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.     

Imperfect space clamp permits electrotonic interactions between inhibitory and excitatory synaptic conductances, distorting voltage clamp recordings

The voltage clamp technique is frequently used to examine the strength and composition of synaptic input to neurons. Even accounting for imperfect voltage control of the entire cell membrane ("space clamp"), it is often assumed that currents measured at the soma are a proportional indicator of the postsynaptic conductance. Here, using NEURON simulation software to model somatic recordings from morphologically realistic neurons, we show that excitatory conductances recorded in voltage clamp mode are distorted significantly by neighboring inhibitory conductances, even when the postsynaptic membrane potential starts at the reversal potential of the inhibitory conductance. Analogous effects are observed when inhibitory postsynaptic currents are recorded at the reversal potential of the excitatory conductance. Escape potentials in poorly clamped dendrites reduce the amplitude of excitatory or inhibitory postsynaptic currents recorded at the reversal potential of the other conductance. In addition, unclamped postsynaptic inhibitory conductances linearize the recorded current-voltage relationship of excitatory inputs comprising AMPAR and NMDAR-mediated components, leading to significant underestimation of the relative contribution by NMDARs, which are particularly sensitive to small perturbations in membrane potential. Voltage clamp accuracy varies substantially between neurons and dendritic arbors of different morphology; as expected, more reliable recordings are obtained from dendrites near the soma, but up to 80% of the synaptic signal on thin, distant dendrites may be lost when postsynaptic interactions are present. These limitations of the voltage clamp technique may explain how postsynaptic effects on synaptic transmission could, in some cases, be attributed incorrectly to presynaptic mechanisms.     

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.     

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.     

Dendritic and Synaptic Effects in Systems of Coupled Cortical Oscillators

We explore the influence of synaptic location and form on the behavior of networks of coupled cortical oscillators. First, we develop a model of two coupled somatic oscillators that includes passive dendritic cables. Using a phase model approach, we show that the synchronous solution can change from a stable solution to an unstable one as the cable lengthens and the synaptic position moves further from the soma. We confirm this prediction using a system of coupled compartmental models. We also demonstrate that when the synchronous solution becomes unstable, a bifurcation occurs and a pair of asynchronous stable solutions appear, causing a phase lag between the cells in the system. Then using a variety of coupling functions and different synaptic positions, we show that distal connections and broad synaptic time courses encourage phase lags that can be reduced, eliminated, or enhanced by the presence of active currents in the dendrite. This mechanism may appear in neural systems where proximal connections could be used to encourage synchrony, and distal connections and broad synaptic time courses could be used to produce phase lags that can be modulated by active currents.     

Spine calcium transients induced by synaptically-evoked action potentials can predict synapse location and establish synaptic democracy

CA1 pyramidal neurons receive hundreds of synaptic inputs at different distances from the soma. Distance-dependent synaptic scaling enables distal and proximal synapses to influence the somatic membrane equally, a phenomenon called "synaptic democracy". How this is established is unclear. The backpropagating action potential (BAP) is hypothesised to provide distance-dependent information to synapses, allowing synaptic strengths to scale accordingly. Experimental measurements show that a BAP evoked by current injection at the soma causes calcium currents in the apical shaft whose amplitudes decay with distance from the soma. However, in vivo action potentials are not induced by somatic current injection but by synaptic inputs along the dendrites, which creates a different excitable state of the dendrites. Due to technical limitations, it is not possible to study experimentally whether distance information can also be provided by synaptically-evoked BAPs. Therefore we adapted a realistic morphological and electrophysiological model to measure BAP-induced voltage and calcium signals in spines after Schaffer collateral synapse stimulation. We show that peak calcium concentration is highly correlated with soma-synapse distance under a number of physiologically-realistic suprathreshold stimulation regimes and for a range of dendritic morphologies. Peak calcium levels also predicted the attenuation of the EPSP across the dendritic tree. Furthermore, we show that peak calcium can be used to set up a synaptic democracy in a homeostatic manner, whereby synapses regulate their synaptic strength on the basis of the difference between peak calcium and a uniform target value. We conclude that information derived from synaptically-generated BAPs can indicate synapse location and can subsequently be utilised to implement a synaptic democracy.     

Changes in dendritic axial resistance alter synaptic integration in cerebellar Purkinje cells

The ability of neurons to process synaptic inputs depends critically on their passive electrical properties. The intracellular resistivity, R_i, is one of the parameters that determine passive properties, yet few experiments have explored how changes in R_i might affect synaptic integration. In this work, I addressed this issue by using targeted dendritic occlusion to locally increase R_i in cerebellar Purkinje cells and examining the consequences of this manipulation for the summation of synaptic inputs. To achieve dendritic occlusion, I used two glass micropipettes to gently pinch the dendritic trunk close to the soma. This pinching produced stereotypical changes in the responses to test pulses applied at the soma under voltage and current clamp. A simple model confirmed that these changes were due to increases in R_i in the dendritic trunk. These localized increases in R_i produced striking alterations in the shapes of postsynaptic potentials at the soma, increasing their amplitude and accelerating their decay kinetics. As a consequence, dendritic occlusion sharpened temporal precision during the summation of synaptic inputs. These findings highlight the importance of local changes in intracellular resistivity for the passive electrical properties of neurons, with implications for their ability to process synaptic information.     

