Impedance profiles of peripheral and central neurons

The electrical impedance of trigeminal ganglion cells (in vivo) and hippocampal CA1 neurons (in vitro) of guinea pigs was measured in the frequency range of 5-1250 Hz using intracellular recording techniques with single microelectrodes and computerized methodology. The transfer functions of the electrode and the electrode-neuron system were computed from the ratio of fast Fourier transforms of the output voltage response from the neuron and input current composed of sine waves with rapidly increasing frequency which displaced membrane potential by 2-5 mV. We believe these to be the first measurements of complex impedance and transfer functions in peripheral and central neurons of vertebrates and the first use of such input current functions. The majority of trigeminal ganglion cells did not exhibit electrical behaviour ascribable to a simple resistance-capacitance (RC) circuit but showed a hump at low frequencies (5-250 Hz) in the computed transfer function, probably attributable to resonance. The transfer function in less than 20% of the trigeminal neurons could be fitted approximately to a theoretical transfer function (resistance in series with a parallel RC circuit model) providing values for electrode resistance, effective input resistance, and effective input capacitance. The transfer functions measured in hippocampal CA1 neurons were characterized by a rapid fall-off in the low frequency range (less than 200 Hz). Impedance locus plots approximate the locus corresponding to a series RC circuit in parallel with a parallel RC circuit.     

The interspike interval of a cable model neuron with white noise input

The firing time of a cable model neuron in response to white noise current injection is investigated with various methods. The Fourier decomposition of the depolarization leads to partial differential equations for the moments of the firing time. These are solved by perturbation and numerical methods, and the results obtained are in excellent agreement with those obtained by Monte Carlo simulation. The convergence of the random Fourier series is found to be very slow for small times so that when the firing time is small it is more efficient to simulate the solution of the stochastic cable equation directly using the two different representations of the Green's function, one which converges rapidly for small times and the other which converges rapidly for large times. The shape of the interspike interval density is found to depend strongly on input position. The various shapes obtained for different input positions resemble those for real neurons. The coefficient of variation of the interspike interval decreases monotonically as the distance between the input and trigger zone increases. A diffusion approximation for a nerve cell receiving Poisson input is considered and input/output frequency relations obtained for different input sites. The cases of multiple trigger zones and multiple input sites are briefly discussed.     

Random currents through nerve membranes

The linear cable equation with uniform Poisson or white noise input current is employed as a model for the voltage across the membrane of a onedimensional nerve cylinder, which may sometimes represent the dendritic tree of a nerve cell. From the Green's function representation of the solutions, the mean, variance and covariance of the voltage are found. At large times, the voltage becomes asymptotically wide-sense stationary and we find the spectral density functions for various cable lengths and boundary conditions. For large frequencies the voltage exhibits “1/f ^3/2 noise”. Using the Fourier series representation of the voltage we study the moments of the firing times for the diffusion model with numerical techniques, employing a simplified threshold criterion. We also simulate the solution of the stochastic cable equation by two different methods in order to estimate the moments and density of the firing time.     

Parameter sensitivity of distributed transfer properties of neuronal dendrites: a passive cable approximation

Geometry and membrane properties of the dendrites crucially determine input–output relations in neurons. Unlike geometry often available in detail from computer reconstruction, the membrane resistivity is fragmentarily known if at all. Moreover, it varies during ongoing activity. In this study we address the question: what is the impact of the variation in membrane resistivity on the transfer properties of dendrites? Following a standard approach of the control system theory, we derive and explore the sensitivity functions complementary to the transfer functions of the passive dendrites with arbitrary geometrical parameters (length and diameter) and boundary conditions. We use the location-dependent somatopetal current transfer ratio (the reciprocal of the somatofugal voltage) as the transfer function, and its membrane resistivity derivatives, as the sensitivity functions. In the dendrites, at every path distance from the origin, the sensitivity function in a common form relates the transfer function, membrane resistivity, characteristic input conductance of semi-infinite cable and directional somatofugal input conductances at the given internal site and origin, and the length. Plotted in membrane resistivity versus path distance coordinates, the sensitivity functions display common features: along any coordinate there are low and high ranges, in which the sensitivity, respectively, increases and decreases. The ranges and corresponding rates depend on morphology and boundary conditions in a characteristic manner. These features predict existence of the geometry-dependent range of membrane resistivity (the earlier unattended mid-conductance state), such that the dendrites with a given metrical asymmetry are most distinguished in their transfer properties and electrical states if membrane resistivity is within the range and are not otherwise.     

