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.     

Spatio-temporal filtering properties of a dendritic cable with active spines: A modeling study in the spike-diffuse-spike framework

The spike-diffuse-spike (SDS) model describes a passive dendritic tree with active dendritic spines. Spine-head dynamics is modeled with a simple integrate-and-fire process, whilst communication between spines is mediated by the cable equation. In this paper we develop a computational framework that allows the study of multiple spiking events in a network of such spines embedded on a simple one-dimensional cable. In the first instance this system is shown to support saltatory waves with the same qualitative features as those observed in a model with Hodgkin-Huxley kinetics in the spine-head. Moreover, there is excellent agreement with the analytically calculated speed for a solitary saltatory pulse. Upon driving the system with time-varying external input we find that the distribution of spines can play a crucial role in determining spatio-temporal filtering properties. In particular, the SDS model in response to periodic pulse train shows a positive correlation between spine density and low-pass temporal filtering that is consistent with the experimental results of Rose and Fortune [1999, ‘Mechanisms for generating temporal filters in the electrosensory system,’ The Journal of Experimental Biology 202: 1281–1289]. Further, we demonstrate the robustness of observed wave properties to natural sources of noise that arise both in the cable and the spine-head, and highlight the possibility of purely noise induced waves and coherent oscillations. Action Editor: Erik De Schutter     

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     

Statistical evaluation of dendritic growth models

A mathematical model (Kliemann, W. 1987.Bull. math. Biol. 49, 135–152.) that predicts the quantitative branching pattern of dendritic tree was evaluated using the apical and basal dendrites of rat hippocampal neurons. The Wald statistics for χ²-test was developed for the branching pattern of dendritic trees and for the distribution of the maximal order of the tree. Using this statistic, we obtained a reasonable, but not excellent, fit of the mathematical model for the dendritic data. The model's predictability of branching patterns was greatly enhanced by replacing one of the assumptions used for the original model “splitting of branches for all dendritic orders is stochastically independent”, with a new assumption “branches are more likely to split in areas where there is already a high density of branches”. The modified model delivered an excellent fit for basal dendrites and for the apical dendrites of hippocampal neurons from young rats (30–34 days postpartum). This indicates that for these cells the development of dendritic patterns is the result of a purely random and a systematic component, where the latter one depends on the density of dendritic branches in the brain area considered. For apical dendrites there is a trend towards decreasing pattern predictability with increasing age. This appears to reflect the late arrival of afferents and subsequent synaptogenesis proximal on the apical dendritic tree of hippocampal neurons.     

Roles of I<Subscript>A</Subscript> and morphology in action potential propagation in CA1 pyramidal cell dendrites

Dendrites of CA1 pyramidal cells of the hippocampus, along with those of a wide range of other cell types, support active backpropagation of axonal action potentials. Consistent with previous work, recent experiments demonstrating that properties of synaptic plasticity are different for distal synapses, suggest an important functional role of bAPs, which are known to be prone to failure in distal locations. Using conductance-based models of CA1 pyramidal cells, we show that underlying “traveling wave attractors” control action potential propagation in the apical dendrites. By computing these attractors, we dissect and quantify the effects of I_A channels and dendritic morphology on bAP amplitudes. We find that non-uniform activation properties of I_A can lead to backpropagation failure similar to that observed experimentally in these cells. Amplitude of forward propagation of dendritic spikes also depends strongly on the activation dynamics of I_A. I_A channel properties also influence transients at dendritic branch points and whether or not propagation failure results. The branching pattern in the distal apical dendrites, combined with I_A channel properties in this region, ensure propagation failure in the apical tuft for a large range of I_A conductance densities. At the same time, these same properties ensure failure of forward propagating dendritic spikes initiated in the distal tuft in the absence of some form of cooperativity of synaptic activation. Electronic supplemary material The online version of this article (doi:) contains supplementary material, which is available to authorized users. Action Editor: Alain Destexhe     

Golgi-type I and Golgi-type II Neurons in the Ventral Anterior Thalamic Nucleus of the Adult Human: Morphological Features and Quantitative Analysis

