Parametric resonance and amplification in excitable membranes. The Hodgkin-Huxley model

It has previously been shown by different investigators that the excitable membrane shows a resonant sensitivity to periodic external perturbations, but its Q-factor is, as a rule, low. The present paper analyses the possible ways of increasing the membrane Q, using a model of the Hodgkin-Huxley type. It is found, in particular, that it can be increased considerably by modulating periodically the membrane capacitance or the activation and inactivation rate constants of ionic channels, with a frequency of about 2 fo (fo being the fundamental frequency of damped oscillations in the membrane), the extent of modulation not exceeding the critical value 2/Q. In this case, a significant parametric amplification of the membrane current takes place. If the modulation coefficient is above 2/Q, the membrane can display a parametric resonance that causes stable self-oscillations in the potential with a frequency approximately fo. The conditions for the realization of parametric amplification and resonance in biological membranes are discussed.     

Frequency domain impedance measurements of erythrocytes. Constant phase angle impedance characteristics and a phase transition.

We report measurements of the electrical impedance of human erythrocytes in the frequency range from 1 Hz to 10 MHz, and for temperatures from 4 to 40 degrees C. In order to achieve high sensitivity in this frequency range, we embedded the cells in the pores of a filter, which constrains the current to pass through the cells in the pores. Based on the geometry of the cells embedded in the filter a circuit model is proposed for the cell-filter saline system. A constant phase angle (CPA) element, i.e., an impedance of the form Z = A/(j omega)alpha, where A is a constant, j = square root of -1, omega is angular frequency, and 0 less than alpha less than 1 has been used to describe the ac response of the interface between the cell surface and the electrolyte solution, i.e., the electrical double layer. The CPA and other elements of the circuit model are determined by a complex nonlinear least squares (CNLS) fit, which simultaneously fits the real and imaginary parts of the experimental data to the circuit model. The specific membrane capacitance is determined to be 0.901 +/- 0.036 microF/cm2, and the specific cytoplasm conductivity to be 0.413 +/- 0.031 S/m at 26 degrees C. The temperature dependence of the cytoplasm conductivity, membrane capacitance, and CPA element has been obtained. The membrane capacitance increases markedly at approximately 37 degrees C, which suggests a phase transition in the cell membrane.     

Evidence of a Hopf bifurcation in frog hair cells

The membrane potential of hair cells in the low-frequency hearing organ of the bullfrog, the amphibian papilla, sinusoidally oscillates at small amplitude in the absence of acoustical input. We stimulate the cell with a series of periodic currents close to this natural frequency and observe that its current-to-voltage transfer function is compressively nonlinear, having a large gain for small stimuli and a smaller gain for larger currents. Along with the spontaneous oscillation, this implies that the cell is poised close to a dynamical instability such as a Hopf bifurcation, because distant from the instability the transfer function becomes linear. The cell's frequency selectivity is enhanced for small stimuli. Simulations show that the cell's membrane capacitance is effectively reduced due to a current gain provided by this dynamical instability. We propose that the Hopf resonance is widely used by transducer cells on the sensory periphery to achieve small-signal amplification.     

Fluctuation of motor charge in the lateral membrane of the cochlear outer hair cell

Functioning of the membrane motor of the outer hair cell is tightly associated with transfer of charge across the membrane. To obtain further insights into the motor mechanism, we examined kinetics of charge transfer across the membrane in two different modes. One is to monitor charge transfer induced by changes in the membrane potential as an excess membrane capacitance. The other is to measure spontaneous flip-flops of charges across the membrane under voltage-clamp conditions as current noise. The noise spectrum of current was inverse Lorentzian, and the capacitance was Lorentzian, as theoretically expected. The characteristic frequency of the capacitance was approximately 10 kHz, and that for current noise was approximately 30 kHz. The difference in the characteristic frequencies seems to reflect the difference in the modes of mechanical movement associated with the two physical quantities.     