An efficient algorithm for cable theory,applied to blowfly photoreceptor cells and LMC's

A both simple and efficient algorithm is presented that yields the voltages and currents in an arbitrary cable structure. The algorithm consists of the following steps: 1. The cable structure is divided into homogeneous cable segments; 2. Each cable segment is considered as a two-port, and replaced by an equivalent circuit consisting of discrete elements; 3. The resulting equivalent scheme of the whole cable structure is solved with an algorithm for ladder networks (or, if the structure is not tree-like, with a network analysis program), which yields the input and output voltages and currents of each cable segment; and if required 4. The voltage and current distribution in each segment is determined from the input and output voltages and currents. The algorithm is applied to blowfly photoreceptor cells that are electrically coupled, and to blowfly Large Monopolar Cells. For LMC's it is shown that the loads at the input and output sides of the axon determine whether unidirectional or bidirectional signal transmission occurs.     

An improved statistical methodology to estimate and analyze impedances and transfer functions

Curran-Everett, Douglas, Yiming Zhang, M. Douglas Jones,Jr., and Richard H. Jones. An improved statistical methodology toestimate and analyze impedances and transfer functions. J. Appl.Physiol. 83: 2146-2157, 1997.Estimating the mathematical relationship between pulsatile time series (e.g., pressure and flow) isan effective technique for studying dynamic systems. Thefrequency-domain relationship between time series, often calculated asan impedance (pressure/flow), is known more generally as a frequency-response or transfer function (output/input). Current statistical methods for transfer function analysis 1) assumeerroneously that repeated observations on a subject are independent,2) have limited statistical value and power, or 3) arerestricted to use in single subjects rather than in an entire sample.This paper develops a regression model for transfer function analysisthat corrects each of these deficiencies. Spectral densities of the input and output time series and the cross-spectral density between them are first estimated from discrete Fourier transforms and then usedto obtain regression estimates of the transfer function. Statisticalcomparisons of the transfer function estimates use a test statisticthat is distributed as ². Confidence intervals foramplitude and phase can also be calculated. By correctly modelingrepeated observations on each subject, this improved statisticalapproach to transfer function estimation and analysis permits thesimultaneous analysis of data from all subjects in a sample, improvesthe power of the transfer function model, and has broad relevance tothe study of dynamic physiological systems.

     

Determination of time-constants in cables of finite length

Current flow in cylindrical nerve and muscle fibre has been analysed in terms of a mathematical model leading to a linear partial differential equation for the voltage as a function of both position and time. In the case of a one-dimensional cable subject to a step input of current, the solution will consist of a steady-state behaviour preceded by an initial transient. The electrical properties of the fibre or cable itself determine a length-constant, λ, which can be determined experimentally from the steady-state response, and a time-constant, τ, which must be found from the initial transient. When the cable is infinite and when there is a single input electrode, an exact solution can be produced which enables ready determination of the time-constant τ. Two complications arise in experimental practice, however. In the first place, the fibre has finite length, and in the second, two spatially separated stimulation electrodes are often required. We thus analyse a more complicated and more general situation. The linearity of the membrane properties, however, allows the solution to the more general case to be built up by superposition of solutions from the simpler case (equivalent to the classical method of images). We also approximate the Hodgkin and Rushton solution by asymptotic formulae in order to allow more tractable expressions for the exact solution. We are thus able to give a method for the ready evaluation of the time constant τ under more general conditions.     

