Subthreshold oscillatory responses of the Hodgkin-Huxley cable model for the squid giant axon

The classical cable equation, in which membrane conductance is considered constant, is modified by including the linearized effect of membrane potential on sodium and potassium ionic currents, as formulated in the Hodgkin-Huxley equations for the squid giant axon. The resulting partial differential equation is solved by numerical inversion of the Laplace transform of the voltage response to current and voltage inputs. The voltage response is computed for voltage step, current step, and current pulse inputs, and the effect of temperature on the response to a current step input is also calculated.The validity of the linearized approximation is examined by comparing the linearized response to a current step input with the solution of the nonlinear partial differential cable equation for various subthreshold current step inputs.All the computed responses for the squid giant axon show oscillatory behavior and depart significantly from what is predicted on the basis of the classical cable equation. The linearization procedure, coupled with numerical inversion of the Laplace transform, proves to be a convenient approach which predicts at least qualitatively the subthreshold behavior of the nonlinear system.     

Nonlinear cable equations for axons. I. Computations and experiments with internal current injection

Steady-state potential and current distributions resulting from internal injection of current in the squid giant axon have been measured experimentally and also computed from nonlinear membrane cable equation models by numerical methods, using the Hodgkin-Huxley equations to give the membrane current density. The solutions obtained by this method satisfactorily reproduce experimental measurements of the steady-state distribution of membrane potential. Computations of the input current-voltage characteristic for a nonlinear cable were in excellent agreement with measurements on axons. Our results demonstrate the power of Cole's equation to extract the nonlinear membrane characteristics simply from measurement of the input resistance.     

A synthetic strand of cardiac muscle: its passive electrical properties

The passive electrical properties of synthetic strands of cardiac muscle, grown in tissue culture, were studied using two intracellular microelectrodes: one to inject a rectangular pulse of current and the other to record the resultant displacement of membrane potential at various distances from the current source. In all preparations, the potential displacement, instead of approaching a steady value as would be expected for a cell with constant electrical properties, increased slowly with time throughout the current step. In such circumstances, the specific electrical constants for the membrane and cytoplasm must not be obtained by applying the usual methods, which are based on the analytical solution of the partial differential equation describing a one-dimensional cell with constant electrical properties. A satisfactory fit of the potential waveforms was, however, obtained with numerical solutions of a modified form of this equation in which the membrane resistance increased linearly with time. Best fits of the waveforms from 12 preparations gave the following values for the membrane resistance times unit length, membrane capacitance per unit length, and for the myoplasmic resistance: 1.22 plus or minus 0.13 x 10-5 omegacm, 0.224 plus or minus 0.023 uF with cm-minus 1, and 1.37 plus or minus 0.13 x 10-7 omegacm-minus 1, respectively. The value of membrane capacitance per unit length was close to that obtained from the time constant of the foot of the action potential and was in keeping with the generally satisfactory fit of the recorded waveforms with solutions of the cable equation in which the membrane impedance is that of a single capacitor and resistor in parallel. The area of membrane per unit length and the cross-sectional area of myoplasm at any given length of the preparation were determined from light and composite electron micrographs, and these were used to calculate the following values for the specific electrical membrane resistance, membrane capacitance, and the resistivity of the cytoplasm: 20.5 plus or minus 3.0 x 10-3 omegacm-2, l.54 plus or minus 0.24 uFWITHcm-minus 2, and 180 plus or minus 34 omegacm, respectively.     

Solutions for transients in arbitrarily branching cables: III. Voltage clamp problems.

Branched cable voltage recording and voltage clamp analytical solutions derived in two previous papers are used to explore practical issues concerning voltage clamp. Single exponentials can be fitted reasonably well to the decay phase of clamped synaptic currents, although they contain many underlying components. The effective time constant depends on the fit interval. The smoothing effects on synaptic clamp currents of dendritic cables and series resistance are explored with a single cylinder + soma model, for inputs with different time courses. "Soma" and "cable" charging currents cannot be separated easily when the soma is much smaller than the dendrites. Subtractive soma capacitance compensation and series resistance compensation are discussed. In a hippocampal CA1 pyramidal neurone model, voltage control at most dendritic sites is extremely poor. Parameter dependencies are illustrated. The effects of series resistance compound those of dendritic cables and depend on the "effective capacitance" of the cell. Plausible combinations of parameters can cause order-of-magnitude distortions to clamp current waveform measures of simulated Schaeffer collateral inputs. These voltage clamp problems are unlikely to be solved by the use of switch clamp methods.     

