Extracellular and intracellular components of the impedance of neural tissue

Electric phenomena in brain tissue can be measured using extracellular potentials, such as the local field potential, or the electro-encephalogram. The interpretation of these signals depends on the electric structure and properties of extracellular media, but the measurements of these electric properties are still debated. Some measurements point to a model in which the extracellular medium is purely resistive, and thus parameters such as electric conductivity and permittivity should be independent of frequency. Other measurements point to a pronounced frequency dependence of these parameters, with scaling laws that are consistent with capacitive or diffusive effects. However, these experiments correspond to different preparations, and it is unclear how to correctly compare them. Here, we provide for the first time, impedance measurements (in the 1–10 kHz frequency range) using the same setup in various preparations, from primary cell cultures to acute brain slices, and a comparison with similar measurements performed in artificial cerebrospinal fluid with no biological material. The measurements show that when the current flows across a cell membrane, the frequency dependence of the macroscopic impedance between intracellular and extracellular electrodes is significant, and cannot be captured by a model with resistive media. Fitting a mean-field model to the data shows that this frequency dependence could be explained by the ionic diffusion mainly associated with Debye layers surrounding the membranes. We conclude that neuronal membranes and their ionic environment induce strong deviations to resistivity that should be taken into account to correctly interpret extracellular potentials generated by neurons.     

MyDEP: A New Computational Tool for Dielectric Modeling of Particles and Cells

Dielectrophoresis (DEP) and electrorotation (ROT) are two electrokinetic phenomena exploiting nonuniform electric fields to exert a force or torque on biological particles suspended in liquid media. They are widely used in lab-on-chip devices for the manipulation, trapping, separation, and characterization of cells, microorganisms, and other particles. The DEP force and ROT torque depend on the respective polarizabilities of the particle and medium, which in turn depend on their dielectric properties and on the field frequency. In this work, we present a new software, MyDEP, which implements several particle models based on concentric shells with adjustable dielectric properties. This tool enables the study of the variation in DEP and ROT spectra according to different parameters, such as the field frequency and medium conductivity. Such predictions of particle behavior are very useful for choosing appropriate parameters in DEP experiments. The software also enables the study of the homogenized properties of spherical or ellipsoidal multishell particles and provides a database containing published cell properties. Equivalent electrical conductivity and relative permittivity of the cell alone and in suspension can be calculated. The software also offers the ability to create graphs of the evolution of the crossover frequencies with the electric field frequency. These graphs can be directly exported from the software.     

Bacterial clogging of porous media: A new modelling approach

A comparison is made between existing mathematical models and experimental data that relate the reduction of the saturated hydraulic conductivity (K) of a porous medium to the porosity reduction caused by microbial growth. The models yielded a realistic prediction of a data set obtained with a model porous medium consisting of millimeter‐size glass spheres, but failed to predict the clogging behaviour observed in smaller‐than‐1‐mm sand. A new modelling approach, semi‐mechanistic in nature, is proposed that gives good predictions of fine sand media as well. It relaxes the assumption about uniformly‐thick biofilms by allowing a second arrangement to occur, i.e. discrete plugs filling the pore lumen. The new model requires input data on two intrinsic properties of the system, which renders it sufficiently flexible as to fit very different data sets. The two model parameters are K_min, the minimum K value when all porosity is filled with microorganisms, and B_c, the biovolume fraction at which most cell detachment from biofilm occurs.     

An ephaptic transmission model of CA3 pyramidal cells: an investigation into electric field effects

Extracellular electric fields existing throughout the living brain affect the neural coding and information processing via ephaptic transmission, independent of synapses. A two-compartment whole field effect model (WFEM) of pyramidal neurons embedded within a resistive array which simulates the extracellular medium i.e. ephapse is developed to study the effects of electric field on neuronal behaviors. We derive the two linearized filed effect models (LFEM-1 and LFEM-2) from WFEM at the stable resting state. Through matching these simplified models to the subthreshold membrane response in experiments of the resting pyramidal cells exposed to applied electric fields, we not only verify our proposed model’s validity but also found the key parameters which dominate subthreshold frequency response characteristic. Moreover, we find and give its underlying biophysical mechanism that the unsymmetrical properties of active ion channels results in the very different low-frequency response of somatic and dendritic compartments. Following, WFEM is used to investigate both direct-current (DC) and alternating-current field effect on the neural firing patterns by bifurcation analyses. We present that DC electric field could modulate neuronal excitability, with the positive field improving the excitability, the modest negative field suppressing the excitability, but interestingly, the larger negative field re-exciting the neuron back into spiking behavior. The neuron exposed to the sinusoidal electric field exhibits abundant firing patterns sensitive to the input frequency and intensity. In addition, the electrical properties of ephapse can modulate the efficacy of field effect. Our simulated results are qualitatively in line with the relevant experimental results and can explain some experimental phenomena. Furthermore, they are helpful to provide the predictions which can be tested in future experiments.     

