Techniques for obtaining analytical solutions to the multicylinder somatic shunt cable model for passive neurones.

The somatic shunt cable model for neurones is extended to the case in which several equivalent cylinders, not necessarily of the same electrotonic length, emanate from the cell soma. The cable equation is assumed to hold in each cylinder and is solved with sealed end conditions and a lumped soma boundary condition at a common origin. A Green's function (G) is defined, corresponding to the voltage response to an instantaneous current pulse at an arbitrary point along one of the cylinders. An eigenfunction expansion for G is obtained where the coefficients are determined using the calculus of residues and compared with an alternative method of derivation using a modified orthogonality condition. This expansion converges quickly for large time, but, for small time, a more convenient alternative expansion is obtained by Laplace transforms. The voltage response to arbitrary currents injected at arbitrary sites in the dendritic tree (including the soma) may then be expressed as a convolution integral involving G. Illustrative examples are presented for a point charge input.     

Analytical solution of the cable equation with synaptic reversal potentials for passive neurons with tip-to-tip dendrodendritic coupling

A passive cable model is presented for a pair of electrotonically coupled neurons in order to investigate the effects of tip-to-tip dendrodendritic gap junctions on the interaction between excitation and either pre or postsynaptic inhibition. The model represents each dendritic tree by a tapered equivalent cylinder attached to an isopotential soma. Analytical solution of the cable equation with synaptic reversal potentials is considered for each neuron to yield a system of Volterra integral equations for the voltage. The solution to the system of linear integral equations (expressed as a Neumann series) is used to determine the current spread within the two coupled neurons, and to re-examine the sensitivity of the soma potentials (in particular) to the coupling resistance for various loci of synaptic inputs. The model is actually posed generally, so that active as well as passive properties could be considered. In the active case, a system of non-linear integral equations is derived for the voltage.     

Solutions for transients in arbitrarily branching cables: II. Voltage clamp theory

Analytical solutions are derived for arbitrarily branching passive neurone models with a soma and somatic shunt, for synaptic inputs and somatic voltage commands, for both perfect and imperfect somatic voltage clamp. The solutions are infinite exponential series. Perfect clamp decouples different dendritic trees at the soma: each exponential component exists only in one tree; its time constant is independent of stimulating and recording position within the tree; its amplitude is the product of a factor constant over that entire tree and factors dependent on stimulating and recording positions. Imperfect clamp to zero is mathematically equivalent to voltage recording with a shunt. As the series resistance increases, different dendritic trees become more strongly coupled. A number of interesting response symmetries are evident. The solutions reveal parameter dependencies, including an insensitivity of the early parts of the responses to specific membrane resistivity and somatic shunt, and an approximately linear dependence of the slower time constants on series resistance, for small series resistances. The solutions are illustrated using a “cartoon” representation of a CA1 pyramidal cell and a two-cylinder + soma model.     

Transient Response in a Dendritic Neuron Model for Current Injected at One Branch

Mathematical expressions are obtained for the response function corresponding to an instantaneous pulse of current injected to a single dendritic branch in a branched dendritic neuron model. The theoretical model assumes passive membrane properties and the equivalent cylinder constraint on branch diameters. The response function when used in a convolution formula enables one to compute the voltage transient at any specified point in the dendritic tree for an arbitrary current injection at a given input location. A particular numerical example, for a brief current injection at a branch terminal, illustrates the attenuation and delay characteristics of the depolarization peak as it spreads throughout the neuron model. In contrast to the severe attenuation of voltage transients from branch input sites to the soma, the fraction of total input charge actually delivered to the soma and other trees is calculated to be about one-half. This fraction is independent of the input time course. Other numerical examples, which compare a branch terminal input site with a soma input site, demonstrate that, for a given transient current injection, the peak depolarization is not proportional to the input resistance at the injection site and, for a given synaptic conductance transient, the effective synaptic driving potential can be significantly reduced, resulting in less synaptic current flow and charge, for a branch input site. Also, for the synaptic case, the two inputs are compared on the basis of the excitatory post-synaptic potential (EPSP) seen at the soma and the total charge delivered to the soma.     

Voltage transients in neuronal dendritic trees.

