Electrotonic conduction of excitation and inhibition along the dendritic fiber with the uniformly distributed synaptic conductance

The existence of an optimum radius of fibre is shown, when the physical parameters of the intracellular medium and the nonexcitable dendritic membrane are known and the length and load resistance are fixed. This provides the maximum potential for the proximal end of the fibre if synaptic conductance is distributed uniformly along the fibre. A formula for calculating the soma potential of the whole neuron is proposed. The optimal ratio l/lambda is 0.9193 ... if the volume of the fibre and synaptic conductivity are fixed.     

Optimum geometric parameters of the dendrite process

Optimal radius of dendritic spine steam is calculated under constant synaptic excitation using three criteria: maximum of excitatory postsynaptic potential (EPSP), maximal rate of EPSP change due to little change of spine steam radius and maximal rate of EPSP change due to little change of synaptic conductance.     

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.     

Analysis of an excitable dendritic spine with an activity-dependent stem conductance

 Dendritic spines are the major target for excitatory synaptic inputs in the vertebrate brain. They are tiny evaginations of the dendritic surface consisting of a bulbous head and a tenuous stem. Spines are considered to be an important locus for plastic changes underlying memory and learning processes. The findings that synaptic morphology may be activity-dependent and that spine head membrane may be endowed with voltage-dependent (excitable) channels is the motivation for this study. We first explore the dynamics, when an excitable, yet morphologically fixed spine receives a constant current input. Two parameter Andronov–Hopf bifurcation diagrams are constructed showing stability boundaries between oscillations and steady-states. We show how these boundaries can change as a function of both the spine stem conductance and the conductance load of the attached dendrite. Building on this reference case an idealized model for an activity-dependent spine is formulated and analyzed. Specifically we examine the possibility that the spine stem resistance, the tunable “synaptic weight” parameter identified by Rall and Rinzel, is activity-dependent. In the model the spine stem conductance depends (slowly) on the local electrical interactions between the spine head and the dendritic cable; parameter regimes are found for bursting, steady states, continuous spiking, and more complex oscillatory behavior. We find that conductance load of the dendrite strongly influences the burst pattern as well as other dynamics. When the spine head membrane potential exhibits relaxation oscillations a simple model is formulated that captures the dynamical features of the full model. Received: 10 January 1997/Revised version: 25 March 1997     

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.     

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.     

Quantitative analysis of electrical properties of dendritic spines

Several sugestions have been made with regard to the functional significance of dendritic spines in connection with synaptic plasticity. We have shown that for a constant synaptic current, when the synaptic resistance is large compared to the spine-stem resistance, a morphological change in the spine does not produce a marked change in the postsynaptic potential (PSP). When the synaptic resistance is comparable to the spine-stem impedance a morphological change in the spine can induce changes in the synaptic current and the PSP due to the so-called nonlinear effect to the synapse (Kawato and Tsukahara, 1983, 1984). Consequently, in a study of the electrical properties of dendritic spines the input impedance of the parent dendrite, the spinestalk conductance and the conductance change associated with synaptic activity must be considered. We quantitatively estimated all three factors. By comparing electrophysiological data with morphological data, we estimated the synaptic conductance which causes corticorubral EPSP. Its maximum amplitude was 43 nS with a time-to-peak value of 0.3 ms. With this value, the effects of the spine were examined using an improved algorithm based on that of Butz and Cowan (1974). It uses a three-dimensional morphology of the rubrospinal (RS) neurons, which was reconstructed from serial sections containing HRP-filled RS cells. As the spine shortens, the amplitude of the EPSP becomes considerably larger, but its time-to-peak value does not markedly change. Moreover, if unitary EPSP in the RS cell is produced by the activation of several synaptic terminals a morphological change of the spine has a smaller effect on the EPSPs.     

