Dynamic Impedance Model of the Skin-Electrode Interface for Transcutaneous Electrical Stimulation

Transcutaneous electrical stimulation can depolarize nerve or muscle cells applying impulses through electrodes attached on the skin. For these applications, the electrode-skin impedance is an important factor which influences effectiveness. Various models describe the interface using constant or current-depending resistive-capacitive equivalent circuit. Here, we develop a dynamic impedance model valid for a wide range stimulation intensities. The model considers electroporation and charge-dependent effects to describe the impedance variation, which allows to describe high-charge pulses. The parameters were adjusted based on rectangular, biphasic stimulation pulses generated by a stimulator, providing optionally current or voltage-controlled impulses, and applied through electrodes of different sizes. Both control methods deliver a different electrical field to the tissue, which is constant throughout the impulse duration for current-controlled mode or have a very current peak for voltage-controlled. The results show a predominant dependence in the current intensity in the case of both stimulation techniques that allows to keep a simple model. A verification simulation using the proposed dynamic model shows coefficient of determination of around 0.99 in both stimulation types. The presented method for fitting electrode-skin impedance can be simple extended to other stimulation waveforms and electrode configuration. Therefore, it can be embedded in optimization algorithms for designing electrical stimulation applications even for pulses with high charges and high current spikes.     

Drift model of the cathode region of a glow discharge

A one-dimensional drift model of the cathode region of a glow discharge with allowance for both electron-impact ionization and charged particle loss is proposed. An exact solution to the model equations is obtained for the case of similar power-law dependences of the ion and electron drift velocities on the electric field strength. It is shown that, even in the drift approximation, a relatively wide transition layer in which the ion-to-electron current ratio approaches a constant value typical of the positive column of a glow discharge should occur between the thin space-charge sheath and the quasineutral plasma, the voltage drop across the space-charge sheath being comparable to that across the transition layer. The calculated parameters of the normal and anomalous glow discharges are in good agreement with available experimental data.     

Attenuation of current and voltage noise signals recorded from epithelia

Transepithelially recorded current and voltage fluctuations are filtered by the impedance of the electrical equivalent parameters of the preparation, in series or in parallel, with the noise source. Fluctuations in voltage and current are assumed to be caused by fluctuations in conductance of the apical membrane. In order to obtain an estimation of the intrinsic noise amplitudes, calculations are presented to correct the transepithelial fluctuations. The influence of different model parameters on the recorded noise spectra is investigated. It is shown that the shape of the transepithelially recorded noise spectra may differ from the intrinsic ones, e.g. “peaking” in the power spectra may be explained by the assumption of a positive (referred to cell inside) e.m.f. at the basolateral membrane or a polarization impedance in series with the epithelium. Furthermore it is demonstrated that the ratio of voltage to current noise power may differ from the squared magnitude of the impedance.     

A model for the polarization of neurons by extrinsically applied electric fields.

A model is presented for the subthreshold polarization of a neuron by an applied electric field. It gives insight into how morphological features of a neuron affect its polarizability. The neuronal model consists of one or more extensively branched dendritic trees, a lumped somatic impedance, and a myelinated axon with nodes of Ranvier. The dendritic trees branch according to the 3/2-power rule of Rall, so that each tree has an equivalent cylinder representation. Equations for the membrane potential at the soma and at the nodes of Ranvier, given an arbitrary specified external potential, are derived. The solutions determine the contributions made by the dendritic tree and the axon to the net polarization at the soma. In the case of a spatially constant electric field, both the magnitude and sign of the polarization depend on simple combinations of parameters describing the neuron. One important combination is given by the ratio of internal resistances for longitudinal current spread along the dendritic tree trunk and along the axon. A second is given by the ratio between the DC space constant for the dendritic tree trunk and the distance between nodes of Ranvier in the axon. A third is given by the product of the electric field and the space constant for the trunk of the dendritic tree. When a neuron with a straight axon is subjected to a constant field, the membrane potential decays exponentially with distance from the soma. Thus, the soma seems to be a likely site for action potential initiation when the field is strong enough to elicit suprathreshold polarization. In a simple example, the way in which orientation of the various parts of the neuron affects its polarization is examined. When an axon with a bend is subjected to a spatially constant field, polarization is focused at the bend, and this is another likely site for action potential initiation.     