Influence of aspect ratios on the creep behaviour of articular cartilage in indentation

Indentation tests are commonly used to determine the mechanical behaviour of articular cartilage with varying properties, thickness, and geometry. This investigation evaluated the effect of changing geometric parameters on the properties determined from creep indentation tests. Finite element analyses simulated the indentation behaviour of two models, an excised cylindrical specimen of cartilage with either normal and repair qualities and an osteochondral defect represented as a cylindrical region of repair cartilage integrated with a surrounding layer of normal tissue. For each model, the ratios of indenter radius to cartilage height (a/h=0.5,1.5) and cartilage radius to indenter radius (r/a=2,5) were varied. The vertical displacement of the cartilage under the indenter obtained through finite element analysis was fitted to a numerical algorithm to determine the aggregate modulus, permeability, and Poisson's ratio. Indentation behaviours of cartilage specimens for either model with a/h=1.5 were not affected by r/a for values of 2 and 5. Aggregate modulus was not greatly affected by the geometric changes studied. Permeability was affected by changes in the ratio of specimen to indenter radii for a/h=0.5. These findings suggest that experimental configurations of excised cylindrical specimens, also representing osteochondral defects with no or unknown degree of integration, where the cartilage layer has a/h=0.5 should not have r/a values on the order of 2 for confidence in the mechanical properties determined. Indentation of osteochondral defects where the repair cartilage is fully integrated to the surrounding cartilage can be performed with confidence for all cases tested.     

An Analytic Solution of the Cable Equation Predicts Frequency Preference of a Passive Shunt-End Cylindrical Cable in Response to Extracellular Oscillating Electric Fields

Under physiological and artificial conditions, the dendrites of neurons can be exposed to electric fields. Recent experimental studies suggested that the membrane resistivity of the distal apical dendrites of cortical and hippocampal pyramidal neurons may be significantly lower than that of the proximal dendrites and the soma. To understand the behavior of dendrites in time-varying extracellular electric fields, we analytically solved cable equations for finite cylindrical cables with and without a leak conductance attached to one end by employing the Green's function method. The solution for a cable with a leak at one end for direct-current step electric fields shows a reversal in polarization at the leaky end, as has been previously shown by employing the separation of variables method and Fourier series expansion. The solution for a cable with a leak at one end for alternating-current electric fields reveals that the leaky end shows frequency preference in the response amplitude. Our results predict that a passive dendrite with low resistivity at the distal end would show frequency preference in response to sinusoidal extracellular local field potentials. The Green's function obtained in our study can be used to calculate response for any extracellular electric field.     

Spectral Analysis of Binary Time Series: Square Waves vs. Sinusoidal Functions

The analysis of signals consisting of discrete and irregular data causes methodological problems for the Fourier spectral Analysis: Since it is based on sinusoidal functions, rectangular signals with unequal periodicities cannot easily be replicated. The Walsh spectral Analysis is based on the so called "Walsh functions", a complete set of orthonormal, rectangular waves and thus seems to be the method of choice for analysing signals consisting of binary or ordinal data. The paper compares the Walsh spectral analysis and the Fourier spectral analysis on the basis of simulated and real binary data sets of various length. Simulated data were derived from signals with defined cyclic patterns that were noised by randomly generated signals of the same length. The Walsh and Fourier spectra of each set were determined and up to 25% of the periodogram coefficients were utilized as input for an inverse transform. Mean square approximation error (MSE) was calculated for each of the series in order to compare the goodness of fit between the original and the reconstructed signal. The same procedure was performed with real data derived from a behavioral observation in pigs. The comparison of the two methods revealed that, in the analysis of discrete and binary time series, Walsh spectral analysis is the more appropriate method, if the time series is rather short. If the length of the signal increases, the difference between the two methods is less substantial.     

Spectral Analysis of Binary Time Series: Square Waves vs. Sinusoidal Functions

The analysis of signals consisting of discrete and irregular data causes methodological problems for the Fourier spectral Analysis: Since it is based on sinusoidal functions, rectangular signals with unequal periodicities cannot easily be replicated. The Walsh spectral Analysis is based on the so called "Walsh functions", a complete set of orthonormal, rectangular waves and thus seems to be the method of choice for analysing signals consisting of binary or ordinal data. The paper compares the Walsh spectral analysis and the Fourier spectral analysis on the basis of simulated and real binary data sets of various length. Simulated data were derived from signals with defined cyclic patterns that were noised by randomly generated signals of the same length. The Walsh and Fourier spectra of each set were determined and up to 25% of the periodogram coefficients were utilized as input for an inverse transform. Mean square approximation error (MSE) was calculated for each of the series in order to compare the goodness of fit between the original and the reconstructed signal. The same procedure was performed with real data derived from a behavioral observation in pigs. The comparison of the two methods revealed that, in the analysis of discrete and binary time series, Walsh spectral analysis is the more appropriate method, if the time series is rather short. If the length of the signal increases, the difference between the two methods is less substantial.     

