The effect of dendritic voltage-gated conductances on the neuronal impedance: a quantitative model

Neuronal impedance characterizes the magnitude and timing of the subthreshold response of a neuron to oscillatory input at a given frequency. It is known to be influenced by both the morphology of the neuron and the presence of voltage-gated conductances in the cell membrane. Most existing theoretical accounts of neuronal impedance considered the effects of voltage-gated conductances but neglected the spatial extent of the cell, while others examined spatially extended dendrites with a passive or spatially uniform quasi-active membrane. We derived an explicit mathematical expression for the somatic input impedance of a model neuron consisting of a somatic compartment coupled to an infinite dendritic cable which contained voltage-gated conductances, in the more general case of non-uniform dendritic membrane potential. The validity and generality of this model was verified through computer simulations of various model neurons. The analytical model was then applied to the analysis of experimental data from real CA1 pyramidal neurons. The model confirmed that the biophysical properties and predominantly dendritic localization of the hyperpolarization-activated cation current I (h) were important determinants of the impedance profile, but also predicted a significant contribution from a depolarization-activated fast inward current. Our calculations also implicated the interaction of I (h) with amplifying currents as the main factor governing the shape of the impedance-frequency profile in two types of hippocampal interneuron. Our results provide not only a theoretical advance in our understanding of the frequency-dependent behavior of nerve cells, but also a practical tool for the identification of candidate mechanisms that determine neuronal response properties.     

Glutamate-Bound NMDARs Arising from In Vivo-like Network Activity Extend Spatio-temporal Integration in a L5 Cortical Pyramidal Cell Model

In vivo, cortical pyramidal cells are bombarded by asynchronous synaptic input arising from ongoing network activity. However, little is known about how such ‘background’ synaptic input interacts with nonlinear dendritic mechanisms. We have modified an existing model of a layer 5 (L5) pyramidal cell to explore how dendritic integration in the apical dendritic tuft could be altered by the levels of network activity observed in vivo. Here we show that asynchronous background excitatory input increases neuronal gain and extends both temporal and spatial integration of stimulus-evoked synaptic input onto the dendritic tuft. Addition of fast and slow inhibitory synaptic conductances, with properties similar to those from dendritic targeting interneurons, that provided a ‘balanced’ background configuration, partially counteracted these effects, suggesting that inhibition can tune spatio-temporal integration in the tuft. Excitatory background input lowered the threshold for NMDA receptor-mediated dendritic spikes, extended their duration and increased the probability of additional regenerative events occurring in neighbouring branches. These effects were also observed in a passive model where all the non-synaptic voltage-gated conductances were removed. Our results show that glutamate-bound NMDA receptors arising from ongoing network activity can provide a powerful spatially distributed nonlinear dendritic conductance. This may enable L5 pyramidal cells to change their integrative properties as a function of local network activity, potentially allowing both clustered and spatially distributed synaptic inputs to be integrated over extended timescales.     

A distributed parameter identification problem in neuronal cable theory models

Dendritic and axonal processes of nerve cells, along with the soma itself, have membranes with spatially distributed densities of ionic channels of various kinds. These ionic channels play a major role in characterizing the types of excitable responses expected of the cell type. These densities are usually represented as constant parameters in neural models because of the difficulty in experimentally estimating them. However, through microelectrode measurements and selective ion staining techniques, it is known that ion channels are non-uniformly spatially distributed. This paper presents a non-optimization approach to recovering a single spatially non-uniform ion density through use of temporal data that can be gotten from recording microelectrode measurements at the ends of a neural fiber segment of interest. The numerical approach is first applied to a linear cable model and a transformed version of the linear model that has closed-form solutions. Then the numerical method is shown to be applicable to non-linear nerve models by showing it can recover the potassium conductance in the Morris-Lecar model for barnacle muscle, and recover the spine density in a continuous dendritic spine model by Baer and Rinzel.     