Electrical signals for chondrocytes in cartilage

An important step toward understanding signal transduction mechanisms modulating cellular activities is the accurate predictions of the mechanical and electro-chemical environment of the cells in well-defined experimental configurations. Although electro-kinetic phenomena in cartilage are well known, few studies have focused on the electric field inside the tissue. In this paper, we present some of our recent calculations of the electric field inside a layer of cartilage (with and without cells) in an open circuit one-dimensional (1D) stress relaxation experiment. The electric field inside the tissue derives from the streaming effects (streaming potential) and the diffusion effect (diffusion potential). Our results show that, for realistic cartilage material parameters, due to deformation-induced inhomogeneity of the fixed charge density, the two potentials compete against each other. For softer tissue, the diffusion potential may dominate over the streaming potential and vice versa for stiffer tissue. These results demonstrate that for proper interpretation of the mechano-electrochemical signal transduction mechanisms, one must not ignore the diffusion potential.     

Electrical Sizing of Particles in Suspensions: I. Theory

The processes involved during the passage of a suspended particle through a small cylindrical orifice across which exists an electric field are considered in detail. Expressions are derived for the resulting change in current in terms of the ratios of particle to orifice volume and particle to suspending fluid resistivity, and particle shape. Graphs are presented of the electric field and of the fluid velocity as functions of position within the orifice, and of the shape factor of spheroids as a function of their axial ratio and orientation in the electric field. The effects of the electric and hydrodynamic fields on the orientation of nonspherical particles and on the deformation of nonrigid spheres is treated, and the migration of particles towards the orifice axis is discussed. Oscillograms of current pulses produced by rigid, nonconducting spheres in various orifices are shown and compared with the theoretical predictions.     

Measuring principles of frictional coefficients in cartilaginous tissues and its substitutes

The frictional properties of cartilaginous tissues, such as the hydraulic permeability, the electro-osmotic permeability, the diffusion coefficients of various ions and solutes, and the electrical conductance, are vital data to characterise the extracellular environment in which chondrocytes reside. This paper analyses one-dimensional measurement principles of these coefficients. Particular attention is given to the deformation dependence of them and the highly deformable nature of the tissues. A suggested strategy is the combination of a diffusion experiment using radiotracer methods, an electro-osmotic flow experiment and an electro-osmotic pressure experiment at low electric current.     

Restricted diffusion of molecules in porous affinity chromatography adsorbents.

A restricted diffusion model is constructed and solved in order to study the permeability of large adsorbate molecules in the pores of affinity chromatography media, when the adsorbate molecules are adsorbed onto immobilized ligands. The combined effects of steric hindrance at the entrance to the pores and frictional resistance within the pores, as well as the effects of pore size distribution, pore connectivity of the adsorbent, molecular size of adsorbate and ligand, and the fractional saturation of adsorption sites (ligands), are considered. Affinity adsorbents with dilute and high ligand concentrations are examined, and the permeability of the adsorbate in porous networks of connectivity nT is studied by means of effective medium approximation (EMA) numerical solutions. As expected, the permeability of the adsorbate decreases as the size of the adsorbate and/or ligand molecule increases. The permeability also decreases when the fractional saturation of the ligands increases, as well as when the pore connectivity of the network decreases. The dependence of the permeability on the pore connectivity tends to be less marked in adsorbents with concentrated ligand than in porous media with dilute ligand concentration. The conditions are also presented for which the percolation threshold is attained in a number of different systems. The restricted diffusion model and results of this work may be of importance in studies involving the modeling, prediction of the dynamic behavior, design, and control of affinity chromatography (biospecific adsorption) systems employing porous adsorbents. The theoretical results may also have important implications in the selection of a ligand as well as in the selection and construction of an affinity porous matrix, so that the adsorbate of interest can be efficiently separated from a given solution. Furthermore, with appropriate modifications this restricted diffusion model may be used in studies involving the immobilization of ligands or enzymes in porous solids.     

Electric fields and surface charges induced by ELF magnetic fields

A method is described for evaluating electric fields induced by ELF magnetic fields into electrically inhomogeneous, low-conductivity (less than 5 S/m) structures. It is applied to cylinders and spheres, and numerical results are given for electrical properties that are representative of some tissues, or of cells embedded either in saline solution or a tissue matrix. Surface currents on spherical cell boundaries are estimated and compared with thermal noise due to ion motion.     