The morphological and quantitative features of neurons in the adult human ventral anterior thalamic nucleus were studied in Golgi preparations. Two neuronal types were found and their quantitative features were studied. Golgi-type I neurons were medium to large cells with dense dendritic trees and dendritic protrusions and short hair-like appendages. They have somatic mean diameter of 30.8 μm (±9.4, n = 85). They have an average 100.3 dendritic branches, 48.97 dendritic branching points, and 58.85 dendritic tips. The mean diameters of their primary, secondary, and tertiary dendrites were 3.1 μm (±1, n = 80), 1.85 μm (±0.8, n = 145), and 1.5 μm (±0.4, n = 160), respectively. Golgi-type II neurons were small to medium cells with few sparsely branching dendrites and dendritic stalked appendages with or without terminal swellings. They have somatic mean diameters of 22.2 μm (±5.8, n = 120). They have an average 33.76 dendritic branches, 16.49 dendritic branching points, and 21.97 dendritic tips. The mean diameters of their primary, secondary, and tertiary dendrites were 1.6 μm (±0.86, n = 70), 1.15 μm (±0.55, n = 118), and 1 μm (±0.70, n = 95), respectively. These quantitative data may form the basis for further quantitative studies involving aging or some degenerative diseases that may affect cell bodies and/or dendritic trees of the Golgi-type I and/or Golgi-type II thalamic neurons.     

Multidimensional flow-cytometric analysis of dendritic cells in peripheral blood of normal donors and cancer patients

 We studied the potential of multidimensional flow cytometry to evaluate the frequency and maturation/activation status of dendritic cells in minimally manipulated peripheral blood mononuclear cell preparations (i.e., only separated on Ficoll-Hypaque) of normal donors and cancer patients. A rare subset of HLA-DR⁺ leukocytes (less than 1% mononuclear cells) was detected in blood of normal donors that displayed all the features of dendritic cells: these cells had high forward-light-scatter characteristics and coexpressed CD4, CD86 and CD54 surface antigens, but lacked the lineage-associated surface markers of T cells, B cells, monocytes, granulocytes or NK i.e. they were CD3^–, CD19^–, CD20^–, CD14^–, CD11b^–, CD16^–, CD56^–). These physical and phenotypic properties were virtually identical to those of immunomagnetically sorted leukocytes characterized as dendritic-cells on the basis of morphology, phenotype and high stimulatory activity in allogeneic mixed-lymphocyte cultures. Using this flow-cytometric approach we observed that the frequency of dendritic cell-like cells in peripheral blood mononuclear cell specimens of cancer patients receiving chemotherapy alone or those recovering from stem cell transplantation was significantly lower than that of normal individuals (mean ± SE: 0.36 ± 0.05%, 0.14 ± 0.06%, and 0.75 ± 0.04% respectively). Multidimensional flow-cytometric analysis of dendritic cells might represent an important new tool for assessing immunocompetence, and for monitoring the effects of therapeutic regimens on the immune system. Received: 20 June 1997 / Accepted: 14 August 1997     

Fast Kalman filtering on quasilinear dendritic trees

Optimal filtering of noisy voltage signals on dendritic trees is a key problem in computational cellular neuroscience. However, the state variable in this problem—the vector of voltages at every compartment—is very high-dimensional: realistic multicompartmental models often have on the order of N = 10⁴ compartments. Standard implementations of the Kalman filter require O(N ³) time and O(N ²) space, and are therefore impractical. Here we take advantage of three special features of the dendritic filtering problem to construct an efficient filter: (1) dendritic dynamics are governed by a cable equation on a tree, which may be solved using sparse matrix methods in O(N) time; and current methods for observing dendritic voltage (2) provide low SNR observations and (3) only image a relatively small number of compartments at a time. The idea is to approximate the Kalman equations in terms of a low-rank perturbation of the steady-state (zero-SNR) solution, which may be obtained in O(N) time using methods that exploit the sparse tree structure of dendritic dynamics. The resulting methods give a very good approximation to the exact Kalman solution, but only require O(N) time and space. We illustrate the method with applications to real and simulated dendritic branching structures, and describe how to extend the techniques to incorporate spatially subsampled, temporally filtered, and nonlinearly transformed observations.     