The response of a nerve cylinder to spatially distributed white noise inputs

The response of a passive nerve cylinder (or dendritic tree in the equivalent cylinder representation) to random white noise input currents is determined. Results for the mean, variance and covariance of the depolarization are obtained for an arbitrary number of independent spatially distributed inputs. The case of a cylinder with sealed ends is considered in detail. The differences that arise when the input currents are distributed over a small but finite region of space instead of concentrated at a point are investigated. In the case of distributed inputs, the expectation is smoother near the stimulus and the variance becomes finite over the entire cable length including the region of the applied stimulus. Away from the stimulus, there are no appreciable differences between the responses for the two cases. The interaction between an excitatory input and an inhibitory input at various locations is examined and one case of more than two inputs is also analysed to study effects which could not have been discerned from point models for a neuron with random inputs.     

Transfer properties of neuronal dendrites with tonically activated conductances

The somatopetal current transfer was studied in the mathematical models of a reconstructed brainstem motoneuron with tonically activated excitatory synaptic inputs uniformly distributed over dendritic arborization. The soma and axon provided a constant passive leak. The extrasynaptic dendritic membrane was either passive or active (of a Hodgkin-Huxley type). The longitudinal membrane current density (per unit path length) was used as an estimate of the current transfer effectiveness of different dendritic paths. Introduction of a steady uniform voltage-independent conductance per unit membrane area simulated such a synaptic activation. This actions always produced a spatially inhomogeneous membrane depolarization decaying from the distal dendritic tips toward the soma. The reason for such an inhomogeneity was the preponderance of somatopetal over somatofugal input conductance at every site in the dendrites with sealed distal ends and a leaky somatic end. In active dendrites, partial voltage-dependent extrasynaptic conductances followed this depolarization according to their activation-inactivation kinetics. The greater the local depolarization, the greater the contribution of the non-inactivating potassium conductance to the total membrane conductance. The contribution of the inactivated sodium conductance was one order of magnitude smaller. Correspondingly, the effective equilibrium potential of the total transmembrane current became spatially inhomogeneous and shifted to the potassium equilibrium potential. In the passive dendrites, the equilibrium potential remained spatially homogeneous. Inhomogeneities of the dendritic geometry (abrupt change in the diameter and, especially, asymmetrical branching) caused characteristic perturbations in the voltage gradient, so that the path profiles of the voltage, conductances, and currents diverged. This indicated a geometry-induced separation of the dendritic paths in their transfer effectiveness. Active dendrites of the same geometry were less effective than passive ones due to the effect of the potassium conductance associated with the hyperpolarizing equilibrium potential.     

A non-uniform equivalent cable model of membrane voltage changes in a passive dendritic tree

A non-uniform equivalent cable model of membrane voltage changes in a passive dendritic tree extending Rall's equivalent cylinder model is presented. It is obtained from a combination of cable theory with the continuum approach. Replacing the fine structure of the branching dendrites by an equivalent, conductive medium characterized by averaged electrical parameters, the one-dimensional cable equations with spatially varying parameters are derived. While these equations can be solved in general only numerically, we were able to formulate a general branching condition (comprising Rall's 3/2 power relationship as a special case) under which analytical solutions can be deduced from those of the equivalent cylinder model. This model allows dendritic trees with a greater variety of branching patterns than before to be analytically treated.     

Linear summation of excitatory inputs by CA1 pyramidal neurons

A fundamental problem in neurobiology is understanding the arithmetic that dendrites use to integrate inputs. The impact of dendritic morphology and active conductances on input summation is still unknown. To study this, we use glutamate iontophoresis and synaptic stimulation to position pairs of excitatory inputs throughout the apical, oblique, and basal dendrites of CA1 pyramidal neurons in rat hippocampal slices. Under a variety of stimulation regimes, we find a linear summation of most input combinations that is implemented by a surprising balance of boosting and shunting mechanisms. Active conductances in dendrites paradoxically serve to make summation linear. This "active linearity" can reconcile predictions from cable theory with the observed linear summation in vivo and suggests that a simple arithmetic is used by apparently complex dendritic trees.     

