Brain stimulation using electromagnetic sources: theoretical aspects.

We prove that, at the frequencies generally proposed for extracranial stimulation of the brain, it is not possible, using any superposition of external current sources, to produce a three-dimensional local maximum of the electric field strength inside the brain. The maximum always occurs on a boundary where the conductivity jumps in value. Nevertheless, it may be possible to achieve greater two-dimensional focusing and shaping of the electric field than is currently available. Towards this goal we have used the reciprocity theorem to present a uniform treatment of the electric field inside a conducting medium produced by a variety of sources: an external magnetic dipole (current loop), an external electric dipole (linear antenna), and surface and depth electrodes. This formulation makes use of the lead fields from magneto- and electroencephalography. For the special case of a system with spherically symmetric conductivity, we derive a simple analytic formula for the electric field due to an external magnetic dipole. This formula is independent of the conductivity profile and therefore embraces spherical models with any number of shells. This explains the "insensitivity" to the skull's conductivity that has been described in numerical studies. We also present analytic formulas for the electric field due to an electric dipole, and also surface and depth electrodes, for the case of a sphere of constant conductivity.     

Displacement currents associated with the insertion of Alzheimer disease amyloid beta-peptide into planar bilayer membranes

The role of endogenous amyloid beta-peptides as causal factors of neurodegenerative diseases is largely unknown. We have previously reported that interactions between Alzheimer's disease A beta P[1-40] peptide in solution and planar bilayer membranes made from anionic phospholipids lead to the formation of cation-selective channels. We now find and report here that the spontaneous insertion of free A beta P[1-40] across the bilayer can be detected as an increase in bilayer capacity. To this end we recorded the displacement currents across planar bilayers (50 mM KCl on both sides) in response to sudden displacements of the membrane potential, from -300 to 300 mV in 20-mV increments. To monitor the A beta P[1-40]-specific displacement currents, we added A beta P[1-40] (1-5 microM) to the solution on either side of the membrane and noted that the direction of the displacement current depended on the side with A beta P[1-40]. The size of the A beta P[1-40]-specific charge displaced during a pulse was always equal to the charge returning to the original configuration after the pulse, suggesting that the dipole molecules are confined to the membrane. As a rule, the steady-state distribution of the A beta P[1-40]-specific charges within the bilayer could be fit by a Boltzmann distribution. The potential at which the charges were found to be equally distributed (V(o)) were approximately -135 mV (peptide added to the solution in the compartment electrically connected to earth) and 135 mV (peptide added to the solution connected to the input of the amplifier). The A beta P[1-40]-specific transfer of charge reached a maximum value (Q(max)) when the electrical potential of the side containing the amyloid beta-protein was taken to either -300 or 300 mV. For a circular membrane of 25-microm radius ( approximately 2000 microm(2)), the total A beta P[1-40]-specific charge Q(max) was estimated as 55 fC, corresponding to some 170 e.c./microm(2). Regardless of the side selected for the addition of A beta P[1-40], at V(o) the charge displaced underwent an e-fold change for a approximately 27-mV change in potential. The effective valence (a) of the A beta P[1-40] dipole (i.e., the actual valence Z multiplied by the fraction of the electric field chi acting on the dipole) varied from 1 to 2 electronic charges. We also tested, with negative results, the amyloid peptide with the reverse sequence (A beta P[40-1]). These data demonstrate that A beta P[1-40] molecules can span the low dielectric domain of the bilayer, exposing charged residues (D(1), E(3), R(5), H(6), D(7), E(11), H(13), and H(14)) to the electric field. Thus the A beta P[1-40] molecules in solution must spontaneously acquire suitable conformations (beta-pleated sheet) allowing specific interactions with charged phospholipids. Interestingly, the domain from residues 676 to 704 in the APP(751) is homologous with the consensus sequence for lipid binding found in other membrane proteins regulated by anionic phospholipids.     