Dynamic characteristics of membrane ions in multifield configurations of low-frequency electromagnetic radiation

We seek to extend the recent suggestion that classical cyclotron resonance of biologically important ions is implicated in weak electromagnetic field-cell interactions. The motion of charged particles in a constant magnetic field and periodic electric field is examined under the simplifying assumption of no damping. Each of the nine terms of the relative dielectric tensor is found to have a dependence on functions that include the factor (omega 2 - omega 2B)-1, where omega B is the gyrofrequency. We also find a plasmalike decomposition of the electric field into oppositely rotating components that could conceivably act to drive oppositely charged ions in the same direction through helical membrane channels. For weak low-frequency magnetic fields, an additional feature arises, namely, periodic reinforcement of the resonance condition with intervals of the order of tens of msec for biological ions such as Li+, Na+, and K+.     

Periodic emission bursts accompanying the interaction of plasma flows propagating across a magnetic field

It is found experimentally that the interaction of plasma flows propagating in opposite directions across a magnetic field is accompanied by periodic local bursts of intense optical and soft X-ray emission. It is shown that periodic variations in the emission intensity from the interaction region are related to the excitation of steady-state self-oscillations in the plasma. By varying the frequency and amplitude of these oscillations, it is possible to satisfy the resonance conditions at certain characteristic plasma frequencies.     

Electrophysiological properties of isolated rat liver cells

The electrophysiological properties of isolated rat liver cells were studied using the patch clamp method in whole-cell configuration. The membrane potential in isolated hepatocytes was -42 +/- 7 mV (n = 20). The input resistance (Rin) and the time constant (tau m) were 51 +/- 17 M (the range of 34 to 180 M omega) (n = 20) and 4.2 +/- 1.0 msec (the range of 3 to 16.5 ms) (n = 20). Assuming that the specific membrane capacitance is 1 microF/cm2, the membrane resistance and membrane capacitance were 42. +/- 9.0 K omega cm2 and 87 +/- 27 pF. These values indicate that isolated rat hepatocytes are not abnormally permeable or leaky. The current-voltage relationship was linear with no rectification. The depolarizing pulse from the resting potential did not induce fast or slow inward currents even when norepinephrine or high Ca2 (3.6 mM) were applied. This indicates that there is no voltage-sensitive Ca2+ channel in the isolated hepatocytes.     

Voltage and frequency dependence of prestin-associated charge transfer

Membrane protein prestin is a critical component of the motor complex that generates forces and dimensional changes in cells in response to changes in the cell membrane potential. In its native cochlear outer hair cell, prestin is crucial to the amplification and frequency selectivity of the mammalian ear up to frequencies of tens of kHz. Other cells transfected with prestin acquire voltage-dependent properties similar to those of the native cell. The protein performance is critically dependent on chloride ions, and intrinsic protein charges also play a role. We propose an electro-diffusion model to reveal the frequency and voltage dependence of electric charge transfer by prestin. The movement of the combined charge (i.e., anion and protein charges) across the membrane is described with a Fokker–Planck equation coupled to a kinetic equation that describes the binding of chloride ions to prestin. We found a voltage- and frequency-dependent phase shift between the transferred charge and the applied electric field that determines capacitive and resistive components of the transferred charge. The phase shift monotonically decreases from zero to −90° as a function of frequency. The capacitive component as a function of voltage is bell-shaped, and decreases with frequency. The resistive component is bell-shaped for both voltage and frequency. The capacitive and resistive components are similar to experimental measurements of charge transfer at high frequencies. The revealed nature of the transferred charge can help reconcile the high-frequency electrical and mechanical observations associated with prestin, and it is important for further analysis of the structure and function of this protein.     