A dendritic cable model for the amplification of synaptic potentials by an ensemble average of persistent sodium channels

The persistent sodium current density (I(NaP)) at the soma measured with the 'whole-cell' patch-clamp recording method is linearized about the resting state and used as a current source along the dendritic cable (depicting the spatial distribution of voltage-dependent persistent sodium ionic channels). This procedure allows time-dependent analytical solutions to be obtained for the membrane depolarization. Computer simulated response to a dendritic current injection in the form of synaptically-induced voltage change located at a distance from the recording site in a cable with unequally distributed persistent sodium ion channel densities per unit length of cable (the so-called 'hot-spots') is used to obtain conclusions on the density and distribution of persistent sodium ion channels. It is shown that the excitatory postsynaptic potentials (EPSPs) are amplified if hot-spots of persistent sodium ion channels are spatially distributed along the dendritic cable, with the local density of I(NaP) with respect to the recording site shown to specifically increase the peak amplitude of the EPSP for a proximally placed synaptic input, while the spatial distribution of I(NaP) serves to broaden the time course of the amplified EPSP. However, in the case of a distally positioned synaptic input, both local and nonlocal densities yield an approximately identical enhancement of EPSPs in contradiction to the computer simulations performed by Lipowsky et al. [J. Neurophysiol. 76 (1996) 2181]. The results indicate that persistent sodium channels produce EPSP amplification even when their distribution is relatively sparse (i.e. , approximately 1-2% of the transient sodium channels are found in dendrites of CA1 hippocampal pyramidal neurons). This gives a strong impetus for the use of the theory as a novel approach in the investigation of synaptic integration of signals in active dendrites represented as ionic cables.     

High-conductance pathways in mitochondrial membranes

The outer and inner membranes of mitochondria have recently been studied with the patch clamp technique. What has emerged is still an ill-defined picture for either membrane, primarily for the wide range of conductances found. Interestingly, however, a few conductances (in the range of 10–80 pS) seem to be ubiquitously distributed. Parallel studiesin situ and in reconstituted systems have allowed the assignment to distinct membrane locations of some conductances, whose physiological role is, however, not yet elucidated.     

Effects of active conductance distribution over dendrites on the synaptic integration in an identified nonspiking interneuron

The synaptic integration in individual central neuron is critically affected by how active conductances are distributed over dendrites. It has been well known that the dendrites of central neurons are richly endowed with voltage- and ligand-regulated ion conductances. Nonspiking interneurons (NSIs), almost exclusively characteristic to arthropod central nervous systems, do not generate action potentials and hence lack voltage-regulated sodium channels, yet having a variety of voltage-regulated potassium conductances on their dendritic membrane including the one similar to the delayed-rectifier type potassium conductance. It remains unknown, however, how the active conductances are distributed over dendrites and how the synaptic integration is affected by those conductances in NSIs and other invertebrate neurons where the cell body is not included in the signal pathway from input synapses to output sites. In the present study, we quantitatively investigated the functional significance of active conductance distribution pattern in the spatio-temporal spread of synaptic potentials over dendrites of an identified NSI in the crayfish central nervous system by computer simulation. We systematically changed the distribution pattern of active conductances in the neuron's multicompartment model and examined how the synaptic potential waveform was affected by each distribution pattern. It was revealed that specific patterns of nonuniform distribution of potassium conductances were consistent, while other patterns were not, with the waveform of compound synaptic potentials recorded physiologically in the major input-output pathway of the cell, suggesting that the possibility of nonuniform distribution of potassium conductances over the dendrite cannot be excluded as well as the possibility of uniform distribution. Local synaptic circuits involving input and output synapses on the same branch or on the same side were found to be potentially affected under the condition of nonuniform distribution while operation of the major input-output pathway from the soma side to the one on the opposite side remained the same under both conditions of uniform and nonuniform distribution of potassium conductances over the NSI dendrite.     