Double-inhibitor and uncoupler-inhibitor titrations. 2. Analysis with a nonlinear model of chemiosmotic energy coupling

The results of double-inhibitor and uncoupler-inhibitor titrations have been simulated and analyzed with a nonlinear model of delocalized protonic coupling obtained by linking two proton pump models of the kind studied by Pietrobon and Caplan [Pietrobon, D., & Caplan, S. R. (1985) Biochemistry 24, 5764-5776] through their common intermediate delta mu H. It is shown that the results predicted by a delocalized chemiosmotic model are highly dependent on the kind of relationships existing between rate of ATP synthesis, Jp, and delta mu H and rate of electron transfer, Je, and delta mu H. With nonlinear flow-force relationships all the results reported so far are not necessarily inconsistent with the delocalized chemiosmotic model provided that the relationships between rates and delta mu H satisfy the following requirements: Jp/delta mu H increases and/or Je/delta mu H decreases as (delta mu H) increases.     

Adiabatic solution for the Ohmic cable equation. An homogeneous cell, prospects for electronic measurements]

Exact and adiabatic electrotonic solutions [1] were calculated for reconstructed motoneurone and hippocampal interneurone in case of linear and exponential ramp stimulation by the fixed current, potential or homogenous electric field. For the rising exponential ramp the solutions are identical. In case of the decaying exponent the adiabatic solution becomes an asymptote for the exact one if the stimulus decays slower than relaxation of the initial conditions in the cell. If the stimulus decays faster, the asymptote is the current or potential axis, depending on the stimulation mode. For electrotonically short cell, the exact solution approaches the asymptote faster. The solution for the exponentially rising field does not depend on the dendritic tree configuration and depends only on the effective electrotonic length of the neurone. It could be useful to apply ramp stimulation, especially exponential ramp of the electric field, to estimate electrotonic parameters of cells.     

Study of the orientation of DNA molecules in an electrical field by anisotropy of electrical conductivity of their aqueous solution

Electro-conductive anisotropy of DNA solution caused by the molecular orientation in the external electric field was investigated. The dependence of a relative change of the conductivity of aqueous-salt DNA solution on the electric field in the amplitude range 0-700 V/cm and in the frequency range 100 Hz-10 kHz was studied. It was pointed out that the field thermal effect is an overcoming factor when the orientation of DNA molecules is investigated by field-free relaxation.     

A finite element model for ice ball evolution in a multi-probe cryosurgery

The ice formation in a water body is examined for the computation of temperature field, phase change and a moving ice-water interface whose location is not known á priori. This is classically referred to as the Stefan problem [Rubinstein, L.I. (1971) The Stefan Problem (American Mathematical Society, Providence, Rhode Island 02904]. Based on the Duvaut [Duvaut, G. (1973) "Résolution d'un probléme Stefan" C.R. Acad Sci. Paris 276, 1461-1463] transformation, the governing equations for heat conduction are formulated within a variational principle that is readily amenable to a standard finite element solution without remeshing. Numerical simulation results pertaining to the freezing of tumour tissue in a multi-cryoprobe cryosurgery are presented. These results lend both quantitative and graphical support to the current empirical standards of "effective therapy" in view of refining clinical applications.     

Kerr effect of charged dipolar macromolecules without condensed counterions in conducting solution.

The theoretical treatment of the Kerr constant of rigid, dipolar, conducting ellipsoidal macromolecules of O'Konski and Krause (1970. J. Phys. Chem. 74:3243) has been extended to very low ionic strength solutions for charged macromolecules. The O'Konski and Krause theoretical treatment postulated a surface conductivity directly on the surface of each macromolecule. For charged macromolecules, this surface conductivity was generally assumed to be caused by movement of condensed counterions on the macromolecules. In the present work, it has been assumed that, at very low ionic strength, the average counterion is at the Debye characteristic distance from the surface of each charged macromolecule and contributes to surface conductivity at that distance, with no additional surface conductivity on the true surface of the macromolecule. Essentially, these considerations change the calculated interaction energy of the macromolecule with an externally applied electric field via a change in both the internal field components and in the reaction field of the macromolecular dipole. The new interaction energy is used to calculate the orientation distribution function of the macromolecules in solution and this distribution function can, in principle, be used to calculate the steady state electric linear or circular dichroism, electric light scattering, anisotropy of conductivity, etc., using the appropriate theoretical treatment for each of these quantities.     