Electroporation in dense cell suspension--theoretical and experimental analysis of ion diffusion and cell permeabilization

Electroporation is a process where increased permeability of cells exposed to an electric field is observed. It is used in many biomedical applications including electrogene transfection and electrochemotherapy. Although the increased permeability of the membrane is believed to be the result of pores due to an induced transmembrane voltage U(m), the exact molecular mechanisms are not fully explained. In this study we analyze transient conductivity changes during the electric pulses and increased membrane permeability for ions and molecules after the pulses in order to determine which parameters affect stabilization of pores, and to analyze the relation between transient pores and long-lived transport pores. By quantifying ion diffusion, fraction of transport pores f(per) was obtained. A simple model, which assumes a quadratic dependence of f(per) on E in the area where U(m)>U(c) very accurately describes experimental values, suggesting that f(per) increases with higher electric field due to larger permeabilized area and due to higher energy available for pore formation. The fraction of transport pores increases also with the number of pulses N, which suggest that each pulse contributes to formation of more and/or larger stable transport pores, whereas the number of transient pores does not depend on N.     

Microgravity dependence of excitable biological and physicochemical media

Neuronal tissue and especially the central nervous system (CNS) is an excitable medium. Self-organisation, pattern formation, and propagating excitation waves as typical characteristics in excitable media consequently have been found in neuronal tissue. The properties of such phenomena in excitable media do critically depend on the parameters (i.e., electromagnetic fields, temperature, chemical drugs) of the system and on small external forces to which gravity belongs. The spreading depression, a propagating excitation depression wave of neuronal activity, is one of the best described of the those wave phenomena in the CNS. Especially in the retina as a true part of the CNS it can be easily observed with optical techniques due to the high intrinsic optical signal of this tissue. Another of such waves in neuronal tissue is the propagating action potential in nerve fibres. In this paper, data from our laboratories concerning the influence of gravity on the velocity of propagating waves in excitable media are summarized mainly in terms of the retinal spreading depression and propagating action potentials. Additionally, we have used waves in gels of the Belousov-Zhabotinsky reaction as the physicochemical model system of biological activity as the properties of these waves follow the same theories as the spreading depression and action potentials and they have some striking similarities in wave behavior. Thus propagating Belousov-Zhabotinsky waves are described by their gravity dependence.     

Macroscopic Models of Local Field Potentials and the Apparent 1/f Noise in Brain Activity

The power spectrum of local field potentials (LFPs) has been reported to scale as the inverse of the frequency, but the origin of this 1/f noise is at present unclear. Macroscopic measurements in cortical tissue demonstrated that electric conductivity (as well as permittivity) is frequency-dependent, while other measurements failed to evidence any dependence on frequency. In this article, we propose a model of the genesis of LFPs that accounts for the above data and contradictions. Starting from first principles (Maxwell equations), we introduce a macroscopic formalism in which macroscopic measurements are naturally incorporated, and also examine different physical causes for the frequency dependence. We suggest that ionic diffusion primes over electric field effects, and is responsible for the frequency dependence. This explains the contradictory observations, and also reproduces the 1/f power spectral structure of LFPs, as well as more complex frequency scaling. Finally, we suggest a measurement method to reveal the frequency dependence of current propagation in biological tissue, and which could be used to directly test the predictions of this formalism.     

Numerical simulation of gel electrophoresis of DNA knots in weak and strong electric fields

Gel electrophoresis allows one to separate knotted DNA (nicked circular) of equal length according to the knot type. At low electric fields, complex knots, being more compact, drift faster than simpler knots. Recent experiments have shown that the drift velocity dependence on the knot type is inverted when changing from low to high electric fields. We present a computer simulation on a lattice of a closed, knotted, charged DNA chain drifting in an external electric field in a topologically restricted medium. Using a Monte Carlo algorithm, the dependence of the electrophoretic migration of the DNA molecules on the knot type and on the electric field intensity is investigated. The results are in qualitative and quantitative agreement with electrophoretic experiments done under conditions of low and high electric fields.     

Transmembrane calcium influx induced by ac electric fields.

