Spike latency and jitter of neuronal membrane patches with stochastic Hodgkin-Huxley channels

Ion channel stochasticity can influence the voltage dynamics of neuronal membrane, with stronger effects for smaller patches of membrane because of the correspondingly smaller number of channels. We examine this question with respect to first spike statistics in response to a periodic input of membrane patches including stochastic Hodgkin-Huxley channels, comparing these responses to spontaneous firing. Without noise, firing threshold of the model depends on frequency—a sinusoidal stimulus is subthreshold for low and high frequencies and suprathreshold for intermediate frequencies. When channel noise is added, a stimulus in the lower range of subthreshold frequencies can influence spike output, while high subthreshold frequencies remain subthreshold. Both input frequency and channel noise strength influence spike timing. Specifically, spike latency and jitter have distinct minima as a function of input frequency, showing a resonance like behavior. With either no input, or low frequency subthreshold input, or input in the low or high suprathreshold frequency range, channel noise reduces latency and jitter, with the strongest impact for the lowest input frequencies. In contrast, for an intermediate range of suprathreshold frequencies, where an optimal input gives a minimum latency, the noise effect reverses, and spike latency and jitter increase with channel noise. Thus, a resonant minimum of the spike response as a function of frequency becomes more pronounced with less noise. Spike latency and jitter also depend on the initial phase of the input, resulting in minimal latencies at an optimal phase, and depend on the membrane time constant, with a longer time constant broadening frequency tuning for minimal latency and jitter. Taken together, these results suggest how stochasticity of ion channels may influence spike timing and thus coding for neurons with functionally localized concentrations of channels, such as in “hot spots” of dendrites, spines or axons.     

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.     

Tonotopical organization and pure tone response characteristics of single units in the auditory cortex of the Greater Horseshoe Bat

Summary Tonotopical organization and frequency representation in the auditory cortex of Greater Horseshoe Bats was studied using multi-unit recordings.The auditory responsive cortical area can be divided into a primary and a secondary region on the basis of response characteristics forming a core/belt structure.In the primary area units with best frequencies in the range of echolocation signals are strongly overrepresented (Figs. 6–8). There are two separate large areas concerned with the processing of the two components of the echolocation signals. In one area frequencies between the individual resting frequency and about 2 kHz above are represented, which normally occur in the constant frequency (CF) part of the echoes (CF-area), in a second one best frequencies between resting frequency and about 8 kHz below are found (FM-area).In the CF-area tonotopical organization differs from the usual mammalian scheme of dorso-ventral isofrequency slabs. Here isofrequency contours are arranged in a semicircular pattern.The representation of the cochlear partition (cochleotopic organization) was calculated. In the inferior colliculus and auditory cortex there is a disproportionate representation of the basilar membrane. This finding is in contradiction to the current opinion that frequency representation in the auditory system of Horseshoe Bats is only determined by the mechanical tuning properties of the basilar membrane.Response characteristics for single units were studied using pure tone stimuli. Most units showed transient responses. In 25% of units response characteristics depended on the combination of frequency and sound pressure level used.Frequency selectivity of units with best frequencies in the range of echolocation sounds is very high. Q-10dB values of up to 400 were found in a small frequency band just above resting frequency.Abbreviations BF best frequency - CF constant frequency - FM frequency modulated - MT minimal threshold     

Parametric resonance and amplification of periodic disturbances in membranes containing ion channels with inactivation

Parametric resonance and amplification of periodic perturbations in the membrane transport of ions through channels with inactivation was studied in computational experiments. It has been shown that a periodic change in the membrane capacitance or in the applied electric current with a frequency approximately 2 omega 0 (omega 0--the own angular frequency of the membrane) may excite stable self-oscillations in the membrane with a frequency of approximately omega 0. For this to occur, the degree of the capacitance modulation m or the amplitude of the applied current i0 must exceed some critical values mcr and i0cr. Excitation of self-oscillations by alternating electric current of the frequency approximately 2 omega 0 has the characteristics of parametric resonance. This can be explained by the fact that the equivalent membrane inductance depends on ionic current and displays periodic changes with a frequency approximately 2 omega 0, as also does the current. Small-amplitude periodic changes in the capacitance (m less than mcr) with frequencies approximately 2 omega 0 may result in significant amplification of periodic perturbations with frequencies approximately omega 0.     