Calcium oscillations in a triplet of pancreatic acinar cells

We use a mathematical model of calcium dynamics in pancreatic acinar cells to investigate calcium oscillations in a ring of three coupled cells. A connected group of cells is modeled in two different ways: 1), as coupled point oscillators, each oscillator being described by a spatially homogeneous model; and 2), as spatially distributed cells coupled along their common boundaries by gap-junctional diffusion of inositol trisphosphate and/or calcium. We show that, although the point-oscillator model gives a reasonably accurate general picture, the behavior of the spatially distributed cells cannot always be predicted from the simpler analysis; spatially distributed diffusion and cell geometry both play important roles in determining behavior. In particular, oscillations in which two cells are in synchrony, with the third phase-locked but not synchronous, appears to be more dominant in the spatially distributed model than in the point-oscillator model. In both types of model, intercellular coupling leads to a variety of synchronous, phase-locked, or asynchronous behaviors. For some parameter values there are multiple, simultaneous stable types of oscillation. We predict 1), that intercellular calcium diffusion is necessary and sufficient to coordinate the responses in neighboring cells; 2), that the function of intercellular inositol trisphosphate diffusion is to smooth out any concentration differences between the cells, thus making it easier for the diffusion of calcium to synchronize the oscillations; 3), that groups of coupled cells will tend to respond in a clumped manner, with groups of synchronized cells, rather than with regular phase-locked periodic intercellular waves; and 4), that enzyme secretion is maximized by the presence of a pacemaker cell in each cluster which drives the other cells at a frequency greater than their intrinsic frequency.     

A biomechanical model of the posterior knee capsule of the cat

A biomechanical model is presented which represents the upper edge of the posterior knee capsule in the cat as a two-segment, vertically loaded catenary suspension cable from which the capsule sheet is suspended. Data are presented which show that the upper edge of the capsule is organized as a cable, which spans the notch between the femoral condyles. When a point load is applied to the cable, measurement of the cable shape allows for calculation of the cable tension and the downward distributed loads acting on the cable. This method was used to measure the in-vivo cable tension and the distributed downward loading acting on the capsule cable. The results show that the lateral side of the posterior joint capsule sustains a higher loading than the medial side.     

NMDA receptor-mediated dendritic spikes and coincident signal amplification

Dendrites of cortical neurons possess active conductances, which contribute to the nonlinear processing of synaptic information. Recently it has been shown that basal dendrites can generate highly localized spikes mediated by NMDA receptor channels. These spikes may serve as a powerful mechanism to detect and amplify synchronously activated spatially clustered excitatory synaptic inputs in individual dendritic segments, and may enable parallel processing in several integrative dendritic subunits.     

Electrical Interactions via the Extracellular Potential Near Cell Bodies

Ephaptic interactions between a neuron and axons or dendrites passing by its cell body can be, in principle, more significant than ephaptic interactions among axons in a fiber tract. Extracellular action potentials outside axons are small in amplitude and spatially spread out, while they are larger in amplitude and much more spatially confined near cell bodies. We estimated the extracellular potentials associated with an action potential in a cortical pyramidal cell using standard one-dimensional cable theory and volume conductor theory. Their spatial and temporal pattern reveal much about the location and timing of currents in the cell, especially in combination with a known morphology, and simple experiments could resolve questions about spike initiation. From the extracellular potential we compute the ephaptically induced polarization in a nearby passive cable. The magnitude of this induced voltage can be several mV, does not spread electrotonically, and depends only weakly on the passive properties of the cable. We discuss their possible functional relevance.     

