestrial plants to react to repetitive mechanical load produced by wind.The second example concerns the role of mechano-gated and voltage-gated channels in the generation and long distance propagation of electrical signal in plant. More precisely, it illustrate how Action Potential and Slow Wave of depolarization are generated and propagate along plant tissues. Eventually the relevance of such electrical signals in plant is illustrated by two examples.