The effect of electric field on callus induction with rape hypocotyls

The influence of electric field treatment on dedifferentiation and calli formation on rape hypocotyls was investigated. Segments, 10 mm long, of the upper part of rape (Brassica napus L., cv. Góczański) hypocotyls were stimulated by different combinations of voltage/time (1.5 V/120 h, 3 V/3 h, 10 V/15 min and 30 V/30 s) under in vitro conditions. With all electric field treatments, segments oriented with their apical part towards the cathode produced more calli as compared to control (non-treated with electric field). Under opposite orientation slight inhibition of callus growth was observed. As the strongest effect on callus growth was observed after treatment with 30 V/30 s, this electric field treatment was selected for following analyses: the incorporation of [14C]-2,4-D (2,4-dichlorophenoxyacetic acid) and [14C]-BAP (benzylaminopurine) from the culture medium, changes in ACC (1-aminocyclopropane-1-carboxylic acid) level and the redox activity in apical and bottom parts of hypocotyls during 18 d of culture. In contrast to changes in fresh weight, electric field treatment (30 V/30 s) stimulated a higher accumulation of 2,4-D and BAP in basal parts of hypocotyls than in apical ones. Moreover, orienting the apical part towards the cathode resulted in lower uptake of hormones as compared with the opposite orientation. The ACC concentration increased, especially in the basal parts of hypocotyls, independently on electric field application. However, the highest level was observed after electric field treatment with orientation of the apical part towards the anode. The distribution of oxidative substances (measured as the amount of ferric ions) between the apical and bottom part of hypocotyls was not changed when the apical parts were oriented towards the cathode. Under these conditions a decrease in apical and an increase in basal parts was observed during culture. Opposite orientation influenced the redistribution of oxidative substances from the first day of electric field treatment. Based on these results we suggest that electric field action can be connected with its influence on specific concentration of oxidative substances and hormone distribution in cells.     

Electric field directed nucleic acid hybridization on microchips.

Selection and adjustment of proper physical parameters enables rapid DNA transport, site selective concentration, and accelerated hybridization reactions to be carried out on active microelectronic arrays. These physical parameters include DC current, voltage, solution conductivity and buffer species. Generally, at any given current and voltage level, the transport or mobility of DNA is inversely proportional to electrolyte or buffer conductivity. However, only a subset of buffer species produce both rapid transport, site specific concentration and accelerated hybridization. These buffers include zwitterionic and low conductivity species such as: d- and l-histidine; 1- and 3-methylhistidines; carnosine; imidazole; pyridine; and collidine. In contrast, buffers such as glycine, beta-alanine and gamma-amino-butyric acid (GABA) produce rapid transport and site selective concentration but do not facilitate hybridization. Our results suggest that the ability of these buffers (histidine, etc.) to facilitate hybridization appears linked to their ability to provide electric field concentration of DNA; to buffer acidic conditions present at the anode; and in this process acquire a net positive charge which then shields or diminishes repulsion between the DNA strands, thus promoting hybridization.     

The propagation of the nerve impulse.

A partial differential equation for the propagated action potential is derived using symmetry, charge conservation, and Ohm's law. Charge conservation analysis explicitly includes the gating charge when applied in the laboratory frame. When applied in the system of reference in which capacitive currents are zero, it yields a relation between orthogonal components of the ionic current allowing us to express the nonlinear ionic current in terms of the voltage-dependent membrane capacitance C(V) and the axial current that satisfies Ohm's law. The ionic current is shown to behave as C(V)V[C(V)V2]' at the foot of the action potential while the gating current behaves as C(V)V[Cg(V)V]' where Cg(V) is the capacitance associated with gating. Improved knowledge of the nonlinear current makes it possible to describe the propagated action potential in an approximated way with quasilinear partial differential equations. These equations have analytical solutions that travel with constant velocity, retain their shape, and account for other properties of the action potential. Furthermore, the quasilinear approximation is shown to be equivalent to the FitzHugh-Nagumo equation without recovery making apparent its physical content.     