Dynamic modification of dendritic cable properties and synaptic transmission by voltage-gated potassium channels

Computer simulations of a dendrite possessing voltage-sensitive potassium conductances were used to determine the effects of these conductances on synaptic transmission and on the propagation of synaptic signals within the dendritic tree. Potassium conductances had two principal effects on voltage transients generated by current injections or synaptic conductances. Locally (near the source of the transient), voltage-gated potassium channels produced a potassium shunt current that reduced the amplitude of voltage transients generated by depolarizing currents. This shunt current increased as the amplitude of the depolarizing transient increased and so acted to prevent large synaptic transients from reaching levels that would saturate due to a reduction in driving force. In the presence of rapidly activating potassium currents, excitatory synapses produced larger synaptic currents that were more linearly related to synaptic conductance, but these produced smaller voltage transients. The maximum amplitudes of the voltage transients were limited by the voltage sensitivity of the K⁺ conductance and the rate at which it could activate. Sufficiently rapid synaptic currents could outrun the K⁺ conductance and thus achieve high local peak amplitudes. These effects of K⁺ conductances were unrelated to whether they were located on dendrites or not, being related only to their proximity to the source of synaptic current. The second class of effects of K⁺ conductances depended on their alteration of the electrotonic structure of the postsynaptic cell and so were observed only when they were located on postsynaptic dendrites. Voltage-gated K⁺ conductances produced voltage-dependent electrotonic expansion of depolarized dendrites, which had the effect of isolating synaptic inputs on depolarized dendrites from events on the rest of the neuron. Thus, synapses on the same dendrite interacted destructively to a degree much greater than that expected from the classical driving force nonlinearity. Synapses located proximally to a depolarized dendritic region were less effected than those located distally, and the range of the nonlinear interaction between synapses was dependent on the kinetics of activation and deactivation of the conductance. When present in conjunction with rapidly activating dendritic sodium conductance, the potassium conductance sharpened the requirement for spatial and temporal coincidence to produce synaptic boosting by inward currents, and suppressed out-of-synchrony synaptic inputs.     

Biphasic Synaptic Ca Influx Arising from Compartmentalized Electrical Signals in Dendritic Spines

Excitatory synapses on mammalian principal neurons are typically formed onto dendritic spines, which consist of a bulbous head separated from the parent dendrite by a thin neck. Although activation of voltage-gated channels in the spine and stimulus-evoked constriction of the spine neck can influence synaptic signals, the contribution of electrical filtering by the spine neck to basal synaptic transmission is largely unknown. Here we use spine and dendrite calcium (Ca) imaging combined with 2-photon laser photolysis of caged glutamate to assess the impact of electrical filtering imposed by the spine morphology on synaptic Ca transients. We find that in apical spines of CA1 hippocampal neurons, the spine neck creates a barrier to the propagation of current, which causes a voltage drop and results in spatially inhomogeneous activation of voltage-gated Ca channels (VGCCs) on a micron length scale. Furthermore, AMPA and NMDA-type glutamate receptors (AMPARs and NMDARs, respectively) that are colocalized on individual spine heads interact to produce two kinetically and mechanistically distinct phases of synaptically evoked Ca influx. Rapid depolarization of the spine triggers a brief and large Ca current whose amplitude is regulated in a graded manner by the number of open AMPARs and whose duration is terminated by the opening of small conductance Ca-activated potassium (SK) channels. A slower phase of Ca influx is independent of AMPAR opening and is determined by the number of open NMDARs and the post-stimulus potential in the spine. Biphasic synaptic Ca influx only occurs when AMPARs and NMDARs are coactive within an individual spine. These results demonstrate that the morphology of dendritic spines endows associated synapses with specialized modes of signaling and permits the graded and independent control of multiple phases of synaptic Ca influx.     

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.     