Axon voltage-clamp simulations. III. Postsynaptic region.

This is the third in a series of four papers in which we present the numerical simulations of the application of the voltage clamp technique to excitable cells. In this paper we discuss the problem of voltage clamping a region of a cylindrical cell using microelectrodes for current injection and voltage recording. A recently developed technique (Llinás et al., 1974) of internal application of oil drops to electrically insulate a short length of the postsynaptic region of the squid giant synapse is evaluated by simulation of the voltage clamp of an excitable cylindrical cell of finite length with variable placement of the current and voltage electrodes. Our results show that ENa can be determined quite accurately with feasible oil gap lengths but that the determination of the reversal potential for the synaptic conductance, ES, can be considerably in error. The error in the determination of ES dependp, and especially the membrane resistance at the time the synaptic conductance occurs. It is shown that the application of tetraethylammonium chloride to block the active potassium conductance very significantly reduces the error in the determination of ES. In addition we discuss the effects of cable length and electrode position on the apparent amplitude and time course of the syn aptic conductance change. These results are particularly relevant to the application of the voltage clamp technique to cells with nonsomatic synapses. The method of simulation presented here provides a tool for evaluation of voltage clamp analysis of synaptic transmission for any cell with known membrane parameters and geometry.     

Dendritic Signals Command Firing Dynamics in a Mathematical Model of Cerebellar Purkinje Cells

Dendrites of cerebellar Purkinje cells (PCs) respond to brief excitations from parallel fibers with lasting plateau depolarizations. It is unknown whether these plateaus are local events that boost the synaptic signals or they propagate to the soma and directly take part in setting the cell firing dynamics. To address this issue, we analyzed a likely mechanism underlying plateaus in three representations of a reconstructed PC with increasing complexity. Analysis in an infinite cable suggests that Ca plateaus triggered by direct excitatory inputs from parallel fibers and their mirror signals, valleys (putatively triggered by the local feed forward inhibitory network), cannot propagate. However, simulations of the model in electrotonic equivalent cables prove that Ca plateaus (resp. valleys) are conducted over the entire cell with velocities typical of passive events once they are triggered by threshold synaptic inputs that turn the membrane current inward (resp. outward) over the whole cell surface. Bifurcation analysis of the model in equivalent cables, and simulations in a fully reconstructed PC both indicate that dendritic Ca plateaus and valleys, respectively, command epochs of firing and silencing of PCs.     

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.     

Computed Membrane Currents in Cardiac Purkinje Fibers during Voltage Clamps

Recent measurements have indicated that some of the cardiac cell electrical capacitance is in series with a resistance. The computations of currents in a voltage clamp presented below show that, in this case, there is a danger that capacitive transient currents recorded during voltage clamp experiments may be confused with currents arising through rapid active membrane conductance changes. Secondly, a voltage clamp technique aimed at avoiding capacitive transients, namely the linear or ramp clamp, has recently been introduced. An attempt has been made here to evaluate the usefulness of ramp clamps in studying membrane electrical properties, by computing ramp clamp results and considering the difficulties in reconstructing the original model from these results. It is concluded that such a reconstruction is not feasible.     

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.     

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.     

Democratization in a passive dendritic tree: an analytical investigation

One way to achieve amplification of distal synaptic inputs on a dendritic tree is to scale the amplitude and/or duration of the synaptic conductance with its distance from the soma. This is an example of what is often referred to as "dendritic democracy". Although well studied experimentally, to date this phenomenon has not been thoroughly explored from a mathematical perspective. In this paper we adopt a passive model of a dendritic tree with distributed excitatory synaptic conductances and analyze a number of key measures of democracy. In particular, via moment methods we derive laws for the transport, from synapse to soma, of strength, characteristic time, and dispersion. These laws lead immediately to synaptic scalings that overcome attenuation with distance. We follow this with a Neumann approximation of Green's representation that readily produces the synaptic scaling that democratizes the peak somatic voltage response. Results are obtained for both idealized geometries and for the more realistic geometry of a rat CA1 pyramidal cell. For each measure of democratization we produce and contrast the synaptic scaling associated with treating the synapse as either a conductance change or a current injection. We find that our respective scalings agree up to a critical distance from the soma and we reveal how this critical distance decreases with decreasing branch radius.     

A cable model for coupled neurons with somatic gap junctions

A cable model is presented for a pair of electrotonically coupled neurons to investigate the spatial effects of soma-somatic gap junctions. The model extends that of Poznanski et al.(1995) in which each neuron is represented by a tapered equivalent cable attached to an isopotential soma with the two somas being electrically coupled. The model is posed generally, so that both active and passive properties can be considered. In the active case a system of nonlinear integral equations is derived for the voltage, whilst in the passive case these have an exact solution that also holds for inputs modelled as synaptic reversal potentials. Analytical and numerical methods are used to examine the sensitivity of the soma potentials (in particular) to the coupling resistance.     