A.C. cable theory and its implications on the measurements of the membrane electric parameters

The solution to the a.c. cable equations for a leaky non-inductive coaxial cable is presented. It is shown that the exact solution can be expressed in terms of a simple Ohm's law formulation modified by multiplicative factors. A numerical analysis of these factors shows that the formulation reduces to a simple Ohm's law for the case of infinite impedance terminations, if the voltage is recorded at a distance 0·421/λ from the centre of a midpoint current injection, 1 is the half-length of the cell and A the space constant. The same result applies to the case of external current injection if the voltage is recorded at a distance of 0·581/λ from the centre of the cell segment of length 21. This thus allows an easy and accurate computation of the membrane resistance and capacitance. It is also shown that the accuracy obtained in these computations can be estimated quantitatively. The actual values of the accuracy obtained in a.c. studies of Nitella translucens are presented.     

RECTIFICATION AND INDUCTANCE IN THE SQUID GIANT AXON

Previous measurements have shown that the electrical properties of the squid axon membrane are approximately equivalent to those of a circuit containing a capacity shunted by an inductance and a rectifier in series. Selective ion permeability of a membrane separating two electrolytes may be expected to give rise to the rectification. A quasi-crystalline piezoelectric structure of the membrane is a plausible explanation of the inductance. Some approximate calculations of behavior of an axon with these membrane characteristics have been made. Fair agreement is obtained with the observed constant current subthreshold potential and impedance during the foot of the action potential. In a simple case a formal analogy is found between the calculated membrane potential and the excitability defined by the two factor formulations of excitation. Several excitation phenomena may then be explained semi-quantitatively by further assuming the excitability proportional to the membrane potential. Some previous measurements and subthreshold potential and excitability observations are not consistent with the circuit considered and indicate that this circuit is only approximately equivalent to the membrane.     

Frequency dispersions of human skin dielectrics.

The electrical properties of many biological materials are known to exhibit frequency dispersions. In the human skin, the impedance measured at various frequencies closely describes a circular locus of the Cole-Cole type in the complex impedance plane. In this report, the formative mechanisms responsible for the anomalous circular-arc behavior of skin impedance were investigated, using data from impedance measurements taken after successive strippings of the skin. The data were analyzed with respect to changes in the parameters of the equivalent Cole-Cole model after each stripping. For an exponential resistivity profile (Tregear, 1966, Physical Functions of Skin; Yamamoto and Yamamoto, 1976, Med. Biol. Eng., 14:151--158), the profile of the dielectric constant was shown to be uniform across the epidermis. Based on these results, a structural model has been formulated in terms of the relaxation theory of Maxwell and Wagner for inhomogeneous dielectric materials. The impedance locus obtained from the model approximates a circular are with phase constant alpha = 0.82, which compares favorably with experimental data. At higher frequencies a constant-phase, frequency-dependent component having the same phase constant alpha is also demonstrated. It is suggested that an approximately rectangular distribution of the relaxation time over the epidermal dielectric sheath is adequate to account for the anomalous frequency characteristics of human skin impedance.     

An approximation method for the determination of diffusion and permeability coefficients in nerve trunks

A recently introduced approximation method is applied in order to obtain an expression for the amount of a substance remaining within a nerve at any time, the nerve having been soaked for a long time in a solution containing the substance until the time zero when it is bathed in the same solution but without the substance. The case of a uniform nerve without a sheath leads to substantially the same results as previously obtained by A. V. Hill (1928) for this case. A solution is given for the case of a nerve without sheath but having fibers which are permeable. In this case it is shown how an effective diffusion coefficient for the interstitial fluid can be obtained, as well as the effective inward and outward fiber permeabilities. A solution is given for the case of a nerve with a sheath in which the substance considered does not penetrate the fibers, and it is shown how the effective diffusion coefficients of the sheath and the interstitial fluid can be obtained.     