An analytical method is outlined for calculating the passive voltage transient at each point in an extensively branched neuron model for arbitrary current injection at a single branch. The method is based on a convolution formula that employs the transient response function, the voltage response to an instantaneous pulse of current. For branching that satisfies Rall's equivalent cylinder constraint, the response function is determined explicitly. Voltage transients, for a brief current injected at a branch terminal, are evaluated at several locations to illustrate the attenuation and delay characteristics of passive spread. A comparison with the same transient input terminal input, the fraction of input charge dissipated by various branches in the neuron model is illustrated. These fractions are independent of the input time course. For transient synaptic conductance change at a single branch terminal, a numerical example demonstrates the nonlinear effect of reduced synaptic driving potential. The branch terminal synaptic input is compared with the same synaptic conductance input applied to the soma on the basis of excitatory postsynaptic potential amplitude at the soma and charge delivered to the soma.     

Axonal contribution to subthreshold currents inaplysia bursting pacemaker neurons

The contribution of axonal activity to the ionic currents which generate bursting pacemaker activity was studied by using the two-electrode voltage-clamp technique in Aplysia bursting neuron somata in conjunction with intraaxonal voltage recordings. Depolarizing voltage-clamp pulses applied to bursting cell somata triggered axonal action potentials. The voltage-clamp current recording exhibited transient inward current "notches" corresponding to each of the axonal spikes. The addition of 50 microM tetrodotoxin (TTX) to the bathing medium blocked the fast axonal spikes and current notches, revealing a slower axonal spike which was blocked by the replacement of external Ca2+ with Co2+. The inward current evoked by applying a depolarizing voltage-clamp pulse in the soma is distorted by the occurrence of the axonal Ca2+ spike. Elimination of the axonal spike, by injecting hyperpolarizing current into the axon, changes both the time course and the magnitude of the inward current. The axonal Ca2+ spikes are followed by a series of Ca2+-dependent afterpotentials: a rapid postspike hyperpolarization, a depolarizing afterpotential (DAP) and, finally, a long-lasting postburst hyperpolarization. The long-lasting hyperpolarization is not blocked by 50 mM external tetraethyl ammonium, an effective blocker of Ca2+-activated K+ current [IK(Ca)], and does not appear to reverse at EK. Hence, the axonal long-lasting hyperpolarization may not be due to IK(Ca). Somatic voltage-clamp pulses in bursting neurons are followed by a slow inward tail current, which is sometimes coincident with a DAP in the axon. In some cells, the amplitude of the slow inward tail current is greatly reduced if axonal spikes and DAPs are prevented by hyperpolarization of the axon, while, in other cells, elimination of axonal activity has little effect. Therefore, the slow inward tail current is not necessarily an artifact of poor voltage-clamp control over the axonal membrane potential but probably results from the activation of an ionic conductance mechanism located partly in the axon and partly in the soma.     

Electrodiffusion model simulation of rectangular current pulses in a voltage-biased biological channel

Numerical methods are presented for simulating stochastic-in-time current pulses for an electrodiffusion model of the biological channel, with a fixed applied voltage across the channel. The electrodiffusion model consists of the parabolic advection-diffusion equation coupled either to Gauss' law or Poisson's equation, depending on the choice of boundary conditions. The TRBDF2 method is employed for the advection-diffusion equation. The rectangular wave shape of previously simulated traveling wave current pulses is preserved by the full set of partial differential equations for electrodiffusion.     

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.     

Branch Input Resistance and Steady Attenuation for Input to One Branch of a Dendritic Neuron Model

Mathematical solutions and numerical illustrations are presented for the steady-state distribution of membrane potential in an extensively branched neuron model, when steady electric current is injected into only one dendritic branch. Explicit expressions are obtained for input resistance at the branch input site and for voltage attenuation from the input site to the soma; expressions for AC steady-state input impedance and attenuation are also presented. The theoretical model assumes passive membrane properties and the equivalent cylinder constraint on branch diameters. Numerical examples illustrate how branch input resistance and steady attenuation depend upon the following: the number of dendritic trees, the orders of dendritic branching, the electrotonic length of the dendritic trees, the location of the dendritic input site, and the input resistance at the soma. The application to cat spinal motoneurons, and to other neuron types, is discussed. The effect of a large dendritic input resistance upon the amount of local membrane depolarization at the synaptic site, and upon the amount of depolarization reaching the soma, is illustrated and discussed; simple proportionality with input resistance does not hold, in general. Also, branch input resistance is shown to exceed the input resistance at the soma by an amount that is always less than the sum of core resistances along the path from the input site to the soma.     