Sperm factor initiates capacitance and conductance changes in mouse eggs that are more similar to fertilization than IP(3)- or Ca(2+)-induced changes

We used patch clamp electrophysiology and concurrent imaging with the Ca(2+)-sensitive dye, fura-2, to study the temporal relationship between membrane capacitance and conductance and intracellular free Ca(2+) concentration ([Ca(2+)](i)) during mouse egg fertilization. We found an approximately 2 pF step increase in egg membrane capacitance and a minor increase in conductance with no change in [Ca(2+)](i) at sperm fusion. This was followed approximately 1 min later by a rise in [Ca(2+)](i) that led to larger changes in capacitance and conductance. The most common pattern for these later capacitance changes was an initial capacitance decrease, followed by a larger increase and eventual return to the approximate starting value. There was some variation in this pattern, and sub-microM peak [Ca(2+)](i) favored capacitance decrease, while higher [Ca(2+)](i) favored capacitance increase. The magnitude of accompanying conductance increases was variable and did not correlate well with peak [Ca(2+)](i). The intracellular introduction of porcine sperm factor reproduced the postfusion capacitance and conductance changes with a similar [Ca(2+)](i) dependence. Raising [Ca(2+)](i) by the intracellular introduction of IP(3) initiated fertilization-like capacitance changes, but the conductance changes were slower to activate. Capacitance decrease could be induced when [Ca(2+)](i) was increased modestly by activation of an endogenous Ca(2+) current, with little effect on resting conductance. These results suggest that net turnover of the mouse egg surface membrane is sensitive to [Ca(2+)](i) and that sperm and the active component of sperm factor may be doing more than initiating the IP(3)-mediated release of intracellular Ca(2+).     

PGE(2) activation of apical membrane Cl(-) channels in A6 epithelia: impedance analysis.

Measurements of transepithelial electrical impedance of continuously short-circuited A6 epithelia were made at audio frequencies (0.244 Hz to 10.45 kHz) to investigate the time course and extent to which prostaglandin E(2) (PGE(2)) modulates Cl(-) transport and apical membrane capacitance in this cell-cultured model epithelium. Apical and basolateral membrane resistances were determined by nonlinear curve-fitting of the impedance vectors at relatively low frequencies (<50 Hz) to equations (P?unescu, T. G., and S. I. Helman. 2001. Biophys. J. 81:838--851) where depressed Nyquist impedance semicircles were characteristic of the membrane impedances under control Na(+)-transporting and amiloride-inhibited conditions. In all tissues (control, amiloride-blocked, and amiloride-blocked and furosemide-pretreated), PGE(2) caused relatively small (< approximately 3 microA/cm(2)) and rapid (<60 s) maximal increase of chloride current due to activation of a rather large increase of apical membrane conductance that preceded significant activation of Na(+) transport through amiloride-sensitive epithelial Na(+) channels (ENaCs). Apical membrane capacitance was frequency-dependent with a Cole-Cole dielectric dispersion whose relaxation frequency was near 150 Hz. Analysis of the time-dependent changes of the complex frequency-dependent equivalent capacitance of the cells at frequencies >1.5 kHz revealed that the mean 9.8% increase of capacitance caused by PGE(2) was not correlated in time with activation of chloride conductance, but rather correlated with activation of apical membrane Na(+) transport.     

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.     

Heptaminol hydrochloride as an epithelium transporter

Electrophysiological and transport effects induced by heptaminol hydrochloride were studied in frog epithelium. This tissue, which can easily be maintained in vitro, is a valuable model for studying sodium active transport with hormone-dependent characteristics that reproduce mammalian nephron behavior (notably in areas with tight gap junctions). The two following techniques were used: the Ussing short-circuit current method and the swept-frequency impedance measurement method. Our findings indicate the following. (i) Heptaminol hydrochloride significantly increases the short-circuit current and transepithelial polarization. (ii) This effect develops progressively as the molecule is introduced on the serous side (3Na+/2K+ active countertransport sites). Time to maximum development is approximately 20 min and the electrophysiological effect lasts from 60 to 90 min. (iii) The mean equivalent cationic current rise is larger in sulfate-Ringer (+23 +/- 4.6 microA, p less than 0.01) than in chloride-Ringer (+14 +/- 4.9 microA, p less than 0.05). The increase in short-circuit current is approximately 0.9 muequiv. cm-2 h-1 in sulfate-Ringer. (iv) The increase in mean polarization is greater in chloride (+21 +/- 6.2 mV, p less than 0.02) than in sulfate (+6 +/- 1.5 mV, p less than 0.01) following a diphasic effect on potential. (v) Changes in apical impedance Z are small (-454 +/- 323 omega, nonsignificant) compared with transepithelial resistance in sulfate (-1065 +/- 359 omega, p less than 0.05). (vi) Changes in membrane capacitance reflect changes in the membrane surface. However, no significant capacitance changes are produced in sulfate and chloride solution by heptaminol hydrochloride (-0.04 +/- 0.11 microF and 0.05 +/- 0.11 microF, respectively).(ABSTRACT TRUNCATED AT 250 WORDS)     