The frequency response function and sinusoidal threshold properties of the Hodgkin-Huxley model of action potential encoding

The behaviour of the space-clamped Hodgkin-Huxley model has been studied using bandlimited white noise (0–50 Hz) as the input membrane current and taking the output as a point process in time given by the peaks of the action potentials. The frequency response and coherence functions were measured by use of the Fourier transform and digital filtering of the spike train. The results obtained are in good agreement with those already published for the simple integrator and leaky integrator models of neuronal encoding, as well as the earlier studies on the response of the Hodgkin-Huxley model to steady currents. In addition, the threshold of the model to sinusoidal membrane currents has been measured as a function of frequency over the range of 0.1–100 Hz. This shows a relatively constant level up to 2 Hz and then a clear minimum at 60 Hz, in agreement with measured thresholds of squid axons. These results are discussed in terms of the possible contributions of action potential encoding mechanisms to the frequency responses and sinusoidal thresholds which have been measured for rapidly adapting receptors.     

Limiting Frequency of the Cochlear Amplifier Based on Electromotility of Outer Hair Cells

Outer hair cells are the critical element for the sensitivity and sharpness of frequency selectivity of the ear. It is believed that fast motility (electromotility) of these cells is essential for this function. Indeed, force produced by outer hair cells follows their membrane potential very closely at least up to 60 kHz. However, it has been pointed out that the cell's receptor potential is attenuated by a low-pass RC circuit inherent to these cells, with the RC roll-off frequencies significantly lower than their operating frequencies. This would render electromotility ineffective in producing force. To address this issue, we assume that multiple degrees of freedom and vibrational modes due to the complex structure of the organ of Corti provide optimal phases for outer hair cells' force to cancel viscous drag. Our derived frequency limit depends on the drag-capacitance product, not directly on the RC time constant. With a reasonable assumption for the viscous drag, the estimated limit is 10–13 kHz, exceeding the RC corner frequency. Our analysis shows that a fast-activating potassium current can substantially extend the frequency limit by counteracting the capacitive current.     

Slow Changes of Potassium Permeability in the Squid Giant Axon

A slow potassium inactivation i.e. decrease of conductance when the inside of the membrane is made more positive with respect to the outside, has been observed for the squid axon. The conductance-potential curve is sigmoid shaped, and the ratio between maximum and minimum potassium conductance is at least 3. The time constant for the change of potassium conductance with potential is independent of the concentration of potassium in the external solution, but dependent upon potential and temperature. At 9 degrees C and at the normal sea water resting potential, the time constant is 11 sec. For lower temperature or more depolarizing potentials, the time constant is greater. The inactivation can be described by modifying the Hodgkin-Huxley equation for potassium current, using one additional parameter. The modified equation is similar in form to the Hodgkin-Huxley equation for sodium current, suggesting that the mechanism for the passive transport of potassium through the axon membrane is similar to that for sodium.     

ELECTRIC IMPEDANCE OF ARBACIA EGGS

The alternating current resistance and capacity of suspensions of unfertilized and fertilized eggs of Arbacia punctulata have been measured at frequencies from 10³ to 1.64 x 10⁷ cycles per second. The unfertilized egg has a static plasma membrane capacity of 0.73 µf./cm.² which is practically independent of frequency. The fertilized egg has a static membrane capacity of 3.1 µf./cm.² at low frequencies which decreases to a value of 0.55 µf./cm.² at high frequencies. The decrease follows closely the relaxation dispersion of the dielectric constant if the dissipation of such a system is ignored. It is considered more probable that the effect is due to a fertilization membrane of 3.1 µf./cm.² capacity lifted 1.5 µ. from the plasma membrane, the interspace having the conductivity of sea water. The suspensions show a frequency-dependent capacity at low frequencies which may be attributable to surface conductance. The equivalent low frequency internal specific resistance of both the unfertilized and fertilized egg is about 186 ohm cm. or about 6 times that of sea water, while the high frequency data extrapolate to a value of about 4 times sea water. There is evidence at the highest frequencies that the current is penetrating the nucleus and other materials in the cytoplasm. If this effect were entirely due to the nucleus it would lead to a very approximate value of 0.1 µf./cm.² for the capacity of the nuclear membrane. The measurements do not indicate any change in this effect on fertilization.     