In situ electroporation of radioactive compounds into adherent cells

We previously developed a technique, termed in situ electroporation, where nonpermeant molecules are introduced through an electrical pulse into adherent cells, while they grow on electrically conductive, optically transparent, indium-tin oxide (ITO). Careful control of the electric field intensity results in essentially 100% of the cells taking up the introduced material, without any detectable effect upon the physiology of the cell, presumably because the pores reseal rapidly so that the cellular interior is restored to its original state. Electroporation of radioactive material is faced with two important considerations: (1) potential for exposure of personnel to irradiation, and (2) the requirement for electroporation of a large number of cells. In this report, we describe a modification in the geometry of the slides and electrodes which permits the use of inexpensive ITO-coated glass of lower conductivity that can be discarded after use, to electroporate large numbers of cells using a minimum volume of radioactive nucleotide solution. The results demonstrate that, using this assembly, the determination of the Ras-bound GTP/GTP+GDP ratios through electroporation of [alpha32P]GTP can be conducted using approximately five times lower amounts of isotope than in previous designs. Moreover, this assembly permits efficient upscaling, which makes the determination of Ras-GTP binding in cells which are deficient in Ras activity possible. In addition, we demonstrate the labeling of two viral phosphoproteins--the Simian Virus 40 Large Tumor antigen, and Adenovirus E1A--through [gamma32P]ATP electroporation using this setup. In both cases, electroporation of the nucleotide can achieve a great increase in the efficiency and specificity of labeling compared to the addition of [32P]-orthophosphate to the culture medium, presumably because the immediate phosphate donor nucleotide itself is introduced, which can directly bind to the target proteins.     

A Finite Element Model for Ice Ball Evolution in a Multi-probe Cryosurgery

The ice formation in a water body is examined for the computation of temperature field, phase change and a moving ice–water interface whose location is not known à priori. This is classically referred to as the Stefan problem [Rubinstein, L.I. (1971) The Stefan Problem (American Mathematical Society, Providence, Rhode Island 02904]. Based on the Duvaut [Duvaut, G. (1973) “Résolution d'un problème de Stefan” C.R. Acad. Sci. Paris 276, 1461–1463] transformation, the governing equations for heat conduction are formulated within a variational principle that is readily amenable to a standard finite element solution without remeshing. Numerical simulation results pertaining to the freezing of tumour tissue in a multi-cryoprobe cryosurgery are presented. These results lend both quantitative and graphical support to the current empirical standards of “effective therapy” in view of refining clinical applications.     

Bifurcation analysis of a nonlinear field model of regeneration

The analysis of bifurcating solutions in the Totafurno and Trainor [23] model of supernumerary limb production in salamanders is re-examined using the symmetry analysis developed by Totafurno [22]. In particular, we show analytically that the appearance of field solutions possessing 2 and 4 singularities (the 2- and 4-centered solutions, respectively) also correspond to true bifurcations with reduced symmetries, just as had been previously found for a solution to the field equations not possessing such singularities (the twist solution). While the results have significance primarily for the biological problem, this work serves as an instructive example of the application of symmetry groups to the bifurcation analysis of nonlinear field equations arising from a variational principle. The relationship between the solutions of the nonlinear equations and the corresponding linear equations is discussed.Supported by the National Sciences and Engineering Research Council and the Medical Research Council of CanadaTo whom correspondence should be sent     

Capacitative pulsed electric stimulation of bone cells. Induction of cyclic-AMP changes and DNA synthesis