Inactivation of HIT cell Ca2+ current by a simulated burst of Ca2+ action potentials.

A novel voltage-clamp protocol was developed to test whether slow inactivation of Ca2+ current occurs during bursting in insulin-secreting cells. Single insulin-secreting HIT cells were patch-clamped and their Ca2+ currents were isolated pharmacologically. A computed beta-cell burst was used as a voltage-clamp command and the net Ca2+ current elicited was determined as a cadmium difference current. Ca2+ current rapidly activated during the computed plateau and spike depolarizations and then slowly decayed. Integration of this Ca2+ current yielded an estimate of total Ca influx. To further analyze Ca2+ current inactivation during a burst, repetitive test pulses to + 10 mV were added to the voltage command. Current elicited by these pulses was constant during the interburst, but then slowly and reversibly decreased during the depolarizing plateau. This inactivation was reduced by replacing external Ca2+ with Ba2+ as a charge carrier, and in some cells inactivation was slower in Ba2+. Experimental results were compared with the predictions of the Keizer-Smolen mathematical model of bursting, after subjecting model equations to identical voltage commands. In this model, bursting is driven by the slow, voltage-dependent inactivation of Ca current during the plateau active phase. The K-S model could account for the slope of the slow decay of spike-elicited Ca current, the waveform of individual Ca current spikes, and the suppression of test pulse-elicited Ca current during a burst command. However, the extent and rate of fast inactivation were underestimated by the model.(ABSTRACT TRUNCATED AT 250 WORDS)     

Voltage clamp calculations for myelinated and demyelinated axons

Exact cable theory is used to calculate voltage distributions along fully myelinated axons and those with various patterns of demyelination. The model employed uses an R-C circuit for the soma, an equivalent cable for the dendrites, a myelinated axon with n internodes and a cable representing telodendria. For the case of a voltage clamp at the soma, a system of 2n + 1 equations must be solved to obtain the potential distribution and this is done for arbitrary n. An explicit calculation is performed for one internode whereas computer-generated solutions are obtained for several internodes. The relative importance of the position of a single demyelinated internode is determined. An approximate expression is given for the critical internodal length necessary for action potential generation.     

Solutions for transients in arbitrarily branching cables: I. Voltage recording with a somatic shunt.

An analytical solution is derived for voltage transients in an arbitrarily branching passive cable neurone model with a soma and somatic shunt. The response to injected currents can be represented as an infinite series of exponentially decaying components with different time constants and amplitudes. The time constants of a given model, obtained from the roots of a recursive transcendental equation, are independent of the stimulating and recording positions. Each amplitude is the product of three factors dependent on the corresponding root: one constant over the cell, one varying with the input site, and one with the recording site. The amplitudes are not altered by interchanging these sites. The solution reveals explicitly some of the parameter dependencies of the responses. An efficient recursive root-finding algorithm is described. Certain regular geometries lead to "lost" roots; difficulties associated with these can be avoided by making small changes to the lengths of affected segments. Complicated cells, such as a CA1 pyramid, produce many closely spaced time constants in the range of interest. Models with large somatic shunts and dendrites of unequal electrotonic lengths can produce large amplitude waveform components with surprisingly slow time constants. This analytic solution should complement existing passive neurone modeling techniques.     