Membrane Characteristics of the Canine Papillary Muscle Fiber

Passive and active responses to intracellular and extracellular stimulation were studied in the canine papillary muscle. The electrotonic potential produced by extracellular polarization with the partition chamber method fitted the time course and the spatial decay expected from the cable theory (the time constant, 3.3 msec; the space constant, 1.2 mm). Contrariwise, spatial decay of the electrotonic potentials produced by intracellular polarization was very short and did not fit the decay curve expected for a simple cable, although only a small difference of time course in the electrotonic potentials produced by intracellular and extracellular polarizations was observed. A similar time course might result from the fact that when current flow results from intracellular polarization, the input resistance is less dependent on the membrane resistance. The foot of the propagated action potential rose exponentially with a time constant of 1.1 msec and a conduction velocity of 0.68 m/sec. The membrane capacity was calculated from the time constant of the foot potential and the conduction velocity to be 0.76 µF/cm². The responses of the papillary muscle membrane to intracellular stimulation differed from those to extracellular stimulation applied with the partition method in the following ways: higher threshold potential, shorter latency for the active response, linearity of the current-voltage relationship, and no reduction in the membrane resistance at the crest of the action potential during current flow.     

A spatial stochastic neuronal model with Ornstein–Uhlenbeck input current

 We consider a spatial neuron model in which the membrane potential satisfies a linear cable equation with an input current which is a dynamical random process of the Ornstein–Uhlenbeck (OU) type. This form of current may represent an approximation to that resulting from the random opening and closing of ion channels on a neuron's surface or to randomly occurring synaptic input currents with exponential decay. We compare the results for the case of an OU input with those for a purely white-noise-driven cable model. The statistical properties, including mean, variance and covariance of the voltage response to an OU process input in the absence of a threshold are determined analytically. The mean and the variance are calculated as a function of time for various synaptic input locations and for values of the ratio of the time constant of decay of the input current to the time constant of decay of the membrane voltage in the physiological range for real neurons. The limiting case of a white-noise input current is obtained as the correlation time of the OU process approaches zero. The results obtained with an OU input current can be substantially different from those in the white-noise case. Using simulation of the terms in the series representation for the solution, we estimate the interspike interval distribution for various parameter values, and determine the effects of the introduction of correlation in the synaptic input stochastic process. Received: 5 March 2001 / Accepted in revised form: 7 August 2001     

Numerical analysis of the voltage-clamp technique applied to frog neuromuscular junctions.

The nonlinear cable equation was solved numerically by means of an implicit procedure. The correlation between end-plate length and fiber diameter was determined in frog (Rana pipiens) sartorius muscles stained with gold chloride (Löwit, 1875). The diameter of the fibers stained by the Löwit method was 80 (74-85) micron (median and its 95% confidence interval for 52 fibers), the length of the end plates in the same fibers was 382 (353-417) micron. The fibers simulated were 80 micron in diameter. To solve the equation the muscle fibers were represented by 500 segments 20 micron long, and the equation was solved in steps of 10 microseconds; a double exponential function was incorporated to the first seven segments to represent the neuromuscular junction. The potential of the first segment of the cable was set to the clamping level and the membrane potential of the remaining segments calculated. The current needed to hold the first segment was estimated by adding the current flowing through the first segment to the current flowing from it to the second segment. Our results indicate that the lack of space clamp in the point voltage-clamp studies of the frog neuromuscular junction introduces serious errors in the estimates of the end-plate conductance value, the kinetics of the conductance changes, and the reversal potential of the end-plate currents. The possibility of an efficient voltage-clamp technique is also explored. Our calculations suggest that the study of end-plate current and conductance is possible with little error if the end-plate potential is controlled at both ends of the synaptic area simultaneously.     

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.     

A.C. cable theory and its implications on the measurements of the membrane electric parameters

The solution to the a.c. cable equations for a leaky non-inductive coaxial cable is presented. It is shown that the exact solution can be expressed in terms of a simple Ohm's law formulation modified by multiplicative factors. A numerical analysis of these factors shows that the formulation reduces to a simple Ohm's law for the case of infinite impedance terminations, if the voltage is recorded at a distance 0·421/λ from the centre of a midpoint current injection, 1 is the half-length of the cell and A the space constant. The same result applies to the case of external current injection if the voltage is recorded at a distance of 0·581/λ from the centre of the cell segment of length 21. This thus allows an easy and accurate computation of the membrane resistance and capacitance. It is also shown that the accuracy obtained in these computations can be estimated quantitatively. The actual values of the accuracy obtained in a.c. studies of Nitella translucens are presented.     