Prestin-driven cochlear amplification is not limited by the outer hair cell membrane time constant

Outer hair cells (OHCs) provide amplification in the mammalian cochlea using somatic force generation underpinned by voltage-dependent conformational changes of the motor protein prestin. However, prestin must be gated by changes in membrane potential on a cycle-by-cycle basis and the periodic component of the receptor potential may be greatly attenuated by low-pass filtering due to the OHC time constant (τ(m)), questioning the functional relevance of this mechanism. Here, we measured τ(m) from OHCs with a range of characteristic frequencies (CF) and found that, at physiological endolymphatic calcium concentrations, approximately half of the mechanotransducer (MT) channels are opened at rest, depolarizing the membrane potential to near -40 mV. The depolarized resting potential activates a voltage-dependent K+ conductance, thus minimizing τ(m) and expanding the membrane filter so there is little receptor potential attenuation at the cell's CF. These data suggest that minimal τ(m) filtering in vivo ensures optimal activation of prestin.     

Compensation for resistance in series with excitable membranes.

Extracellular resistance in series (Rs) with excitable membranes can give rise to significant voltage errors that distort the current records in voltage-clamped membranes. Electrical methods for measurement of and compensation for such resistances are described and evaluated. Measurement of Rs by the conventional voltage jump in response to a current step is accurate but the measurement of sine-wave admittance under voltage-clamp conditions is better, having about a fivefold improvement in resolution (+/- 0.1 omega cm2) over the conventional method. Conventional feedback of the membrane current signal to correct the Rs error signal leads to instability of the voltage clamp when approximately two-thirds of the error is corrected. We describe an active electronic bridge circuit that subtracts membrane capacitance from the total membrane current and allows full, yet stable, compensation for the voltage error due to ionic currents. Furthermore, this method provides not only fast and accurate control of the membrane potential in response to a command step, but also fast recovery following an abrupt change in the membrane conductance. Marked changes in the kinetics and amplitude of ionic currents resulting from full compensation for Rs are shown for several typical potential patterns.     

Reduction of electrode polarization capacitance in low-frequency impedance spectroscopy by using mesh electrodes

Dielectric measurements of biological samples are obscured by electrode polarization, which at low frequencies dominates over the actual sample response. Reduction of this artifact is especially necessary in studying interactions of electric field with biological systems in the α-dispersion range. We developed a method to reduce the influence of electrode polarization by employing mesh instead of solid electrodes as sensing probes, thereby reducing the area of the double layer. The design decreases the electrode-electrolyte contact area by almost 40% while keeping the bulk sample capacitance the same. Interrogation electric fields away from the electrode surface and sensitivity are unaffected. Electrodes were microfabricated (600μm×50μm, spacing of 100μm) with and without mesh holes 7.5μm×7.5μm in size. Simulations of electric field performed using Comsol Multiphysics showed non-uniformity of the electric field within less than 1.5μm from the electrode surface, which encompasses the double layer region, but at greater distance the solid and mesh electrodes gave the same results. Mesh electrodes reduced capacitance measurements for water and KCl solutions of different concentrations at low frequencies (<10kHz), while higher frequency capacitance remained the same for both electrode types, confirming our hypothesis that this design leaves the electric field mainly unaffected. Impedance measurements at low frequencies for water and mice heart mitochondrial suspension were lower for mesh than for solid electrodes. Comsol simulations confirmed these results by showing that mesh electrodes have a greater charge density than solid electrodes, which affects conductance. These electrodes are being used for mitochondrial membrane potential studies.     