Exogenous electric fields induce cellular responses including redistribution of integral membrane proteins, reorganization of microfilament structures, and changes in intracellular calcium ion concentration ([Ca2+]i). Although increases in [Ca2+]i caused by application of direct current electric fields have been documented, quantitative measurements of the effects of alternating current (ac) electric fields on [Ca2+]i are lacking and the Ca2+ pathways that mediate such effects remain to be identified. Using epifluorescence microscopy, we have examined in a model cell type the [Ca2+]i response to ac electric fields. Application of a 1 or 10 Hz electric field to human hepatoma (Hep3B) cells induces a fourfold increase in [Ca2+]i (from 50 nM to 200 nM) within 30 min of continuous field exposure. Depletion of Ca2+ in the extracellular medium prevents the electric field-induced increase in [Ca2+]i, suggesting that Ca2+ influx across the plasma membrane is responsible for the [Ca2+]i increase. Incubation of cells with the phospholipase C inhibitor U73122 does not inhibit ac electric field-induced increases in [Ca2+]i, suggesting that receptor-regulated release of intracellular Ca2+ is not important for this effect. Treatment of cells with either the stretch-activated cation channel inhibitor GdCl3 or the nonspecific calcium channel blocker CoCl2 partially inhibits the [Ca2+]i increase induced by ac electric fields, and concomitant treatment with both GdCl3 and CoCl2 completely inhibits the field-induced [Ca2+]i increase. Since neither Gd3+ nor Co2+ is efficiently transported across the plasma membrane, these data suggest that the increase in [Ca2+]i induced by ac electric fields depends entirely on Ca2+ influx from the extracellular medium.     

Synchronization of neuron population subject to steady DC electric field induced by magnetic stimulation

Electric fields, which are ubiquitous in the context of neurons, are induced either by external electromagnetic fields or by endogenous electric activities. Clinical evidences point out that magnetic stimulation can induce an electric field that modulates rhythmic activity of special brain tissue, which are associated with most brain functions, including normal and pathological physiological mechanisms. Recently, the studies about the relationship between clinical treatment for psychiatric disorders and magnetic stimulation have been investigated extensively. However, further development of these techniques is limited due to the lack of understanding of the underlying mechanisms supporting the interaction between the electric field induced by magnetic stimulus and brain tissue. In this paper, the effects of steady DC electric field induced by magnetic stimulation on the coherence of an interneuronal network are investigated. Different behaviors have been observed in the network with different topologies (i.e., random and small-world network, modular network). It is found that the coherence displays a peak or a plateau when the induced electric field varies between the parameter range we defined. The coherence of the neuronal systems depends extensively on the network structure and parameters. All these parameters play a key role in determining the range for the induced electric field to synchronize network activities. The presented results could have important implications for the scientific theoretical studies regarding the effects of magnetic stimulation on human brain.     

Interaction between two conducting spheres in weakly ionized collisional plasma

Charging of two conducting spheres in a weakly ionized collisional plasma flow is considered. The spheres are arranged along the flow, and the plasma is assumed to consist of two ion species with the charges equal in magnitude but opposite in sign. The problem is analyzed with allowance for the external electric field, charging of the spheres due to the sedimentation of plasma ions on them, the fields of the sphere charges, the space charge field, and the processes of recombination and molecular diffusion. The interaction between the spheres and plasma is studied by numerically solving a time-dependent problem in a bispherical coordinate system by the finite difference method. The steady-state values of the sphere charges and the distributions of the space charge and ion densities in the ambient plasma are found as functions of the plasma parameters and the distance between the spheres. The electrostatic forces acting on the spheres are determined, and the effects of the external field, the space charge fields, and the fields of the sphere charges are comparatively analyzed. It is shown that, for the considered plasma parameters, the main electrostatic effect in the interaction between two spheres is their mutual approach in the external field due to the difference in their charges (one sphere catches up with the other). Due to the friction force with the neutral gas, this mutual approach is much slower than all other processes in the system. For widely spaced spheres, the results coincide with the solution obtained previously for a solitary sphere.     

Formation and properties of aqueous leaks induced in human erythrocytes by electrical breakdown