An improved model of heat transfer through penguin feathers and down

Penguins, mostly live in the extremely cold Antarctic, are known to have feathers and down, which are light weight, compact and extremely efficient in preventing heat loss. Nevertheless, the mechanisms of heat transfer through the penguin feathers and down, and how the unique characteristics of penguin feathers and down make them such good thermal insulators are not fully understood. In this paper, an integrated model of heat transfer through the penguin feathers and down is developed and computed using finite volume method, with the geometrical structure of the barbules being considered. Monte-Carlo method is adopted to determine the radiative absorption and emission constant in the integrated model. The effective thermal conductance of penguin feathers and down computed from our model compared well with the experimentally measured value reported in the literature. Three models (penguin model, random fibre model (fibre radius=3microm) and random fibre model (fibre radius=10microm)) are further simulated and compared. Results showed that the relative small radius of the barbules of penguin feather and their geometrical structure are responsible for the reduction of heat loss in cold environment.     

Motoneuron dendrites: role in synaptic integration.

Dendrites constitute over 80 per cent of the receptive surface area in cat motoneurons. Calculations based on matched electrical and gemoetrical measurements in these neurons indicate that the specific resistance of dendritic membranes in resting motoneurons is at least 2,000 ohm-cm2. When the specific membrane resistance is this high, even the most distal dendritic synapses can contribute significantly to the depolarization of the soma, and hence influence the rate of action potential generation. However, dendritic membrane resistance depends strongly on the level of background synaptic activity. The conductance changes associated with excitatory synaptic activity on a dendrite can be great enough to reduce significantly both the excitatory synaptic driving potential and the effective membrane resistance on that dendrite, and thus greatly reduce the effectiveness of synapses on the dendrite. Inhibitory synaptic activity produces an even greater reduction in dendritic membrane resistance. Thus the relative effectiveness of dendritic synapses depends on the type, distribution, and intensity of background synaptic activity, as well as on dendritic geometry and resting membrane properties.     

Functions and Modulation of Neuronal SK Channels

Small conductance (SK) channels are calcium-activated potassium channels that, when cloned in 1996, were thought solely to contribute to the afterhyperpolarisation that follows action potentials, and to control repetitive firing patterns of neurons. However, discoveries over the past few years have identified novel roles for SK channels in controlling dendritic excitability, synaptic transmission and synaptic plasticity. More recently, modulation of SK channel calcium sensitivity by casein kinase 2, and of SK channel trafficking by protein kinase A, have been demonstrated. This article will discuss recent findings regarding the function and modulation of SK channels in central neurons.     

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.     

On shunting inhibition

The interaction between excitation and inhibition is analyzed for nerve cylinders when reversal potentials for synaptic action are included. Both impulsive and sustained conductance changes are employed to model synaptic action.Exact results, in terms of Green's functions are obtained for the solutions of the cable equation with reversal potentials when there are impulsive conductance changes. The amplification factor for an inhibitory input due to a prior excitatory input is found exactly. In the case of an infinite cylinder, the dependence of this factor on the spatial separation of the excitatory and inhibitory synapses is one plus a Gaussian density function. Similar results aply when excitation follows inhibition. There is shunting inhibition even for impulsive conductance changes in the cable, but it is very different from that for sustained conductance changes. The interaction of excitation and inhibition is also studied in the full cable equation with reversal potentials and sustained conductance changes. An exact result is obtained for the potential in response to simultaneous excitation and inhibition at the same space point in an infinite cable. The effects of timing and spatial separation of inputs is analyzed in a finite nerve cylinder by numerically integrating the cable equation by the Crank-Nicolson method. Shunting inhibition is found to be most effective, for the chosen parameter values, when inhibition quickly foolows excitation. The EPSP amplitude at the soma is found to be roughly proportional to the distance from the soma to the site of inhibition when the excitation is at the center of the nerve cylinder.Dedicated to Jane Pauley     

Backpropagating action potentials enable detection of extrasynaptic glutamate by NMDA receptors

Synaptic NMDA receptors (NMDARs) are crucial for neural coding and plasticity. However, little is known about the adaptive function of extrasynaptic NMDARs occurring mainly on dendritic shafts. Here, we find that in CA1 pyramidal neurons, back-propagating action potentials (bAPs) recruit shaft NMDARs exposed to ambient glutamate. In contrast, spine NMDARs are "protected," under baseline conditions, from such glutamate influences by peri-synaptic transporters: we detect bAP-evoked Ca(2+) entry through these receptors upon local synaptic or photolytic glutamate release. During theta-burst firing, NMDAR-dependent Ca(2+) entry either downregulates or upregulates an h-channel conductance (G(h)) of the cell depending on whether synaptic glutamate release is intact or blocked. Thus, the balance between activation of synaptic and extrasynaptic NMDARs can determine the sign of G(h) plasticity. G(h) plasticity in turn regulates dendritic input probed by local glutamate uncaging. These results uncover a metaplasticity mechanism potentially important for neural coding and memory formation.     