Magnetic probe measurements of the current sheath on the PF-3 facility

Results are presented from experimental studies of the dynamics of the current sheath (CS) on the PF-3 plasma focus facility. The parameters of the sheath, including the current distribution in it, were measured using absolutely calibrated magnetic probes installed at different positions with respect to the facility axis and the anode surface. The CS dynamics in discharges operating in argon and neon was investigated, and the skin depth in different stages of the discharge was determined. One of the probes was installed at a distance of ≈2 cm from the facility axis, which made it possible to estimate the efficiency of current transfer to the region of pinch formation. Operating modes were obtained in which the current dynamics detected by magnetic probes at different distances from the axis agreed well with the dynamics of the total discharge current until the instant of singularity in the current time derivative. It is shown that shunting breakdowns can lead to the formation of closed current loops. The shunting of the discharge current by the residual plasma is directly related to the efficiency of snowplowing of the working gas by the CS as it propagates from the insulator toward the facility axis.     

Electrical Impedance of Isolated Amnion

The electrical impedance of the guinea pig amniotic membrane was measured, under standardized conditions, over the frequency range of 20 to 7000 cycles/second. This impedance can be represented analytically by a simple frequency-dependent function which is precisely of the form of the Debye relaxation equation. The observed data exhibit a broad dispersion centered at a frequency of 1050 cycles/second and a narrow distribution of time constants centered about 152 microseconds, both effects being due to the polydisperse nature of amniotic tissue. If the narrow time-constant distribution is approximated by a single time constant, amnion impedance can be simulated by a simple electrical circuit of frequency-independent elements. The Maxwell-Wagner interfacial treatment, although successfully adapted for cell suspensions, is shown to lose its quantitative significance in the case of the tightly structured amnion. In addition, determinations were made on the chemical composition of amniotic fluid, fetal blood and urine, and maternal blood and urine; the DC potential across the amniotic membrane was also measured.     

The Transient Response and Impedance Locus of a Mobile Site Membrane

The kinetic properties of a mobile site membrane are analyzed theoretically and the results are compared with experimental data obtained from a simple model system. In the model system the measured response to a step change in electric current is a single relaxation process, in excellent agreement with the theoretically predicted behavior. To this relaxation process corresponds an anomalous capacitance whose properties are obtained by performing a frequency analysis of the system. By plotting the impedance locus of this mobile site membrane, the locus is shown to exhibit a 45° phase angle at high frequencies, a phenomenon which is generally associated with depletion layers.     

The effect of dendritic voltage-gated conductances on the neuronal impedance: a quantitative model

Neuronal impedance characterizes the magnitude and timing of the subthreshold response of a neuron to oscillatory input at a given frequency. It is known to be influenced by both the morphology of the neuron and the presence of voltage-gated conductances in the cell membrane. Most existing theoretical accounts of neuronal impedance considered the effects of voltage-gated conductances but neglected the spatial extent of the cell, while others examined spatially extended dendrites with a passive or spatially uniform quasi-active membrane. We derived an explicit mathematical expression for the somatic input impedance of a model neuron consisting of a somatic compartment coupled to an infinite dendritic cable which contained voltage-gated conductances, in the more general case of non-uniform dendritic membrane potential. The validity and generality of this model was verified through computer simulations of various model neurons. The analytical model was then applied to the analysis of experimental data from real CA1 pyramidal neurons. The model confirmed that the biophysical properties and predominantly dendritic localization of the hyperpolarization-activated cation current I (h) were important determinants of the impedance profile, but also predicted a significant contribution from a depolarization-activated fast inward current. Our calculations also implicated the interaction of I (h) with amplifying currents as the main factor governing the shape of the impedance-frequency profile in two types of hippocampal interneuron. Our results provide not only a theoretical advance in our understanding of the frequency-dependent behavior of nerve cells, but also a practical tool for the identification of candidate mechanisms that determine neuronal response properties.     