An optical determination of the series resistance in Loligo

The resistance in series with the membrane capacitance in the giant axon of the squid Loligo pealei was measured using potentiometric probes that exhibit absorbance changes proportional to the voltage across the plasma membrane proper. The method relies upon the fact that a voltage drop across the series resistance produces a deviation in the true transmembrane voltage from that imposed by a voltage clamp. Optical measurement of the true transmembrane potential, together with electrical measurement of the ionic current, permits the immediate determination of the series resistance by Ohm's law. An alternative method monitored the amount of electronic series resistance compensation required to force the optical signal to match the shape of the reference potential. The value of the series resistance measured in artificial seawater was 3.78 +/- 0.95 omega X cm2. The estimated value of the contribution of the Schwann cell layer to the series resistance was 2.57 +/- 0.89 omega X cm2.     

A cleft model for cardiac Purkinje strands.

Conduction of the action potential in cardiac muscle is complicated by its multicellular structure, with narrow intercellular clefts and cell-to-cell coupling. A model is developed from anatomical data to describe cardiac Purkinje strands of variable diameter and different internal arrangements of cells. The admittance of the model is solved analytically and fit to results of cable analysis. Using the extracted specific membrane and cell electrical parameters (Rm = 13 K omega cm2, Cm = 1.5 mu F/cm2, Ri = 100 mu cm, and Re = 50 omega cm), the model correctly predicted conduction velocity and filling of capacitance at the onset of a voltage step. The analysis permits more complete studies of the factors controlling conduction velocity; for instance, the effect on conduction velocity of a capacity in the longitudinal current circuit is discussed. Predictions of the impedance and phase angle were also made. Measurements of the frequency dependence of phase angle may provide a basis for separating cleft membrane properties from those of the surface membrane and may aid the measurement of nonlinear membrane properties in muscle.     

Membrane capacity of squid giant axon during hyper- and depolarizations

Summary The change in membrane capacitance and conductance of squid giant axons during hyper- and depolarizations was investigated. The measurements of capacitance and conductance were performed using an admittance bridge with resting, hyperpolarized and depolarized membranes. The duration of DC pulses is 20–40 msec and is long enough to permit the admittance measurements between 1 and 50 kHz. The amplitudes of DC pulses were varied between 0 and 40mV for both depolarization and hyperpolarization. Within these limited experimental conditions, we found a substantial increase in membrane capacitance with depolarization and a decrease with hyperpolarization. Our results indicate that the change in membrane capacitance will increase further if low frequencies are used with larger depolarizing pulses. The change in membrane capacitance is frequency dependent and it increases with decreasing frequencies. The analyses based on an equivalent circuit (vide infra) gives rise to a time constant of active membrane capacitance close to that of sodium currents. This result indicates that the observed capacitance changes may arise from sodium channels. A brief discussion is given on the nature of frequency-dependent membrane capacitance of nerve axons.     

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.     

The noise of membrane capacitance measurements in the whole-cell recording configuration

High-resolution measurement of membrane capacitance in the whole-cell-recording configuration can be used to detect small changes in membrane surface area that accompany exocytosis and endocytosis. We have investigated the noise of membrane capacitance measurements to determine the fundamental limits of resolution in actual cells in the whole-cell mode. Two previously overlooked sources of noise are particularly evident at low frequencies. The first noise source is accompanied by a correlation between capacitance estimates, whereas the second noise source is due to "1/f-like" current noise. An analytic expression that summarizes the noise from thermal and 1/f sources is derived, which agrees with experimental measurements from actual cells over a large frequency range. Our results demonstrate that the optimal frequencies for capacitance measurements are higher than previously believed. Finally, we demonstrate that the capacitance noise at high frequencies can be reduced by compensating for the voltage drop of the sine wave across the series resistance.