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.     

Theoretical analysis of the amplification of synaptic potentials by small clusters of persistent sodium channels in dendrites

We extend on the work developed by R.R. Poznanski and J. Bell from a linearized somatic persistent sodium current source to a non-linear representation of the dendritic Na(+)P current source associated with a small number of persistent sodium channels. The main objective is to investigate the modulation in the amplification of excitatory postsynaptic potentials (EPSPs) in dendrites studded with persistent sodium channels. The relation between membrane potential (V) and persistent sodium current density (I(NaP)) is approximated heuristically with a sigmoidal function and the resultant cable equation is solved analytically using a regular perturbation expansion and Green's function techniques. The transient simulated (non-evoked) response is found as a result of current injection in the form of synaptically induced voltage change located at a distance from the recording site in a cable with a uniform distribution of ion channel densities per unit length of cable (the so-called 'hot-spots') and with the conductance of each hot-spot (i.e., number of channels per hot-spot) assumed to be a constant. The results show an amplification in the observed EPSPs to be compatible with the experimentally derived estimates, and in addition a saturation in the amplification is observed indicating an optimum number of ionic channels.     

Measurement of GABA-evoked conductance changes of lobster muscle fibres by a three-microelectrode voltage clamp technique

The effective membrane conductance and capacity of lobster muscle fibres was measured by a three-intracellular-microelectrode voltage clamp technique. Conductance values agreed well with those determined under current clamp, by means of the 'short' cable equations. Reversible increases in conductance evoked by gamma-aminobutyric acid (GABA) were reflected by differences (delta V) in electrotonic potential amplitude recorded at the centre, and midway between the centre and fibre end respectively. GABA dose--conductance curves derived from cable theory or from delta V measurements were virtually identical. The effective capacity (ceff), determined from the area beneath the 'on' delta V capacity transient, yielded values of the membrane time constant consistently lower than those obtained by the graphical method of E. Stefani & A.B. Steinbach (J. Physiol., London. 203, 383-401 (1969)); one possible explanation for this discrepancy is discussed. In the presence of GABA, the effective capacity was reduced in a dose-related manner. The results were interpreted in terms of an equivalent circuit in which surface membrane was arranged in parallel with cleft-tubular membrane of finite conductance, charged through an access resistance. GABA was though to be decreasing ceff by selectively increasing the conductance of the cleft-tubular membranes.     

Voltage clamp simulations for multifiber bundles in a double sucrose gap: cable complications

A theoretical model is presented for voltage clamp of a bundle of cylindrical excitable cells in a double sucrose gap. The preparation in the test node is represented by a single one-dimensional cable (length/diameter ratio approximately) with standard Hodgkin-Huxley kinetics for transmembrane Na current. Imperfections of voltage control due to internal (longitudinal) resistivity and external (radial) resistance in series to the membrane are analysed. The electrical behavior of a fiber is described by the cable equation with appropriate boundary conditions and subsidiary equations reflecting the membrane characteristics. Membrane voltage and current distribution in response to a step command was obtained by numerical integration. The results are described in two papers. The present paper deals with the effect of internal resistivity with the external resistance being neglected. The closed loop response of a fiber displays a strong tendency to oscillate. To stabilize the system a phase lead was inserted and the gain of the control amplifier was reduced. Conditions for stability were examined by Nyquist analysis. When the Na system was activated by a command pulse below ENa, a voltage gradient developed between a depolarization (relative to the command signal) at the end where voltage was monitored and a hyperpolarization at the site of current injection. In spite of a poor voltage control the total measured current appeared to have a smooth transient. With large voltage gradients a small, second inward current was seen. At a low (high) Na conductance maximum peak inward current was larger (smaller) that the current expected from ideal space clamping.     