Kinetics of light-sensitive channels in vertebrate photoreceptors

We have studied the ion channels mediating the light response of vertebrate rod photoreceptors by analysing fluctuations in the current across the rod membrane, using the whole cell patch-clamp technique on rods isolated from the axolotl retina. Light decreases the membrane current fluctuations. Noise analysis reveals two components to this decrease: a low frequency component due to biochemical noise in the transduction mechanism, and a high frequency component we attribute to the random opening and closing of the ion channels in the dark. The probability of any one channel being open in the dark is low. The spectrum of the high frequency component of the current fluctuations indicates that the current through an open channel is 4 X 10(-15)A, that the mean channel open time is 2 ms, and that about 10000 channels are open in each rod in the dark. The effect of light is to reduce the opening rate constant of these channels, with no effect on the closing rate constant.     

Urban noise and the cultural evolution of bird songs

In urban environments, anthropogenic noise can interfere with animal communication. Here we study the influence of urban noise on the cultural evolution of bird songs. We studied three adjacent dialects of white-crowned sparrow songs over a 30-year time span. Urban noise, which is louder at low frequencies, increased during our study period and therefore should have created a selection pressure for songs with higher frequencies. We found that the minimum frequency of songs increased both within and between dialects during the 30-year time span. For example, the dialect with the highest minimum frequency is in the process of replacing another dialect that has lower frequency songs. Songs with the highest minimum frequency were favoured in this environment and should have the most effective transmission properties. We suggest that one mechanism that influences how dialects, and cultural traits in general, are selected and transmitted from one generation to the next is the dialect''s ability to be effectively communicated in the local environment.     

Fluctuation and linear analysis of Na-current kinetics in squid axon

The power spectrum of current fluctuations and the complex admittance of squid axon were determined in the frequency range 12.5 to 5,000 Hx during membrane voltage clamps to the same potentials in the same axon during internal perfusion with cesium. The complex admittance was determined rapidly and with high resolution by a fast Fourier transform computation of the current response, acquired after a steady state was attained, to a synthesized signal with predetermined spectral characteristics superposed as a continuous, repetitive, small perturbation on step voltage clamps. Linear conduction parameters were estimated directly from admittance data by fitting an admittance model, derived from the linearized Hodgkin-Huxley equations modified by replacing the membrane capacitance with a "constant-phase-angle" capacitance, to the data. The constant phase angle obtained was approximately 80 degrees. At depolarizations the phase of the admittance was 180 degrees, and the real part of the impedance locus was in the left-half complex plane for frequencies below 1 kHz, which indicates a steady-state negative Na conductance. The fits also yielded estimates of the natural frequencies of Na "activation" and "inactivation" processes. By fitting Na-current noise spectra with a double Lorentzian function, a lower and an upper corner frequency were obtained; these were compared with the two natural frequencies determined from admittance analysis at the corresponding potentials. The frequencies from fluctuation analyses ranged from 1.0 to 10.3 times higher than those from linear (admittance) analysis. This discrepancy is consistent with the concept that the fluctuations reflect a nonlinear rate process that cannot be fully characterized by linear perturbation analysis. Comparison of the real part of the admittance and the current noise spectrum shows that the Nyquist relation, which generally applies to equilibrium conductors, does not hold for the Na process in squid axon. The Na-channel conductance, gamma Na, was found to increase monotonically from 0.1 to 4.8 pS for depolarizations up to 50 mV from a holding potential of -60 mV, with no indication of a maximum value.     

Comparison of membrane electroporation and protein denature in response to pulsed electric field with different durations

In this paper, we compared the minimum potential differences in the electroporation of membrane lipid bilayers and the denaturation of membrane proteins in response to an intensive pulsed electric field with various pulse durations. Single skeletal muscle fibers were exposed to a pulsed external electric field. The field‐induced changes in the membrane integrity (leakage current) and the Na channel currents were monitored to identify the minimum electric field needed to damage the membrane lipid bilayer and the membrane proteins, respectively. We found that in response to a relatively long pulsed electric shock (longer than the membrane intrinsic time constant), a lower membrane potential was needed to electroporate the cell membrane than for denaturing the membrane proteins, while for a short pulse a higher membrane potential was needed. In other words, phospholipid bilayers are more sensitive to the electric field than the membrane proteins for a long pulsed shock, while for a short pulse the proteins become more vulnerable. We can predict that for a short or ultrashort pulsed electric shock, the minimum membrane potential required to start to denature the protein functions in the cell plasma membrane is lower than that which starts to reduce the membrane integrity. Bioelectromagnetics 34:253–263, 2013. © 2012 Wiley Periodicals, Inc.     