Leaks were induced in human erythrocytes by brief (tau = 1-40 microseconds) discharges of high electric fields (3-20 kV/cm). Leak permeabilities were characterized by measuring (a) net and tracer fluxes of K+ and nonelectrolytes under protection of the cells against colloid-osmotic lysis, or (b) rates of colloid osmotic lysis in various salt solutions. The induced permeabilities are essentially stable for hours at 0-2 degrees C. Leak permeability P increases exponentially with the breakdown voltage ED according to a function of the general type P = bED. The basis b varies with the pulse length. A log-linear presentation reveals a biphasic linear relationship with a break at which the slope (= log b) decreases markedly. Elevated ionic strengths of the suspension medium during the electric discharge enhance leak formation. Leak permeability exhibits an apparent activation energy of 29 +/- 5 kJ/mol, indicative of diffusion through aqueous pathways. Somewhat differing equivalent pore radii emerge from measurements with different probes: 0.6-0.8 nm from tracer fluxes of polyols (Mr = 3600, ED = 4-7 kV/cm) and 0.8-1.9 nm from osmotic protection studies with polyethylene glycols (Mr = 200-3300, ED = 6-10 kV/cm). These numbers and the non-monoexponential increase of leak permeability with the field strength suggest a dual mechanism for the increase of leak permeability: an increase of the number of pores at low breakdown voltage and an additional increase of pore size at higher voltage. Estimated numbers of pores range from 1 to 10 per cell, which suggests dynamic fluctuating structural defects to be involved. The leaks discriminate small monovalent inorganic ions in the sequence of free solution mobility. Organic anions are discriminated according to size and charge. Common properties of these electrically induced defects and of chemically induced leaks (diamide, periodate, t-butylhydroperoxide) in the erythrocyte membrane suggest close similarities in the molecular organization.     

Brownian diffusion and electrophoretic transport of double‐stranded DNA in agarose gels

The field free diffusion constant and the electric field dependence of the electrophoretic mobility and molecular orientation of DNA samples from 5 to 164 kilobase pairs in agarose gels from 0.5 to 2% have been measured by fluorescence recovery after photobleaching and birefringence. In conditions where the reptation predictions hold for the field free diffusion, they partially fail for the DNA size dependence of the low field limit of the electrophoretic mobility. The linear field dependencies of the electrophoretic mobility and orientation factor seem to favor the biased reptation model with fluctuations over the standard biased reptation model, which predicts a quadratic field dependence. The quantitative analysis of the molecular parameters shows, however, that most experiments have been carried out at values of the field where the difference between the two models may be less conclusive. The pore size dependence of the different quantities has been given a particular attention and the role of the distribution of pore sizes in the departures from the reptation predictions is discussed. © 1999 John Wiley & Sons, Inc. Biopoly 50: 45–59, 1999     

Tensorial electrokinetics in articular cartilage

Electrokinetic phenomena contribute to biomechanical functions of articular cartilage and underlie promising methods for early detection of osteoarthritic lesions. Although some transport properties, such as hydraulic permeability, are known to become anisotropic with compression, the direction-dependence of cartilage electrokinetic properties remains unknown. Electroosmosis experiments were therefore performed on adult bovine articular cartilage samples, whereby fluid flows were driven by electric currents in directions parallel and perpendicular to the articular surface of statically compressed explants. Magnitudes of electrokinetic coefficients decreased slightly with compression (from approximately -7.5 microL/As in the range of 0-20% compression to -6.0 microL/As in the 35-50% range) consistent with predictions of microstructure-based models of cartilage material properties. However, no significant dependence on direction of the electrokinetic coupling coefficient was detected, even for conditions where the hydraulic permeability tensor is known to be anisotropic. This contrast may also be interpreted using microstructure-based models, and provides insights into structure-function relationships in cartilage extracellular matrix and physical mediators of cell responses to tissue compression. Findings support the use of relatively simple isotropic modeling approaches for electrokinetic phenomena in cartilage and related materials, and indicate that measurement of electrokinetic properties may provide particularly robust means for clinical evaluation of cartilage matrix integrity.     

Orientation behavior of retinal photoreceptors in alternating electric fields

In alternating electric (AC) fields, particles experience polarizing effects that induce dipoles that orient elongated specimens either parallel or perpendicular to the field lines. In this work we studied the behavior of photoreceptor cells' rod outer segments (ROS) in AC fields of different frequencies. We showed that at low frequencies, ROS orient parallel to the field, whereas at higher frequencies they orient perpendicular to the field lines (in the frequency range from 100 Hz to 10 MHz). We found this behavior to be dependent on the physiological state of cells (due to modifications in their electrical properties). To simulate cell damage, the membrane conductivity was changed by treating the cell with gramicidin A, which resulted in a decrease of cytosol conductivity and, consequently, in a change of the orientation behavior of the treated cells. The change of cell orientation with cytosol conductivity is rather sharp, suggesting the potential of the method for accurate evaluation of the cell physiological status. We modeled the interaction between ROS and AC fields approximating the rod cell by a prolate spheroid with a very long axis. The internal compartment of the ellipsoid was considered to be filled with an inhomogeneous medium consisting of alternating layers of membrane and cytoplasm as media modeling the disks. This theoretical model proved to be in good agreement with the experimental results and enabled the derivation (by fitting with the experimental results) of the membrane and cytosol parameters for normal and damaged cells.