A voltage-dependent persistent sodium current in mammalian hippocampal neurons

Currents generated by depolarizing voltage pulses were recorded in neurons from the pyramidal cell layer of the CA1 region of rat or guinea pig hippocampus with single electrode voltage-clamp or tight-seal whole-cell voltage-clamp techniques. In neurons in situ in slices, and in dissociated neurons, subtraction of currents generated by identical depolarizing voltage pulses before and after exposure to tetrodotoxin revealed a small, persistent current after the transient current. These currents could also be recorded directly in dissociated neurons in which other ionic currents were effectively suppressed. It was concluded that the persistent current was carried by sodium ions because it was blocked by TTX, decreased in amplitude when extracellular sodium concentration was reduced, and was not blocked by cadmium. The amplitude of the persistent sodium current varied with clamp potential, being detectable at potentials as negative as -70 mV and reaching a maximum at approximately -40 mV. The maximum amplitude at -40 mV in 21 cells in slices was -0.34 +/- 0.05 nA (mean +/- 1 SEM) and -0.21 +/- 0.05 nA in 10 dissociated neurons. Persistent sodium conductance increased sigmoidally with a potential between -70 and -30 mV and could be fitted with the Boltzmann equation, g = gmax/(1 + exp[(V' - V)/k)]). The average gmax was 7.8 +/- 1.1 nS in the 21 neurons in slices and 4.4 +/- 1.6 nS in the 10 dissociated cells that had lost their processes indicating that the channels responsible are probably most densely aggregated on or close to the soma. The half-maximum conductance occurred close to -50 mV, both in neurons in slices and in dissociated neurons, and the slope factor (k) was 5-9 mV. The persistent sodium current was much more resistant to inactivation by depolarization than the transient current and could be recorded at greater than 50% of its normal amplitude when the transient current was completely inactivated. Because the persistent sodium current activates at potentials close to the resting membrane potential and is very resistant to inactivation, it probably plays an important role in the repetitive firing of action potentials caused by prolonged depolarizations such as those that occur during barrages of synaptic inputs into these cells.     

"Creep currents" in single frog atrial cells may be generated by electrogenic Na/Ca exchange

The objective of these experiments was to test the hypothesis that the "creep currents" induced by Na loading of single frog atrial cells (Hume, J. R., and A. Uehara. 1986. Journal of General Physiology. 87:833) may be generated by an electrogenic Na/Ca exchanger. Creep currents induced by Na loading were examined over a wide range of membrane potentials. During depolarizing voltage-clamp pulses, outward creep currents were observed, followed by inward creep currents upon the return to the holding potential. During hyperpolarizing voltage-clamp pulses, creep currents of the opposite polarity were observed: inward creep currents were observed during the pulses, followed by outward creep currents upon the return to the holding potential. The current-voltage relations for inward and outward creep currents in response to depolarizing or hyperpolarizing voltage displacements away from the holding potential all intersect the voltage axis at a common potential, which indicates that inward and outward creep currents may have a common reversal potential under equilibrium conditions and may therefore be generated by a common mechanism. Measurements of inward creep currents confirm that voltage displacements away from the holding potential rapidly alter equilibrium conditions. Current-voltage relationships of inward creep currents after depolarizing voltage-clamp pulses are extremely labile and depend critically upon the amplitude and duration of outward creep currents elicited during preceding voltage-clamp pulses. An optical monitor of mechanical activity in single cells revealed (a) a similar voltage dependence for the outward creep currents induced by Na loading and tonic contraction, and (b) a close correlation between the time course of the decay of the inward creep current and the time course of mechanical relaxation. A mathematical model of electrogenic Na/Ca exchange (Mullins, L.J. 1979. Federation Proceedings. 35:2583; Noble, D. 1986. Cardiac Muscle. 171-200) can adequately account for many of the properties of creep currents. It is concluded that creep currents in single frog atrial cells may be attributed to the operation of an electrogenic Na/Ca exchange mechanism.     

Voltage-imaging and simulation of effects of voltage- and agonist-activated conductances on soma-dendritic voltage coupling in cerebellar Purkinje cells

We investigated the spread of membrane voltage changes from the soma into the dendrites of cerebellar Purkinje cells by using voltage-imaging techniques in combination with intracellular recordings and by performing computer simulations using a detailed compartmental model of a cerebellar Purkinje cell. Fluorescence signals from single Purkinje cells in cerebellar cultures stained with the styryl dye di-4-ANEPPS were detected with a 10 × 10 photodiode array and a charge coupled device (CCD). Fluorescence intensity decreased and increased with membrane depolarization and hyperpolarization, respectively. The relation between fractional fluorescence change (F/F) and membrane potential could be described by a linear function with a slope of up to – 3%/100 mV. Hyperpolarizing and depolarizing voltage jumps applied to Purkinje cells voltage-clamped with an intrasomatic recording electrode induced dendritic dye signals, demonstrating that these voltage transients invaded the dendrites. Dye signals induced by depolarizing somatic voltage jumps were weaker in the dendrites, when compared with those induced by hyperpolarizing voltage jumps. Dendritic responses to hyperpolarizing voltage steps applied at the soma were attenuated when membrane conductance was increased by muscimol, an agonist for GABA_Areceptors.Corresponding experimental protocols were applied to a previously developed detailed compartmental model of a Purkinje cell. In the model, as in the electrophysiological recordings, voltage attenuation from soma to dendrites increased under conditions where membrane conductance is increased by depolarization or by activation of GABA_A receptors, respectively.We discuss how these results affect voltage clamp studies of synaptic currents and synaptic integration in Purkinje cells.     