Sinusoidal voltage clamp of the Hodgkin-Huxley model.

A voltage clamp consisting of a sinusoidal voltage of amplitude V1 and frequency f, superimposed on a steady voltage level V0, is applied to the Hodgkin-Huxley model of the squid giant axon membrane. The steady-state response is a current composed of sinusoidal components of frequencies O, f, 2f, 3f,... The frequencies greater than f arise from the nonlinearity of the membrane. The total current is described by a power series in V1; each coefficient of this series is composed of current components for one or more frequencies. For different frequencies one can derive higher-order generalized admittances characterizing the nonlinear as well as the linear properties of the membrane. Formulas for the generalized admittances are derived from the Hodgkin-Huxley equations for frequencies up to 3f, using a perturbation technique. Some of the resulting theoretical curves are compared with experimental results, with good qualitative agreement.     

Electrical Impedance Analysis in Plant Tissues3

Based on the electrical model for plant tissue proposed by Hayden,Moyse, Calder, Crawford, and Fensom (1969), a method is describedfor calculating symplasmic resistance and cell membrane capacitancefrom impedances measured over a range of alternating current(AC) frequencies. The method of calculation has been appliedto ten different plant organs using frequencies from 20 Hz to300 KHz. In contrast with previous assumptions, it was foundthat both the symplasmic resistance and the membrane capacitancewere not constant but decreased with increasing frequency, giventhe constraints of the Hayden model. In cucumber fruit tissue,the symplasmic resistance was 20 000 ohms at 3 KHz but only1200 ohms at 200 KHz; the capacitance was 2.4 nF at 3 KHz butonly 0.8 nF at 200 KHz. The changes were similar in other materials,such as carrot root and cabbage leaf. It is concluded that theHayden model does not represent plant tissues accurately. Itis suggested that a better representation would be obtainedby including a capacitor in the component of the circuit whichrepresents the symplasm, in order to make allowance for membranesof organelles, particularly the vacuole. Key words: Electrical impedance, electrical modelling, membrane capacitance     

Comparison of three methods for measuring electrical resistances of plant cell membranes

The reliability of two different membrane resistance-measuring methods that use a single intracellular microelectrode was tested against a conventional method that uses two intracellular microelectrodes. The first single-electrode method used a single square current pulse and required a constant microelectrode resistance. This method was unreliable because the electrode resistance changed markedly on cell penetration and changed with time within the cell. The second method used a high frequency square wave for injecting current into the cell and depended upon the membrane having a much longer RC (resistance × capacitance)-time constant than the microelectrode. The resistance values obtained by this latter method were usually different from membrane resistances obtained at the same time on the same cells using two intracellular microelectrodes. Therefore, neither single intracellular microelectrode method was as reliable as the conventional method. All tests were with coleoptile cells of Avena sativa var. Victory.     

Measurement of cell microrheology by magnetic twisting cytometry with frequency domain demodulation.

Magnetic twisting cytometry (MTC) (Wang N, Butler JP, and Ingber DE, Science 260: 1124-1127, 1993) is a useful technique for probing cell micromechanics. The technique is based on twisting ligand-coated magnetic microbeads bound to membrane receptors and measuring the resulting bead rotation with a magnetometer. Owing to the low signal-to-noise ratio, however, the magnetic signal must be modulated, which is accomplished by spinning the sample at approximately 10 Hz. Present demodulation approaches limit the MTC range to frequencies <0.5 Hz. We propose a novel demodulation algorithm to expand the frequency range of MTC measurements to higher frequencies. The algorithm is based on coherent demodulation in the frequency domain, and its frequency range is limited only by the dynamic response of the magnetometer. Using the new algorithm, we measured the complex modulus of elasticity (G*) of cultured human bronchial epithelial cells (BEAS-2B) from 0.03 to 16 Hz. Cells were cultured in supplemented RPMI medium, and ferromagnetic beads (approximately 5 microm) coated with an RGD peptide were bound to the cell membrane. Both the storage (G', real part of G*) and loss (G", imaginary part of G*) moduli increased with frequency as omega(alpha) (2 pi x frequency) with alpha approximately equal to 1/4. The ratio G"/G' was approximately 0.5 and varied little with frequency. Thus the cells exhibited a predominantly elastic behavior with a weak power law of frequency and a nearly constant proportion of elastic vs. frictional stresses, implying that the mechanical behavior conformed to the so-called structural damping (or constant-phase) law (Maksym GN, Fabry B, Butler JP, Navajas D, Tschumperlin DJ, LaPorte JD, and Fredberg JJ, J Appl Physiol 89: 1619-1632, 2000). We conclude that frequency domain demodulation dramatically increases the frequency range that can be probed with MTC and reveals that the mechanics of these cells conforms to constant-phase behavior over a range of frequencies approaching three decades.     