Mapping the distribution of outer hair cell voltage-dependent conductances by electrical amputation.

The mammalian outer hair cell (OHC) functions not only as sensory receptor, but also as mechanical effector; this unique union is believed to enhance our ability to discriminate among acoustic frequencies, especially in the kilohertz range. An electrical technique designed to isolate restricted portions of the plasma membrane was used to map the distribution of voltage-dependent conductances along the cylindrical extent of the cell. We show that three voltage-dependent currents, outward K, I(K,n), and I(Ca) are localized to the basal, synaptic pole of the OHC. Previously we showed that the lateral membrane of the OHC harbors a dense population of voltage sensor-motor elements responsible for OHC motility. This segregation of membrane molecules may have important implications for auditory function. The distribution of OHC conductances will influence the cable properties of the cell, thereby potentially controlling the voltage magnitudes experienced by the motility voltage sensors in the lateral membrane, and thus the output of the "cochlear amplifier."     

Noninvasive measurements of transmural myocardial metabolites using 3-D (31)P NMR spectroscopy

A completely noninvasive three-dimensional (3-D) static magnetic field magnitude spatially localized (31)P spectroscopy technique has been developed and applied to study the in vivo canine myocardium at 9.4 T. The technique incorporates both Fourier series windows and selective Fourier transform methods utilizing all three orthogonal gradients for 3-D phase encoding. The number of data acquisitions for each phase-encoding step was weighted according to the Fourier coefficients to define cylindrical voxels. Spatially localized (31)P spectra can be generated for voxels of desired location within the field of view as a postprocessing step. The quality of localization was first demonstrated by using a three-compartment phantom. The technique was then applied to in vivo canine models and yielded (31)P cardiac spectra with an excellent signal-to-noise ratio. The in vivo validation experiments, using an implanted 2-phosphoenolpyruvate-containing marker, demonstrated that the technique is capable of measuring at least two transmural layers of left ventricular myocardium representing the subepicardium and subendocardium.     

Axon voltage-clamp simulations. I. Methods and tests.

This is the first in a series of four papers in which we present the numerical simulation of the application of the voltage clamp technique to excitable cells. In this paper we describe the application of the Crank-Nicolson (1947) method for the solution of the parabolic partial differential equations that describe a cylindrical cell in which the ionic conductances are functions of voltage and time (Hodgkin and Huxley, 1952). This method is compared with other methods in terms of accuracy and speed of solution for a propagated action potential. In addition, differential equations representing a simple voltage-clamp electronic circuit are presented. Using the voltage clamp circuit equations, we simulate the voltage clamp of a single isopotential membrane patch and show how the parameters of the circuit affect the transient response of the patch to a step change in the control potential.The stimulation methods presented in this series of papers allow the evaluation of voltage clamp control of an excitable cell or a syncytium of excitable cells. To the extent that membrane parameters and geometrical factors can be determined, the methods presented here provide solutions for the voltage profile as a function of time.     

A differentiable,periodic function for pulsatile cardiac output based on heart rate and stroke volume

Many mathematical models of human hemodynamics, particularly those which describe pressure and flow pulses throughout the circulatory system, require as specified input a modeling function which describes cardiac output in terms of volume per unit time. To be realistic, this cardiac output function should capture, to the greatest extent possible, all relevant features observed in measured physical data. For model analysis purposes, it is also highly desirable to have a model function that is continuous, differentiable, and periodic. This paper addresses both classes of needs by developing such a function. Physically, the present function provides an accurate model for flow into the ascending aorta. It is completely specified by a minimal number of standard input parameters associated with left ventricle dynamics, including heart rate, mean cardiac output, and an estimation of the peak-to-mean flow ratio. Analytically, it can be expressed as a product of two continuous, differentiable and periodic factors. Further, the Fourier expansion of this model function is shown to be a finite Fourier series, and explicit closed-form expressions are given for the non-zero coefficients in this series.     