Corrections for space-clamp errors in measured parameters of voltage-dependent conductances in a cylindrical neurite

The ability to correct parameters of voltage-gated conductances measured under poor spatial control by point voltage clamp could rescue much flawed experimental data. We explore a strategy for correcting errors in experiments that employs a full-trace approach to parameter determination. Simulated soma voltage-clamp runs are made on a model neuron with a single voltage-gated, Hodgkin-Huxley channel type distributed uniformly along an elongate process. Estimates for both kinetic and I(V) parameters are obtained by fitting a form of the Hodgkin-Huxley equations to the complete time course of leak-subtracted current curves. The fitted parameters are used to determine how much correction in each parameter is needed to regenerate the set actually belonging to the channel. Corrections are generated for a range of neurite lengths, conductance densities, and channel characteristics.     

Deexcitation of cardiac cells.

Excitation and deexcitation are fundamental phenomena in the electrophysiology of excitable cells. Both of them can be induced by stimulating a cell with intracellularly injected currents. With extracellular stimulation, deexcitation was never observed; only cell excitation was found. Why? A generic model with two variables (FitzHugh) predicts that an extracellular stimulus can both excite the cell and terminate the action potential (AP). Our experiments with single mouse myocytes have shown that short (2-5 ms) extracellular pulses never terminated the AP. This result agrees with our numerical experiments with the Beeler-Reuter model. To analyze the problem, we exploit the separation of time scales to derive simplified models with fewer equations. Our analysis has shown that the very specific form of the current-voltage (I-V) characteristics of the time-independent potassium current (almost no dependence on voltage for positive membrane potentials) is responsible here. When the shape of the I-V characteristics of potassium currents was modified to resemble that in ischemic tissues, or when the external potassium concentration (K0) is increased, the AP was terminated by extracellular pulses. These results may be important for understanding the mechanisms of defibrillation.     

Experimental study of the processes accompanying argon breakdown in a long discharge tube at a reduced pressure

Results are presented from experimental studies of the breakdown stage of a low-pressure discharge (1 and 5 Torr) in a glass tube the length of which (75 cm) is much larger than its diameter (2.8 cm). Breakdowns occurred under the action of positive voltage pulses with an amplitude of up to 9.4 kV and a characteristic rise time of 2–50 μs. The discharge current in the steady-state mode was 10–120 mA. The electrode voltage, discharge current, and radiation from the discharge gap were detected simultaneously. The dynamic breakdown voltage was measured, the prebreakdown ionization wave was recorded, and its velocity was determined. The dependence of the discharge parameters on the time interval between voltage pulses (the socalled “memory effect”) was analyzed. The memory effect manifests itself in a decrease or an increase in the breakdown voltage and a substantial decrease in its statistical scatter. The time interval between pulses in this case can reach 0.5 s. The effect of illumination of the discharge tube with a light source on the breakdown was studied. It is found that the irradiation of the anode region of the tube by radiation with wavelengths of ≤500 nm substantially reduces the dynamic breakdown voltage. Qualitative explanations of the obtained results are offered.     

Analytical solutions to the multicylinder somatic shunt cable model for passive neurones with differing dendritic electrical parameters

The multicylinder somatic shunt cable model for passive neurones with differing time constants in each cylinder is considered in this paper. The solution to the model with general inputs is developed, and the parametric dependence of the voltage response is investigated. The method of analysis is straightforward and follows that laid out in Evans et al. (1992, 1994): (i) The dimensional problem is stated with general boundary and initial conditions, (ii) The model is fully non-dimensionalised, and a dimensionless parameter family which uniquely governs the behaviour of the dimensionless voltage response is obtained, (iii) The fundamental unit impulse problem is solved, and the solutions to problems involving general inputs are written in terms of the unit impulse solution, (iv) The large and small time behaviour of the unit impulse solution is examined, (v) The parametric dependence of the unit impulse upon the dimensionless parameter family is explored for two limits of practical interest. A simple expression for the principle relationship between the dimensionless parameter family is derived and provides insight into the interaction between soma and cylinders. A well-posed method for the solution of the dimensional inverse problem is presented.