Enzyme localization as a mechanism of apparent active transport

A model based on enzyme localization is developed which gives rise to an apparent active transport of a metabolite into or out of cells. The model is applied to three simple situations, using Fick's equation and the Rashevsky approximation. It is shown that the apparent efficiency can be made as large as desired if, for constant reaction, the outer cell region is made sufficiently small, or, for autocatalytic reaction, if the metabolite concentration in the outer region is sufficiently small. The physical limitations imposed by this mechanism are developed for all three situations.     

Breakdown in helium in high-voltage open discharge with subnanosecond current front rise

Investigations of high-voltage open discharge in helium have shown a possibility of generation of current pulses with subnanosecond front rise, due to ultra-fast breakdown development. The open discharge is ignited between two planar cathodes with mesh anode in the middle between them. For gas pressure 6 Torr and 20 kV applied voltage, the rate of current rise reaches 500 A/(cm² ns) for current density 200 A/cm² and more. The time of breakdown development was measured for different helium pressures and a kinetic model of breakdown in open discharge is presented, based on elementary reactions for electrons, ions and fast atoms. The model also includes various cathode emission processes due to cathode bombardment by ions, fast atoms, electrons and photons of resonant radiation with Doppler shift of frequency. It is shown, that the dominating emission processes depend on the evolution of the discharge voltage during the breakdown. In the simulations, two cases of voltage behavior were considered: (i) the voltage is kept constant during the breakdown; (ii) the voltage is reduced with the growth of current. For the first case, the exponentially growing current is maintained due to photoemission by the resonant photons with Doppler-shifted frequency. For the second case, the dominating factor of current growth is the secondary electron emission. In both cases, the subnanosecond rise of discharge current was obtained. Also the effect of gas pressure on breakdown development was considered. It was found that for 20 Torr gas pressure the time of current rise decreases to 0.1 ns, which is in agreement with experimental data.     

面向血液葡萄糖测量的生物组织阻抗模型

本文推导了细胞膜电容与血糖浓度之间的关系,并将这种关系应用到Pauly和Schwan提出的弛豫模型中,再将弛豫模型应用于多重Cole-Cole介电模型,最后通过介电模型得出阻抗模型,建立起人体血糖浓度与组织介电谱,阻抗谱之间的联系。通过分析发现,细胞膜电容随着血糖浓度增加而增大,且当血糖浓度趋于无穷大时,细胞膜电容等于血糖浓度为0时细胞膜电容的1.53倍。对介电谱的仿真得出,当外加信号频率小于1 MHz时,介电常数实部随着葡萄糖浓度改变而发生明显变化。阻抗谱的研究结果表明,随着葡萄糖浓度的升高,阻抗的实部减小,阻抗虚部模增大,且分别在频率为1.48 MHz和0.55 MHz时变化得最快。本文建立的模型为血糖无创检测提供了一种新的方法和手段。     

Generation of capacitance current harmonics during electrostriction of inhomogeneous bilayers

The mechanisms of generation of capacitance current harmonics arising from bilayer electrostriction are investigated theoretically in the case when fixed charges or dipoles are located inside the bilayer. If the elastic properties of the bilayer are inhomogeneous in a transversal direction, these charges or dipoles affect substantially the amplitude A2 of the second current harmonic and accordingly the magnitude of compensatory constant voltage VC, which shold be applied to the membrane to reduce A2 to zero. In membranes whose Young's modulus is minimum at the center of bilayer, the dependence of VC on depth of immersion of the charge xA into the bilayer represents a function with alternating signs, which is reduced to zero at the center of bilayer. In the case of a dipole source immersed into the bilayer, the appropriate dependence is oscillating as well, and its shape is a derivative of the corresponding dependence of VC for fixed charges with respect to a coordinate. Possible applications of the results are discussed.     

Does direction of induced electric field or current provide a test of mechanism involved in nerve regeneration?