Intracellular Na+ and K+ activities and membrane conductances in the collecting tubule of Amphiuma

Membrane potentials and conductances, and intracellular ionic activities were studied in isolated perfused collecting tubules of K+-adapted Amphiuma. Intracellular Na+ (aNai) and K+ (aKi) activities were measured, using liquid ion-exchanger double-barreled microelectrodes. Apical and basolateral membrane conductances were estimated by cable analysis. The effects of inhibition of the apical conductance by amiloride (10(-5) M) and of inhibition of the basolateral Na-K pump by either a low K+ (0.1 mM) bath or by ouabain (10(-4) M) were studied. Under control conditions, aNai was 8.4 +/- 1.9 mM and aKi 56 +/- 3 mM. With luminal amiloride, aNai decreased to 2.2 +/- 0.4 mM and aKi increased to 66 +/- 3 mM. Ouabain produced an increase of aNai to 44 +/- 4 mM, and a decrease of aKi to 22 +/- 6, and similar changes were observed when the tubule was exposed to a low K+ bath solution. During pump inhibition, there was a progressive decrease of the K+-selective basolateral membrane conductance and of the Na+ permeability of the apical membrane. A similar inhibition of both membrane conductances was observed after pump inhibition by low K+ solution. Upon reintroduction of K+, a basolateral membrane hyperpolarization of -23 +/- 4 mV was observed, indicating an immediate reactivation of the electrogenic Na-K pump. However, the recovery of the membrane conductances occurred over a slower time course. These data imply that both membrane conductances are regulated according to the intracellular ionic composition, but that the basolateral K+ conductance is not directly linked to the pump activity.     

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."     

Ladder network prediction of transients in linear spatially inhomogeneous cables

A numerical method is described for finding steady state and transient responses in electrically linear, spatially inhomogeneous cables. Spatial inhomogeneities are incorporated by representing the cable by a number of finite length uniform cylindrical segments, each having the radius and electrical characteristics of a small region along the cable. Input waveforms are approximated by truncated Fourier series of sinusoidal components. Output waveforms are produced by multiplying the input Fourier series sinusoids by their respective transfer functions between input and output points on the cable and summing the resultant output point sinusoids. The transfer functions, representing attenuation and phase shift for each input sinusoid, are obtained by numerical analysis of an electrical ladder network derived from the cylindrical segment model of the cable. Results are shown for application of this method to both cylindrical and expanding radius cable geometries.     

Chloride conductance of denervated gastrocnemius fibers from normal goats.

Membrane potentials, cable parameters, and component resting ionic conductances of gastrocnemius fibers from normal goats were measured in vitro at six to 32 days following denervation by section of the tibial nerve. Denervated fibers were depolarized an average of 11.6 +/- 1.5 mV (six preparations) from the control mean of 62.1 +/- 1.0 mV (124 fibers) over the period studied. Fibrillation, tetrodotoxin-resistant action potentials, and anode-break excitation were present in the denervated preparations after 13 days. The control cable parameters from 124 fibers (13 preparations) were membrane resistance, 1052 +/- 70 omega-cm2 and membrane capacitance, 6.2 muF/cm2. In denervated fibers membrane resistance increased two to three times in the 13 to 32 day period; membrane capacitance increased about 50% in normal solution at eight to nine, 27-28, and 32 days. Myoplasmic resistivity was assumed to be 112 omega-cm. Measurements were made at 38 degrees C. Component resting conductances were determined from the cable parameters in normal and chloride-free solution. Mean chloride conducantance GC1 and mean potassium conductance GK of control fibers were 776 +/- 49 mumhos/cm2 and 175 +/- 15 mumhos/cm2 (92 fibers), respectively. Following denervation GC1 increased slightly at six to nine days then fell to low values at 16 to 32 days that were close to or indistinguishable from zero. GK increased significantly to 372 +/- 40 mumhos/cm2 and 499 +/- 90 mumhos/cm2 at 16 to 20 and 32 days, respectively. It was concluded from these findings that GC1 and GK of mammalian skeletal muscle are controlled by factors from the nerve and/or muscle action potentials. Goat muscle is different from frog muscle in which GC1 does not change and GK decreases during denervation.     

MEG (Model Extender for Gepasi): a program for the modelling of complex, heterogeneous, cellular systems

SUMMARY: We describe a program for the construction of spatially distributed metabolic models, which may then be simulated using the metabolic simulator GEPASI: This is useful for the modelling of heterogeneous systems whether as liquid cultures or as spatially organised systems with specified interconnections.