Experimental study of the conducted action potential in cardiac Purkinje strands

Conduction velocity is a complex physiological process that integrates the active and passive properties of the excitable cell. The relations between these properties in determining the conduction velocity are not intuitively obvious, and models have been used frequently to illustrate important relationships. To study the relationships of important parameters and to evaluate commonly used models, we changed conduction velocity experimentally in sheep cardiac Purkinje strands by reducing extracellular Na systematically. Cable analyses were also performed to obtain passive membrane and cable properties. Resting membrane resistance and capacitance did not change, nor did core resistance. Active properties measured in addition to conduction velocity included maximal upstroke velocity, action potential height, time constant of the foot, peak inward current, and upstroke power. With reduction in extracellular Na, all of these parameters of the action potential changed nonlinearly and not in direct proportion to the change in conduction velocity. The only simple relation found was a linear relationship between maximal upstroke velocity and peak inward current, normalized by the capacity of the foot. Models based on the cable equation and the wave equation offer a basis for quantitative analysis of conduction, and these data can be used to test the models.     

A continuous cable method for determining the transient potential in passive dendritic trees of known geometry

Models using cable equations are increasingly employed in neurophysiological analyses, but the amount of computer time and memory required for their implementation are prohibitively large for many purposes and many laboratories. A mathematical procedure for determining the transient voltage response to injected current or synaptic input in a passive dendritic tree of known geometry is presented that is simple to implement since it is based on one equation. It proved to be highly accurate when results were compared to those obtained analytically for dendritic trees satisfying equivalent cylinder constraints. In this method the passive cable equation is used to express the potential for each interbranch segment of the dendritic tree. After applying boundary conditions at branch points and terminations, a system of equations for the Laplace transform of the potential at the ends of the segments can be readily obtained by inspection of the dendritic tree. Except for the starting equation, all of the equations have a simple format that varies only with the number of branches meeting at a branch point. The system of equations is solved in the Laplace domain, and the result is numerically inverted back to the time domain for each specified time point (the method is independent of any time increment t). The potential at any selected location in the dendritic tree can be obtained using this method. Since only one equation is required for each interbranch segment, this procedure uses far fewer equations than comparable compartmental approaches. By using significantly less computer memory and time than other methods to attain similar accuracy, this method permits extensive analyses to be performed rapidly on small computers. One hopes that this will involve more investigators in modeling studies and will facilitate their motivation to undertake realistically complex dendritic models.     

A Study on the Electrical Resistance of the Frog Sartorius Muscle

Four different methods of measuring the resistance of a muscle fiber have been applied to the frog sartorius muscle. The methods, in which the resistance of the microelectrode entered the calculation of the effective resistance of the fiber, resulted in values which were 8 times higher than the resistance values obtained with methods independent of the electrode resistance. A simple cable model of a muscle fiber could not account for the discrepancy in the effective resistance found in these measurements; therefore, an enlarged cable model for a muscle fiber has been proposed, and its biological implications have been discussed. The effective resistance (measured with the two different groups of methods) decreased when the potassium concentration in the bath increased. Using the proposed enlarged cable model for the interpretation of these results, it is shown that not only the membrane resistance but also the myoplasmic resistance decreases with an increasing potassium concentration in the Ringer solution.     

Equivalent Circuit of Frog Atrial Tissue as Determined by Voltage Clamp-Unclamp Experiments

The equivalent circuit that has been used in the analysis of nerve voltage-clamp data is that of the membrane capacity in parallel with the membrane resistance. Voltage-clamp experiments on frog atrial tissue indicate that this circuit will not suffice for this cardiac tissue. The change in membrane current associated with a step change in membrane potential does not show a rapid spike of capacitive current as would be expected for the simple parallel resistance-capacitance network. Rather, there is a step change in current followed by an exponential decay in current with a time constant of about 1 msec. This relatively slow capacitive charging current suggests that there is a resistance in series with the membrane capacity. A possible equivalent circuit is that of a series resistance external to the parallel resistance-capacitance network of the cell membranes. Another possible circuit assumes that the series resistance is an integral part of the cell membrane. The data presented in this paper demonstrate that the equivalent circuit of a bundle of frog atrial muscle is that of an external resistance in series with the cell membranes.     

Further study of the two-dimensional cable theory: An electric model for a flat thin association of cells with a directional intercellular communication

Summary The two-dimensional cable theory originally presented in relation to the electrotonus along a flat tissue of the rat atrial appendage is improved by taking account of double space constants instead of a single one and an explicit boundary condition at the tip of the current injecting microelectrode. A differential equation is formulated for the membrane potential change which is produced along the tissue by the intracellular injection of a current. The solution is formally expressed in terms of the Green's function. Specific solutions corresponding to the injection of a unit current step or a linearly rising current are discussed in detail.