We suggest an experimental comparison of two directions for applying the time-varying magnetic fields which have been found to speed spontaneous regeneration of injured peripheral nerves and in attempts to repair spinal cord injuries. Time-varying magnetic fields induce currents in a plane perpendicular to the magnetic field direction. The lower conductivity of the spinal cord's sheath (dura matter) or of the myelin sheath of peripheral nerves would seem to confine the induced electric fields and currents to the spinal cord or nerve itself. The proposed comparison could allow choosing between two possible modes of action of the fields: (1) Magnetically-induced electric fields or currents may be encouraging ion flow or otherwise helping enzyme, channel or other interactions at the cell membrane, as is thought to be the case in field stimulation of healing in bone. This mechanism should be independent of field direction. (2) Work in developing organisms and with fields applied to nerve cells in vitro has shown that neurite growth is guided parallel to both endogenous and external electric fields. This mechanism would be effective when induced electric fields are parallel, but not when they are perpendicular to the nerve. Any experimental test should seek to produce as close as possible to the same induced current intensity with both field directions. Possible confounding factors, as well as breakdowns in the assumptions of the simple model presented here, would have to be considered. This proposal was motivated by a recent report in which the authors listed a changed field direction as one of several possible reasons for an unsuccessful experiment.     

Dynamics of the structure of the plasma current sheath in a plasma focus discharge

The study is aimed at investigating the fine structure of the plasma current sheath (PCS) in the PF-3 plasma focus facility. The PCS dynamics in a deuterium discharge was studied. The PCS parameters were measured using absolutely calibrated magnetic probes installed at different positions with respect to the facility axis and the anode surface. A magneto-optical probe recording both the magnetic signal and the PCS optical luminosity was first applied to analyze the PCS structure. This made it possible to spatially resolve the current and shock-wave regions. It is demonstrated that the current distribution is different in different discharge stages. It is shown that the neutron yield is determined by the value of the current compressed toward the axis, rather then the amplitude of the total discharge current.     

Modeling and measuring the elastic properties of an archaeal surface, the sheath of Methanospirillum hungatei, and the implication of methane production.

We describe a technique for probing the elastic properties of biological membranes by using an atomic force microscope (AFM) tip to press the biological material into a groove in a solid surface. A simple model is developed to relate the applied force and observed depression distance to the elastic modulus of the material. A measurement on the proteinaceous sheath of the archaebacterium Methanospirillum hungatei GP1 gave a Young's modulus of 2 x 10(10) to 4 x 10(10) N/m2. The measurements suggested that the maximum sustainable tension in the sheath was 3.5 to 5 N/m. This finding implied a maximum possible internal pressure for the bacterium of between 300 and 400 atm. Since the cell membrane and S-layer (wall) which surround each cell should be freely permeable to methane and since we demonstrate that the sheath undergoes creep (expansion) with pressure increase, it is possible that the sheath acts as a pressure regulator by stretching, allowing the gas to escape only after a certain pressure is reached. This creep would increase the permeability of the sheath to diffusible substances.     

Towards the realization of label-free biosensors through impedance spectroscopy integrated with IDES technology

Impedance spectroscopy (IS) is a powerful technique for analysis of the complex electrical impedance of a large variety of biological systems, because it is sensitive both to surface phenomena and to changes of bulk properties. A simple and convenient method of analysis of cell properties by IS is described. An interdigitated electrodes configuration was used for the measurements; human epithelial cells were grown on the device to investigate the complex dielectric response as a function of frequency, in order to test the suitability of the device for use as a label-free biosensor. To test the ability of the device to detect channels in the cell membrane, the effect of drugs known to affect membrane integrity was also investigated. The frequency response of the admittance (i.e. the reciprocal of the impedance) can be well fitted by a model based on very simple assumptions about the cells coating the device surface and the current flow; from the calculations, membrane-specific capacitance and information about cell adhesion can be inferred. These preliminary efforts have shown that our configuration could lead to a label-free non-invasive technique for biosensing and cellular behavior monitoring which might prove useful in investigation of the basic properties of cells and the effect of drugs by estimation of some fundamental properties and modification of the electrical characteristics of the device.