The uncoupled chloride conductance of a bacterial glutamate transporter homolog

Glutamate transporters (EAATs) are pivotal in mammalian synaptic transmission, tightly regulating synaptic levels of this excitatory neurotransmitter. In addition to coupled glutamate transport, the EAATs also show an uncoupled Cl(-) conductance, whose physiological importance has recently been demonstrated. Little is yet known about the molecular mechanism of chloride permeation. Here we show that Glt(Ph), a bacterial EAAT homolog whose structure has been determined, displays an uncoupled Cl(-) conductance that can determine the rate of substrate uptake. A mutation analogous to one known to specifically affect Cl(-) movement in EAAT1 has similar effects on Glt(Ph), suggesting that this protein is an excellent structural model for understanding Cl(-) permeation through the EAATs. We also observed an uncoupled Cl(-) conductance in another bacterial EAAT homolog but not in a homolog of the Na(+)/Cl(-)-coupled neurotransmitter transporters.     

Electrical properties of dendritic spines with bulbous end terminals

Several suggestions have been made about the functional significance of dendritic spines in connection with synaptic plasticity. We investigated transient electrical behavior of spines with bulbous terminals in neurons with arbitrary dendritic geometries. It is shown that postsynaptic potential transform caused by a synapse on a spine can be resolved into a product of two transfer functions and the synaptic input current transform. The first transfer function was determined to be independent of the spine. The second transfer function represents the straightforward attenuation effect of the spine, which determines the effective synaptic current reaching the parent dendrite. Using what is known of the size and the shape of spines from histology, we conclude that almost all of the synaptic current flow into the parent dendrite, and that therefore the straightforward attenuation effect is negligible. Consequently, when the synaptic current remained unaltered, as was the case for a large synaptic resistance as compared with the spine stem resistance, a morphological change of the spine did not produce an effective change in the postsynaptic potential. On the other hand, when the synaptic resistance is compared with the spine stem impedance, the morphological change of the spine might induce changes of the synaptic current and the postsynaptic potential.     

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.     

Bilinearity in Spatiotemporal Integration of Synaptic Inputs

Neurons process information via integration of synaptic inputs from dendrites. Many experimental results demonstrate dendritic integration could be highly nonlinear, yet few theoretical analyses have been performed to obtain a precise quantitative characterization analytically. Based on asymptotic analysis of a two-compartment passive cable model, given a pair of time-dependent synaptic conductance inputs, we derive a bilinear spatiotemporal dendritic integration rule. The summed somatic potential can be well approximated by the linear summation of the two postsynaptic potentials elicited separately, plus a third additional bilinear term proportional to their product with a proportionality coefficient . The rule is valid for a pair of synaptic inputs of all types, including excitation-inhibition, excitation-excitation, and inhibition-inhibition. In addition, the rule is valid during the whole dendritic integration process for a pair of synaptic inputs with arbitrary input time differences and input locations. The coefficient is demonstrated to be nearly independent of the input strengths but is dependent on input times and input locations. This rule is then verified through simulation of a realistic pyramidal neuron model and in electrophysiological experiments of rat hippocampal CA1 neurons. The rule is further generalized to describe the spatiotemporal dendritic integration of multiple excitatory and inhibitory synaptic inputs. The integration of multiple inputs can be decomposed into the sum of all possible pairwise integration, where each paired integration obeys the bilinear rule. This decomposition leads to a graph representation of dendritic integration, which can be viewed as functionally sparse.