Identification of nonlinear systems using random impulse train inputs

Nonlinear systems that require discrete inputs can be characterized by using random impulse train (Poisson process) inputs. The method is analagous to the Wiener method for continuous input systems, where Gaussian white-noise is the input. In place of the Wiener functional expansion for the output of a continuous input system, a new series for discrete input systems is created by making certain restrictions on the integrals in a Volterra series. The kernels in the new series differ from the Wiener kernels, but also serve to identify a system and are simpler to compute. For systems whose impulse responses vary in amplitude but maintain a similar shape, one argument may be held fixed in each kernel. This simplifies the identification problem. As a test of the theory presented, the output of a hypothetical second order nonlinear system in response to a random impulse train stimulus was computer simulated. Kernels calculated from the simulated data agreed with theoretical predictions. The Poisson impulse train method is applicable to any system whose input can be delivered in discrete pulses. It is particularly suited to neuronal synaptic systems when the pattern of input nerve impulses can be made random.     

The steady-state finite cable: Numerical method for non-linear membrane

An iterative numerical method is described for finding steady-state solutions to the one-dimensional cable equations for finite cable lengths and open-circuit termination. The method is suitable for any non-linear membrane with a single positive-slope crossing of the zero current axis, including those with regions of negative slope conductance. The method generates the current necessary to cause any desired voltage displacement at the input end of the cable as well as solutions for the transmembrane potential and axial current along the cable.     

Subtraction of background fluorescence in multiharmonic frequency-domain fluorimetry.

Background fluorescence is a major concern in time-resolved microfluorimetry studies of biological samples. A general method for subtraction of an arbitrary background signal in measurements of lifetime and anisotropy decay by multiharmonic Fourier transform spectroscopy is presented. Multifrequency phase and modulation values are measured in parallel by transformation of digitized time-domain waveforms into the frequency domain. For subtraction of background, time-domain waveforms are acquired for emission and reference photomultipliers for sample (e.g., cell containing fluorophore) and blank (e.g., unlabeled cell). Time-domain waveforms obtained in a series of measurements (e.g., sample and blank for parallel and perpendicular orientations of an emission polarizer) are time-justified by least-squares fitting of reference channel waveforms or by phase comparison of the first Fourier harmonics of the reference channel. Background is then subtracted directly in the time domain, and the subtracted waveform is Fourier transformed to the frequency domain for analysis of lifetime or anisotropy decay. This approach yielded excellent background correction over a wide range of background intensities and decay profiles. The method was tested in cuvette fluorimetry with fluorescein and acridine orange and in fluorescence microscopy with living MDCK cells loaded with the pH indicator BCECF. Sample lifetimes and rotational parameters could be recovered accurately with greater than 50% of the signal arising from background. These results establish a direct and practical approach to subtraction of background in complex biological and chemical samples studied by frequency-domain fluorimetry.     

Modeling the electrical behavior of anatomically complex neurons using a network analysis program: Passive membrane

We describe the application of a popular and widely available electrical circuit simulation program called SPICE to modeling the electrical behavior of neurons with passive membrane properties and arbitrarily complex dendritic trees. Transient responses may be calculated at any location in the cell model following current, voltage or conductance perturbations at any point. A numbering method is described for binary trees which is helpful in transforming complex dendritic structures into a coded list of short cylindrical dendritic segments suitable for input to SPICE. Individual segments are modeled as isopotential compartments comprised of a parallel resistor and capacitor, representing the transmembrane impedance, in series with one or two core resistors. Synaptic current is modeled by a current source controlled by the local membrane potential and an alpha-shaped voltage, thus simulating a conductance change in series with a driving potential. Extensively branched test cell circuits were constructed which satisfied the equivalent cylinder constraints (Rall 1959). These model neurons were perturbed by independent current sources and by synaptic currents. Responses calculated by SPICE are compared with analytical results. With appropriately chosen model parameters, extremely accurate transient calculations may be obtained. Details of the SPICE circuit elements are presented, along with illustrative examples sufficient to allow implementation of passive nerve cell models on a number of common computers. Methods for modeling excitable membrane are presented in the companion paper (Bunow et al. 1985).