A distributed-parameter model of the myelinated human motor nerve fibre: temporal and spatial distributions of action potentials and ionic currents

 A double cable model of the myelinated human motor nerve fibre is presented. The model is based on the nodal and internodal channels in a previous, two-component model of human motor axons (Bostock et al. 1991), added to a complex extended cable structure of nodal, paranodal and internodal segments. The model assumes a high-resistance myelin sheath and a leakage pathway to the internodal axolemma via the paranodal seal resistance and periaxonal space. The parameter values of the model were adjusted to match the recordings of threshold electrotonus in human motor fibres from Bostock et al. (1991). Kirchoff ’s current law was used to derive a system of partial differential equations for the electrical equivalent circuit, and numerical integration was performed with a fixed time increment and non-uniform spatial step sizes, in accordance with the complex structure of the fibre. The model calculations provide estimates of the spatial and temporal distributions of action potentials and their transaxonal and transmyelin components, both in different segments of the fibre and at different moments during action potential propagation. The distribution of transaxonal and transmyelin currents along the fibre and their contributions from different ionic channels are also explored. Received: 14 July 1994/Accepted in revised form: 4 April 1995     

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.     

Molecular Evolution of the Aldo-keto Reductase Gene Superfamily

The aldo-keto reductase enzymes comprise a functionally diverse gene family which catalyze the NADPH-dependant reduction of a variety of carbonyl compounds. The protein sequences of 45 members of this family were aligned and phylogenetic trees were deduced from this alignment using the neighbor-joining and Fitch algorithms. The branching order of these trees indicates that the vertebrate enzymes cluster in three groups, which have a monophyletic origin distinct from the bacterial, plant, and invertebrate enzymes. A high level of conservation was observed between the vertebrate hydroxysteroid dehydrogenase enzymes, prostaglandin F synthase, and ρ-crystallin of Xenopus laevis. We infer from the phylogenetic analysis that prostaglandin F synthase may represent a recent recruit to the eicosanoid biosynthetic pathway from the hydroxysteroid dehydrogenase pathway and furthermore that, in the context of gene recruitment, Xenopus laevisρ-crystallin may represent a shared gene. Received: 26 August 1996 / Accepted: 5 June 1997     

Topography of cell traces studied by atomic force microscopy

Migrating adherent cells release material onto artificial substrates like glass and silicon while moving. Traces of mouse fibroblasts (L929) have been visualised by atomic force microscopy (AFM). “Non-contact” mode AFM in a liquid environment can extract topographic information from these traces. This dynamic mode allows the study of these soft structures without damage or compression. The AFM images show crossing and branching networks (with specific angles of branching), structured patches, nodular elements, linear elements with irregular height and other features. Fourier analysis of segment spacing in the strands is presented. These spatial features of fibroblast traces are strong indications that actin linked to structural proteins is involved in the formation of cell traces. We also give methods for trace preparation and undistorted imaging and discuss further perspectives. Received: 11 January 1999 / Revised version: 1 April 1999 / Accepted: 8 April 1999     

A generalized tapering equivalent cable model for dendritic neurons

A mathematical model has been developed which collapses a dendritic neuron of complex geometry into a single electrotonically tapering equivalent cable. The modified cable equation governing the transient distribution of subthreshold membrane potential in a branching tree is transformed, becoming amenable to analytic solution. This transformation results in a Riccati differential equation whose six solutions (expressed in terms of elementary functions) control the amount and degree of taper found in the equivalent cable model. To illustrate the theory, an analytic solution (in series form) of the modified cable equation is obtained for a voltage-clamp present at the soma of a quadratically tapering equivalent cable whose distal end is sealed.     