Photoelectric effects in Characean cells I. The influence of light intensity

Summary The kinetic features of the action of light on the membrane potential ofNitella mucronata were investigated by measuring the frequency responses at different light intensities ranging from 0.2 to 80 W/m². Frequencies from 1 cycle/3 h to 32 cycles/min were applied. This range exceeded that of earlier investigations and resulted in the demonstration of allpass elements at low frequencies. From the all-pass elements it was concluded that the system comprises parallel pathways.From the influence of the light intensity on the frequency responses it was seen that the kinetic data depend on the intensity. By means of the Laplace transformation the squarewave responses were calculated from the frequency responses, and it could be demonstrated that a single cell is able of exhibiting all those different types of curve shapes reported in literature, if only one parameter, the light intensity, is changed. With constant modulation depth the amplitude of the evoked changes in potential varied only little with the light intensity. This is in line with a logarithmic dose-effect function as known from many light effects.     

Fast activation of a time-dependent outward current in protoplasts derived from coats of developing Phaseolus vulgaris seeds

An outward current that appeared to activate instantaneously in response to depolarising voltage pulses at low sampling frequencies predominated in the plasma membrane of ground-parenchyma protoplasts derived from coats of developing Phaseolus vulgaris L. (cv. Redland Pioneer) seeds. However, the outward current showed time-dependent activation when higher sampling frequencies were used to measure the current. Activation of the current was best described as a double-exponential time course with the fast and slow time constants being 1 and 20 ms, respectively. The current also exhibited a rapid deactivation that followed a double-exponential time course with time constants of approximately 2 and 30 ms, respectively. “Tail-current” analysis allowed us to show that this current exhibited a low selectivity between K⁺ and Cl⁻ (P _K:Cl=1.8). Such a fast-activating current may account for some of the reports of time-independent, instantaneous currents that have been observed in plasma membranes of plant cells digitised at low sampling frequencies. Therefore, when “instantaneous” currents appear it is advisable to characterise these currents using higher sampling frequencies with correspondingly higher filtering frequency cut-offs. Received: 12 May 2000 / Accepted: 26 June 2000     

High standard one-loop potential clamp device for Ranvier nodes

Starting from the observation that using a conventional potential clamp device for membrane current measurements in Ranvier nodes neither the kinetics of sodium currents nor the constant field concept agree satisfactorily with the Hodgkin-Huxley-Frankenhaeuser (HHF)-formalism, an extendend measuring system has been developed. The extensions introduced base largely on physical implications of myelinated nerve fibres which give rise 1. to systematic distortions of any current records at the high frequency end and 2. to current proportional deviations of the membrane potential from desired potential values. In addition, we provided to meet any unwanted current load during membrane current measurements and to push the time resolution of the measuring system to the highest possible value. After having tested thoroughly the new circuitry by appropriate physical methods, from sodium current measurements the following conclusions were drawn: 1. Occasional deviations of sodium current kinetics near the sodium equilibrium potential from the predictions of the HHF-formalism are measurement errors. 2. The constant field formalism holds for sodium currents in the potential range of biological relevance only. 3. Instantaneous sodium current measurements, however, are of unsatisfactory significance because for this kind of experiments the time resolution of the measuring system used might be still too low.     

心肌细胞晚钠通道的四种开放模式及其对动作电位的影响

应用膜片箝技术记录豚鼠游离心室肌细胞钠通道电流,发现晚钠通道电流可分为四种开放模式;单个短暂开放,散在开放,长时间长放和爆发型开放。它们的开放机率不一样,其中爆发型的机率为１／２０００,其开放时间常数比前三种大。离细胞体的小片膜电压箝制实验中,亦可观察到晚钠通道的这四种开放模式,它们均可被ＴＴＸ使波动现象基本消失,动作电位时程和有效不应期缩短,静息电位增加,表明晚钠通道的活动在动作电位平台期的形成