Pulsed electric stimulation, coupled capacitively to bone cells isolated from rat embryo calvaria, caused changes in the intracellular level of cyclic AMP and enhanced DNA synthesis. The capacitive method of electrical stimulation was characterized in terms of displacement currents (0.7-4.0 A) and voltages (10-54 V/cm) prevailing in the stimulation chamber. Changes, both in cyclic AMP and in incorporation of [3H]thymidine into DNA, were correlated with the strength of the applied electric field. Unlike the mechanical stimulation of bone cells, the electrical stimulus was not mediated by de novo synthesis of prostaglandins. The findings suggest that cyclic-AMP changes, induced by the capacitive electrical stimulation of bone cells, trigger DNA synthesis.     

Effects of 50 Hz electric currents and magnetic fields on the prokaryote Propionibacterium acnes

The effects of 50 Hz sinusoidal electric currents and magnetic fields on the Gram-positive skin bacterium Propionibacterium acnes were investigated. Intracellular free calcium ([Ca(2+)](i)), intracellular pH (pH(i)), and cell viability were examined, based on their relevance to ELF field studies and on previous studies conducted on P. acnes (UVA irradiation, photosensitization using porphyrin-based sensitizers, and broad-band red light). The [Ca(2+)](i) and the pH(i) were measured spectrofluorimetrically using the fluorescent probes fura-2 and BCECF, respectively. Sham-exposed controls were used to assess the field exposed samples. Cell suspensions were exposed to 50 Hz, 0.2 mT sinusoidal magnetic fields generated by using Helmholtz coils for up to 30 min. The estimated maximum induced electric field was 0.2 mV/m. Changes in [Ca(2+)](i) and cell viability were not detected. Ag/AgCl electrodes were used to expose cell suspensions to 50 Hz sinusoidal electric currents. The current densities were in the range 0.015-1500 A/m(2) (corresponding electric fields congruent with0.01-1000 V/m). Changes in [Ca(2+)](i) were not observed after current exposure. Current densities of 800 A/m(2) (electric field E congruent with550 V/m) were required for a 50% reduction in cell viability. Current densities greater than 800 A/m(2) were required for a reduction in pH(i). However, a pH gradient across the cell membrane (inside alkaline) was maintained even when exposure resulted in less than 0. 2% survival (1400 A/m(2), E congruent with950 V/m). Thus, dissipation of the pH gradient across the cell membrane and changes in [Ca(2+)](i) were not a consequence of cell inactivation by 50 Hz electric currents. This is in contrast to inactivation of P. acnes by UVA irradiation or photosensitization, where such changes have been obtained.     

Effect of continuous current during pauses between successive strokes on the decay of the lightning channel

A one-dimensional model is used to study the dynamics of the hydrodynamic parameters of the lightning channel in the return stroke and after the pulse current is damped. The effect of the continuous residual electric current during pauses between the successive strokes on the plasma cooling in the channel is analyzed. It is shown that a continuous electric current, which is several orders of magnitude lower than the peak current in the return stroke, is capable of maintaining the channel conductivity. This effect cannot be explained merely by Joule heating but is largely governed by the fact that the turbulent heat transport is substantially suppressed. In this case, even a continuous current as low as 50–100 A is capable of maintaining the conductivity of the lightning channel at a level at which only M-components can develop in the channel rather than the dart leader of the subsequent stroke.     

The effect of temperature gradient on the transport phenomenon in roots of maize plants grown under salinity conditions. conductivity and filtration properties