A novel theoretical approach to the analysis of dendritic transients.

A novel theoretical framework for analyzing dendritic transients is introduced. This approach, called the method of moments, is an extension of Rall's cable theory for dendrites. It provides analytic investigation of voltage attenuation, signal delay, and synchronization problems in passive dendritic trees. In this method, the various moments of a transient signal are used to characterize the properties of the transient. The strength of the signal is measured by the time integral of the signal, its characteristic time is determined by its centroid ("center of gravity"), and the width of the signal is determined by a measure similar to the standard deviation in probability theory. Using these signal properties, the method of moments provides theorems, expressions, and efficient algorithms for analyzing the voltage response in arbitrary passive trees. The method yields new insights into spatiotemporal integration, coincidence detection mechanisms, and the properties of local interactions between synaptic inputs in dendritic trees. The method can also be used for matching dendritic neuron models to experimental data and for the analysis of synaptic inputs recorded experimentally.     

Synaptic integration in spinal motoneurones.

Spinal motoneurones receive thousands of presynaptic excitatory and inhibitory synaptic contacts distributed throughout their dendritic trees. Despite this extensive convergence, there have been very few studies of how synaptic inputs interact in mammalian motoneurones when they are activated concurrently. In the experiments reported here, we measured the effective synaptic currents and the changes in firing rate evoked in cat spinal motoneurones by concurrent repetitive activation of two separate sets of presynaptic neurons. We compared these effects to those predicted by a linear sum of the effects produced by activating each set of presynaptic neurons separately. We generally found that when two inputs were activated concurrently, both the effective synaptic currents and the synaptically-evoked changes in firing rate they produced in motoneurones were generally linear, or slightly less than the linear sum of the effects produced by activating each input alone. The results suggest that the spatial distribution synaptic terminals on the dendritic trees of motoneurones may help isolate synapses from one another, minimizing non-linear interactions.     

Supercharging: a method for improving patch-clamp performance.

Patch-clamp performance can be improved without altering the normal headstage configuration described by (Hamill, O. P., A. Marty, E. Neher, B. Sakmann, and F. J. Sigworth, 1981, Pfluegers Arch. Eur. J. Physiol., 391:85-100). The "supercharging" method permits resolution of such fast events as calcium and sodium tail currents. Digital computer modeling and analog electronic simulation were used to identify appropriate shapes for the command voltage and the voltage applied to a capacitor tied to the input of the headstage. The voltage command pulse consists of a step with a brief (5-15 microseconds) rectangular spike on its leading edge. Spike amplitude is a function of the membrane capacitance and the access resistance. The spike drives current through the access resistance and speeds charging of the membrane capacitance, making it possible to complete a voltage step within 5-15 microseconds. Clamping speed is independent of the electrode and feedback resistance over a wide range. The second function of the patch clamp amplifier is current measurement, and good time resolution requires suppression of the capacity transient. This can be accomplished by applying an appropriately shaped voltage to the small capacitor tied to the input of the headstage. Series resistance compensation for ionic current transients does not interfere with supercharging. Although the focus of this paper is on whole cell recording, the supercharging concept may prove useful for single channel and bilayer recording techniques.     

Properties of internally perfused, voltage-clamped, isolated nerve cell bodies

The membrane properties of isolated neurons from Helix aspersa were examined by using a new suction pipette method. The method combines internal perfusion with voltage clamp of nerve cell bodies separated from their axons. Pretreatment with enzymes such as trypsin that alter membrane function is not required. A platinized platinum wire which ruptures the soma membrane allows low resistance access directly to the cell's interior improving the time resolution under voltage clamp by two orders of magnitude. The shunt resistance of the suction pipette was 10-50 times the neuronal membrane resistance, and the series resistance of the system, which was largely due to the tip diameter, was about 10(5) omega. However, the peak clamp currents were only about 20 nA for a 60-mV voltage step so that measurements of membrane voltage were accurate to within at least 3%. Spatial control of voltage was achieved only after somal separation, and nerve cell bodies isolated in this way do not generate all-or-none action potentials. Measurements of membrane potential, membrane resistance, and membrane time constant are equivalent to those obtained using intracellular micropipettes, the customary method. With the axon attached, comparable all-or-none action potentials were also measured by either method. Complete exchange of Cs+ for K+ was accomplished by internal perfusion and allowed K+ currents to be blocked. Na+ currents could then be blocked by TTX or suppressed by Tris-substituted snail Ringer solution. Ca2+ currents could be blocked using Ni2+ and other divalent cations as well as organic Ca2+ blockers. The most favorable intracellular anion was aspartate-, and the sequence of favorability was inverted from that found in squid axon.