The prior probabilities of phylogenetic trees

Bayesian methods have become among the most popular methods in phylogenetics, but theoretical opposition to this methodology remains. After providing an introduction to Bayesian theory in this context, I attempt to tackle the problem mentioned most often in the literature: the “problem of the priors”—how to assign prior probabilities to tree hypotheses. I first argue that a recent objection—that an appropriate assignment of priors is impossible—is based on a misunderstanding of what ignorance and bias are. I then consider different methods of assigning prior probabilities to trees. I argue that priors need to be derived from an understanding of how distinct taxa have evolved and that the appropriate evolutionary model is captured by the Yule birth–death process. This process leads to a well-known statistical distribution over trees. Though further modifications may be necessary to model more complex aspects of the branching process, they must be modifications to parameters in an underlying Yule model. Ignoring these Yule priors commits a fallacy leading to mistaken inferences both about the trees themselves and about macroevolutionary processes more generally.     

Electro-geometrical coupling in non-uniform branching dendrites

The relationships between somatofugal electronic voltage spread, somatopetal charge transfer and non-uniform geometry of the neuronal dendrites were studied on the basis of the linear cable theory. It is demonstrated that for the dendritic arborization of arbitrary geometry, the path distribution of the relative effectiveness of somatopetal synaptic charge transfer defined as in Barrett and Crill (1974) is identical to that of the somatofugal steady electrotonic voltage normalized to the voltage at the soma. The features of both distributions are determined by breaks in the voltage gradient (the slope of monotonic voltage decay) at the sites of local non-uniformity of the dendritic geometry, such as abrupt change in diameter and asymmetric branching. If the membrane- and cytoplasm-specific electrical parameters are assumed as uniform and the branch diameter as piece-wise uniform, then at any site of step change the square reciprocal ratio of the pre- and poststep diameters determines the ratio of the pre- and poststep electronic gradients. At branching points this ratio is modulated by partition of the core current between the daughter branches in proportion to their input conductances depending on global geometries of the daughter subtrees originating there. Thus, simply computed steady somatofugal voltages provide a physiologically meaningful estimation of the relative influence of synaptic inputs in different parts of the dendritic arborization on the output of the neuron.     

Theoretical relationship between the optimal models of the vascular tree

Two models of optimal branching structure of the vascular tree are compared. Murray’s minimum work model derived from minimum energy loss due to flow and volume in the duct system is proved to be included as a mathematical group in the authors’ model defined by the minimum volume under determinant pressure, flow and position at the terminals. The problem about heterotypical trees which are identical at the terminal conditions but different in the topological order of branch combinations are discussed, applying the results of analyses on the equivalent duct of uniform terminal pressure trees. It is proved that the minimum work tree has the least energy loss compared with its heterotypical minimum volume trees and is a better model of branching structure of the vascular tree.     

Relationships among root branch order, carbon, and nitrogen in four temperate species

The objective of this study was to examine how root length, diameter, specific root length, and root carbon and nitrogen concentrations were related to root branching patterns. The branching root systems of two temperate tree species, Acer saccharum Marsh. and Fraxinus americana L., and two perennial herbs from horizontal rhizomes, Hydrophyllum canadense L. and Viola pubescens Ait., were quantified by dissecting entire root systems collected from the understory of an A. saccharum-Fagus grandifolia Ehrh. forest. The root systems of each species grew according to a simple branching process, with laterals emerging from the main roots some distance behind the tip. Root systems normally consisted of only 4–6 branches (orders). Root diameter, length, and number of branches declined with increasing order and there were significant differences among species. Specific root length increased with order in all species. Nitrogen concentration increased with order in the trees, but remained constant in the perennial herbs. More than 75% of the cumulative root length of tree seedling root systems was accounted for by short (2–10 mm) lateral roots almost always <0.3 mm in diameter. Simple assumptions suggest that many tree roots normally considered part of the dynamic fine-root pool (e.g., all roots <2.0 mm in diameter) are too large to exhibit rapid rates of production and mortality. The smallest tree roots may be the least expensive to construct but the most expensive to maintain based on an increase in N concentration with order. Received: 25 November 1996 / Accepted: 27 March 1997