An analysis of flows through primary root and first node root tissues of plants grown under conditions of salinity and nutrient deficiency induced by temperature gradients was carried out using. a mathematical model. The results obtained show that high KNO₃ concentration in Knop’s nutrient solution (salinity) causes an inhibition of volume and heat flows and that the omission of KNO₃ from Knop’s nutrient solution (deficiency) stimulates these flows. The causes of the inhibition lay in the fact that salinity reduced hydraulic, electric, and osmotic conductivity when compared with the control (Knop’s solution), but relative to nutrient deficiency, it increased osmotic conductivity, electrodiffusion, diffusion, and filtration of heat flow induced by the electric and heat power. The causes of the stimulation were that deficiency partially decreased conductivities, similarly as salinity when compared with the control, and also decreased osmotic abilities of the system. By contrast, it increased heat conductivity and corresponding filtrations (diffusion-thermal, thermoosmotic). In first node root tissues, it increased all conductivities with the exception of electric conductivity, then osmotic, electroosmotic, diffusion, electrodiffusion, and filtration of heat flow and current flow, that is the number of possible ways of solution transport through root tissues increased.     

The Effect of temperature gradient on the transport phenomenon in roots of maize plants grown under salinity conditions. substance,heat, and ion flows

An analysis of flows through primary root and first node root tissues of plants grown under conditions of salinity and nutrient deficiency induced by temperature gradients was carried out using. a mathematical model. The results obtained show that high KNO₃ concentration in Knop’s nutrient solution (salinity) causes an inhibition of volume and heat flows and that the omission of KNO₃ from Knop’s nutrient solution (deficiency) stimulates these flows. The causes of the inhibition lay in the fact that salinity reduced hydraulic, electric, and osmotic conductivity when compared with the control (Knop’s solution), but relative to nutrient deficiency, it increased osmotic conductivity, electrodiffusion, diffusion, and filtration of heat flow induced by the electric and heat power. The causes of the stimulation were that deficiency partially decreased conductivities, similarly as salinity when compared with the control, and also decreased osmotic abilities of the system. By contrast, it increased heat conductivity and corresponding filtrations (diffusion-thermal, thermoosmotic). In first node root tissues, it increased all conductivities with the exception of electric conductivity, then osmotic, electroosmotic, diffusion, electrodiffusion, and filtration of heat flow and current flow, that is the number of possible ways of solution transport through root tissues increased. Part II.     

Correlation between dielectric property by dielectrophoretic levitation and growth activity of cells exposed to electric field

The purpose of this study is to develop a system analyzing cell activity by the dielectrophoresis method. Our previous studies revealed a correlation between the growth activity and dielectric property (Re[K(ω)]) of mouse hybridoma 3-2H3 cells using dielectrophoretic levitation. Furthermore, it was clarified that the differentiation activity of many stem cells could be evaluated by the Re[K(ω)] without differentiation induction. In this paper, 3-2H3 cells exposed to an alternating current (AC) electric field or a direct current (DC) electric field were cultivated, and the influence of damage by the electric field on the growth activity of the cells was examined. To evaluate the activity of the cells by measuring the Re[K(ω)], the correlation between the growth activity and the Re[K(ω)] of the cells exposed to the electric field was examined. The relations between the cell viability, growth activity, and Re[K(ω)] in the cells exposed to the AC electric field were obtained. The growth activity of the cells exposed to the AC electric field could be evaluated by the Re[K(ω)]. Furthermore, it was found that the adverse effects of the electric field on the cell viability and the growth activity were smaller in the AC electric field than the DC electric field.     

3D Stationary electric current density in a spherical tumor treated with low direct current: an analytical solution

Electrotherapy with direct current delivered through implanted electrodes is used for local control of solid tumors in both preclinical and clinical studies. The aim of this research is to develop a solution method for obtaining a three-dimensional analytical expression for potential and electric current density as functions of direct electric current intensity, differences in conductivities between the tumor and the surrounding healthy tissue, and length, number and polarity of electrodes. The influence of these parameters on electric current density in both media is analyzed. The results show that the electric current density in the tumor is higher than that in the surrounding healthy tissue for any value of these parameters. The conclusion is that the solution method presented in this study is of practical interest because it provides, in a few minutes, a convenient way to visualize in 3D the electric current densities generated by a radial electrode array by means of the adequate selection of direct current intensity, length, number, and polarity of electrodes, and the difference in conductivity between the solid tumor and its surrounding healthy tissue.